Global Optimization Toolbox

Global Optimization Toolbox
‫معرفی چند منبع در زمینه آموزش برنامه نویسی ‪ MATLAB‬یا متلب‬
‫کتاب های به زبان انگلیسی‬
‫عنوان‪Matlab, Third Edition: A Practical Introduction to :‬‬
‫‪Programming and Problem Solving‬‬
‫ترجمه عنوان‪ :‬متلب‪ :‬مقدمه ای عملی بر برنامه نويسی و حل مساله‪ ،‬چاپ سوم‬
‫مولفین‪Stormy Attaway :‬‬
‫سال چاپ‪2013 :‬‬
‫انتشارات‪Butterworth-Heinemann :‬‬
‫کتاب های به زبان فارسی‬
‫عنوان‪ :‬اصول و مبانی متلب برای علوم مهندسی‬
‫مولفین‪ :‬برايان هان‪ ،‬دانیل تی‪ ،‬والنتین‬
‫مترجمین‪ :‬رامین موالنا پور‪ ،‬سارا موالناپور‪ ،‬نینا اسدی پور‬
‫انتشارات‪ :‬سها دانش‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪MATLAB For Dummies :‬‬
‫ترجمه عنوان‪ :‬تلب به زبان ساده‬
‫مولفین‪Jim Sizemore, John Paul Mueller :‬‬
‫سال چاپ‪2014 :‬‬
‫انتشارات‪For Dummies :‬‬
‫عنوان‪ :‬کاربرد ‪ MATLAB‬در علوم مهندسی‬
‫مولفین‪ :‬حیدرعلی شايانفر‪ ،‬حسین شايقی‬
‫انتشارات‪ :‬ياوريان‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Essential MATLAB for Engineers and Scientists :‬‬
‫عنوان‪ :‬برنامه نويسی ‪ MATLAB‬برای مهندسان‬
‫ترجمه عنوان‪ :‬آنچه بايد مهندسین و دانشمندان از متلب بدانند‬
‫مولفین‪ :‬محمود کشاورز مهر‪ ،‬بهزاد عبدی‬
‫مولفین‪Brian Hahn, Daniel Valentine:‬‬
‫سال چاپ‪2013 :‬‬
‫انتشارات‪Academic Press :‬‬
‫انتشارات‪ :‬نوپردازان‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪MATLAB: An Introduction with Applications :‬‬
‫عنوان‪ :‬آموزش کاربردی مباحث پیشرفته با ‪MATLAB‬‬
‫ترجمه عنوان‪ :‬مقدمه ای بر متلب و کاربردهای آن‬
‫مولفین‪ :‬نیما جمشیدی‪ ،‬علی ابويی مهريزی‪ ،‬رسول مواليی‬
‫مولف‪Amos Gilat :‬‬
‫انتشارات‪ :‬عابد‬
‫سال چاپ‪2014 :‬‬
‫انتشارات‪Wiley :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪MATLAB For Beginners: A Gentle Approach:‬‬
‫عنوان‪ :‬کاملترين مرجع آموزشی و کاربردی ‪MATLAB‬‬
‫ترجمه عنوان‪ :‬متلب برای افراد مبتدی با يک رويکرد تدريجی‬
‫مولفین‪ :‬علی اکبر علمداری‪ ،‬نسرين علمداری‬
‫مولف‪Peter I. Kattan:‬‬
‫انتشارات‪ :‬نگارنده دانش‬
‫سال چاپ‪2008 :‬‬
‫انتشارات‪CreateSpace Independent Publishing Platform :‬‬
‫لینک دسترسی‪ :‬لینک‬
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‫عنوان‪MATLAB for Engineers :‬‬
‫عنوان‪ :‬برنامه نويسی ‪ MATLAB‬برای مهندسین‬
‫ترجمه عنوان‪ :‬متلب برای مهندسین‬
‫مولف‪ :‬استفن چاپمن‬
‫مولف‪Holly Moore :‬‬
‫سال چاپ‪2011 :‬‬
‫انتشارات‪Prentice Hall :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Mastering MATLAB :‬‬
‫ترجمه عنوان‪ :‬تسلط بر متلب‬
‫مولفین‪Duane C. Hanselman, Bruce L. Littlefield :‬‬
‫سال چاپ‪2011 :‬‬
‫انتشارات‪Prentice Hall :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫مترجم‪ :‬سعدان زکائی‬
‫انتشارات‪ :‬دانشگاه صنعتی خواجه نصیرالدين طوسی‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪ :‬آموزش گام به گام محاسبات عددی با متلب‬
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‫مترجم‪ :‬رسول نصیری‬
‫انتشارات‪ :‬نشر گستر‬
‫لینک دسترسی‪ :‬لینک‬
‫منابع آموزشی آنالین‬
‫عنوان‪ :‬مجموعه فرادرسهای برنامهنويسی متلب‬
‫مدرس‪ :‬دکتر سید مصطفی کالمی هريس‬
‫مدت زمان‪ ۹ :‬ساعت و ‪ ۳‬دقیقه‬
‫زبان‪ :‬فارسی‬
‫ارائه دهنده‪ :‬فرادرس‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪ :‬مجموعه فرادرسهای متلب برای علوم و مهندسی‬
‫مدرس‪ :‬دکتر سید مصطفی کالمی هريس‬
‫مدت زمان‪ 14 :‬ساعت و ‪ 2۲‬دقیقه‬
‫زبان‪ :‬فارسی‬
‫ارائه دهنده‪ :‬فرادرس‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪ :‬مجموعه فرادرسهای برنامه نويسی متلب پیشرفته‬
‫مدرس‪ :‬دکتر سید مصطفی کالمی هريس‬
‫مدت زمان‪ ۲ :‬ساعت و ‪ 12‬دقیقه‬
‫زبان‪ :‬فارسی‬
‫ارائه دهنده‪ :‬فرادرس‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Introduction to Programming with MATLAB :‬‬
‫ترجمه عنوان‪ :‬آشنايی با برنامهنويسی متلب‬
‫مدرسین‪Akos Ledeczi, Michael Fitzpatrick, Robert Tairas :‬‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪Vanderbilt University :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Introduction to MATLAB :‬‬
‫ترجمه عنوان‪ :‬مقدمهای بر متلب‬
‫مدرس‪Danilo Šćepanović :‬‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪MIT OCW :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Up and Running with MATLAB :‬‬
‫ترجمه عنوان‪ :‬شروع سريع کار با متلب‬
‫مدرس‪Patrick Royal :‬‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪lynda.com :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Modelling and Simulation using MATLAB :‬‬
‫ترجمه عنوان‪ :‬مدلسازی و شبیهسازی با استفاده از متلب‬
‫مدرسین‪ Prof. Dr.-Ing. Georg Fries :‬و دیگران‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪iversity.org :‬‬
‫لینک دسترسی‪ :‬لینک‬
Global Optimization Toolbox
User’s Guide
R2014a
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Global Optimization Toolbox User’s Guide
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New for Version 1.0 (Release 13SP1+)
Revised for Version 1.0.1 (Release 14)
Revised for Version 1.0.2 (Release 14SP1)
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Contents
Getting Started
Introducing Global Optimization Toolbox
Functions
1
Global Optimization Toolbox Product Description . . . .
Key Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-2
1-2
Solution Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-4
Comparison of Four Solvers . . . . . . . . . . . . . . . . . . . . . . . .
Function to Optimize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Four Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Compare Syntax and Solutions . . . . . . . . . . . . . . . . . . . . . .
1-5
1-5
1-6
1-11
What Is Global Optimization? . . . . . . . . . . . . . . . . . . . . . . .
Local vs. Global Optima . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basins of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-13
1-13
1-14
............................
1-18
Choose a Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table for Choosing a Solver . . . . . . . . . . . . . . . . . . . . . . . . .
Solver Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Why Are Some Solvers Objects? . . . . . . . . . . . . . . . . . . . . . .
1-19
1-19
1-24
1-26
Optimization Workflow
Write Files for Optimization Functions
2
Compute Objective Functions . . . . . . . . . . . . . . . . . . . . . .
Objective (Fitness) Functions . . . . . . . . . . . . . . . . . . . . . . . .
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2-2
2-2
v
Write a Function File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Write a Vectorized Function . . . . . . . . . . . . . . . . . . . . . . . . .
Gradients and Hessians . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Maximizing vs. Minimizing . . . . . . . . . . . . . . . . . . . . . . . . .
2-2
2-3
2-5
2-5
Write Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Consult Optimization Toolbox Documentation . . . . . . . . . .
Set Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ensure ga Options Maintain Feasibility . . . . . . . . . . . . . . .
Gradients and Hessians . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vectorized Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-7
2-7
2-7
2-8
2-8
2-8
Using GlobalSearch and MultiStart
3
Problems That GlobalSearch and MultiStart Can
Solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
............................
3-4
Inputs for Problem Structure . . . . . . . . . . . . . . . . . . . . . . .
3-6
Create Problem Structure . . . . . . . . . . . . . . . . . . . . . . . . . .
About Problem Structures . . . . . . . . . . . . . . . . . . . . . . . . . .
Using the createOptimProblem Function . . . . . . . . . . . . . .
Exporting from the Optimization app . . . . . . . . . . . . . . . . .
3-7
3-7
3-7
3-10
Create Solver Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What Is a Solver Object? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties (Global Options) of Solver Objects . . . . . . . . . . .
Creating a Nondefault GlobalSearch Object . . . . . . . . . . . .
Creating a Nondefault MultiStart Object . . . . . . . . . . . . . .
3-17
3-17
3-17
3-19
3-19
Set Start Points for MultiStart . . . . . . . . . . . . . . . . . . . . . .
Four Ways to Set Start Points . . . . . . . . . . . . . . . . . . . . . . .
Positive Integer for Start Points . . . . . . . . . . . . . . . . . . . . .
RandomStartPointSet Object for Start Points . . . . . . . . . .
CustomStartPointSet Object for Start Points . . . . . . . . . . .
3-20
3-20
3-20
3-21
3-21
Optimization Workflow
vi
Contents
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Cell Array of Objects for Start Points . . . . . . . . . . . . . . . . .
3-22
Run the Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimize by Calling run . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of Run with GlobalSearch . . . . . . . . . . . . . . . . . . .
Example of Run with MultiStart . . . . . . . . . . . . . . . . . . . . .
3-24
3-24
3-25
3-26
Single Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-28
Multiple Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
About Multiple Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Change the Definition of Distinct Solutions . . . . . . . . . . . .
3-30
3-30
3-32
Iterative Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Types of Iterative Display . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examine Types of Iterative Display . . . . . . . . . . . . . . . . . . .
3-35
3-35
3-36
..........................
3-39
Visualize the Basins of Attraction . . . . . . . . . . . . . . . . . . .
3-40
Output Functions for GlobalSearch and MultiStart . . .
What Are Output Functions? . . . . . . . . . . . . . . . . . . . . . . . .
GlobalSearch Output Function . . . . . . . . . . . . . . . . . . . . . .
No Parallel Output Functions . . . . . . . . . . . . . . . . . . . . . . .
3-43
3-43
3-43
3-46
Plot Functions for GlobalSearch and MultiStart . . . . . .
What Are Plot Functions? . . . . . . . . . . . . . . . . . . . . . . . . . . .
MultiStart Plot Function . . . . . . . . . . . . . . . . . . . . . . . . . . .
No Parallel Plot Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
3-47
3-47
3-48
3-51
How GlobalSearch and MultiStart Work . . . . . . . . . . . . .
Multiple Runs of a Local Solver . . . . . . . . . . . . . . . . . . . . . .
Differences Between the Solver Objects . . . . . . . . . . . . . . .
GlobalSearch Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MultiStart Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-52
3-52
3-52
3-54
3-59
3-61
...............
3-62
Global Output Structures
Can You Certify a Solution Is Global?
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vii
No Guarantees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Check if a Solution Is a Local Solution with
patternsearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Identify a Bounded Region That Contains a Global
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Use MultiStart with More Start Points . . . . . . . . . . . . . . . .
3-62
Refine Start Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
About Refining Start Points . . . . . . . . . . . . . . . . . . . . . . . . .
Methods of Generating Start Points . . . . . . . . . . . . . . . . . .
Example: Searching for a Better Solution . . . . . . . . . . . . . .
3-66
3-66
3-67
3-70
Change Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How to Determine Which Options to Change . . . . . . . . . . .
Changing Local Solver Options . . . . . . . . . . . . . . . . . . . . . .
Changing Global Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-75
3-75
3-76
3-77
Reproduce Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Identical Answers with Pseudorandom Numbers . . . . . . . .
Steps to Take in Reproducing Results . . . . . . . . . . . . . . . . .
Example: Reproducing a GlobalSearch or MultiStart
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parallel Processing and Random Number Streams . . . . . .
3-79
3-79
3-79
Find Global or Multiple Local Minima . . . . . . . . . . . . . . .
Function to Optimize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Single Global Minimum Via GlobalSearch . . . . . . . . . . . . .
Multiple Local Minima Via MultiStart . . . . . . . . . . . . . . . .
3-83
3-83
3-85
3-87
Optimize Using Only Feasible Start Points . . . . . . . . . . .
3-91
MultiStart Using lsqcurvefit or lsqnonlin . . . . . . . . . . . .
3-96
3-62
3-63
3-64
3-80
3-82
Parallel MultiStart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-100
Steps for Parallel MultiStart . . . . . . . . . . . . . . . . . . . . . . . . 3-100
Speedup with Parallel Computing . . . . . . . . . . . . . . . . . . . . 3-102
Isolated Global Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-104
Difficult-To-Locate Global Minimum . . . . . . . . . . . . . . . . . . 3-104
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Default Settings Cannot Find the Global Minimum — Add
Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
GlobalSearch with Bounds and More Start Points . . . . . . .
MultiStart with Bounds and Many Start Points . . . . . . . . .
MultiStart Without Bounds, Widely Dispersed Start
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MultiStart with a Regular Grid of Start Points . . . . . . . . .
MultiStart with Regular Grid and Promising Start
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-106
3-106
3-107
3-108
3-109
3-109
Using Direct Search
4
What Is Direct Search? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2
Optimize Using Pattern Search . . . . . . . . . . . . . . . . . . . . .
Call patternsearch at the Command Line . . . . . . . . . . . . . .
Pattern Search on Unconstrained Problems . . . . . . . . . . . .
Pattern Search on Constrained Problems . . . . . . . . . . . . . .
Additional Output Arguments . . . . . . . . . . . . . . . . . . . . . . .
Use the Optimization App for Pattern Search . . . . . . . . . .
4-3
4-3
4-3
4-4
4-5
4-5
Optimize Using the GPS Algorithm . . . . . . . . . . . . . . . . . .
Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finding the Minimum of the Function . . . . . . . . . . . . . . . . .
Plotting the Objective Function Values and Mesh Sizes . .
4-8
4-8
4-8
4-10
Pattern Search Terminology . . . . . . . . . . . . . . . . . . . . . . . .
Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Expanding and Contracting . . . . . . . . . . . . . . . . . . . . . . . . .
4-12
4-12
4-13
4-14
4-14
How Pattern Search Polling Works . . . . . . . . . . . . . . . . . .
Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Successful Polls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Unsuccessful Poll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Displaying the Results at Each Iteration . . . . . . . . . . . . . .
More Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-15
4-15
4-16
4-19
4-20
4-21
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Contents
Poll Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Complete Poll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stopping Conditions for the Pattern Search . . . . . . . . . . . .
Robustness of Pattern Search . . . . . . . . . . . . . . . . . . . . . . .
4-22
4-24
4-24
4-26
Searching and Polling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definition of Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How to Use a Search Method . . . . . . . . . . . . . . . . . . . . . . . .
Search Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
When to Use Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-27
4-27
4-29
4-30
4-30
Setting Solver Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . .
4-32
Search and Poll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using a Search Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Search Using a Different Solver . . . . . . . . . . . . . . . . . . . . . .
4-33
4-33
4-37
Nonlinear Constraint Solver Algorithm . . . . . . . . . . . . . .
4-39
Custom Plot Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
About Custom Plot Functions . . . . . . . . . . . . . . . . . . . . . . . .
Creating the Custom Plot Function . . . . . . . . . . . . . . . . . . .
Setting Up the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using the Custom Plot Function . . . . . . . . . . . . . . . . . . . . .
How the Plot Function Works . . . . . . . . . . . . . . . . . . . . . . .
4-42
4-42
4-42
4-43
4-44
4-46
Set Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Set Options Using psoptimset . . . . . . . . . . . . . . . . . . . . . . .
Create Options and Problems Using the Optimization
App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-48
4-48
Polling Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using a Complete Poll in a Generalized Pattern Search . .
Compare the Efficiency of Poll Options . . . . . . . . . . . . . . . .
4-51
4-51
4-56
Set Mesh Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mesh Expansion and Contraction . . . . . . . . . . . . . . . . . . . .
Mesh Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-63
4-63
4-70
Constrained Minimization Using patternsearch . . . . . .
4-74
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4-50
Linearly Constrained Problem . . . . . . . . . . . . . . . . . . . . . . .
Nonlinearly Constrained Problem . . . . . . . . . . . . . . . . . . . .
4-74
4-77
Use Cache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-81
Vectorize the Objective and Constraint Functions . . . .
Vectorize for Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vectorized Objective Function . . . . . . . . . . . . . . . . . . . . . . .
Vectorized Constraint Functions . . . . . . . . . . . . . . . . . . . . .
Example of Vectorized Objective and Constraints . . . . . . .
4-85
4-85
4-85
4-88
4-89
Optimize an ODE in Parallel . . . . . . . . . . . . . . . . . . . . . . . .
4-91
Using the Genetic Algorithm
5
What Is the Genetic Algorithm? . . . . . . . . . . . . . . . . . . . . .
5-3
Optimize Using ga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calling the Function ga at the Command Line . . . . . . . . . .
Use the Optimization App . . . . . . . . . . . . . . . . . . . . . . . . . .
5-4
5-4
5-4
Minimize Rastrigin’s Function . . . . . . . . . . . . . . . . . . . . . .
Rastrigin’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finding the Minimum of Rastrigin’s Function . . . . . . . . . .
Finding the Minimum from the Command Line . . . . . . . . .
Displaying Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-8
5-8
5-10
5-12
5-13
Genetic Algorithm Terminology . . . . . . . . . . . . . . . . . . . . .
Fitness Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Populations and Generations . . . . . . . . . . . . . . . . . . . . . . . .
Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitness Values and Best Fitness Values . . . . . . . . . . . . . . .
Parents and Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-18
5-18
5-18
5-19
5-19
5-20
5-20
How the Genetic Algorithm Works . . . . . . . . . . . . . . . . . .
5-21
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Contents
Outline of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initial Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating the Next Generation . . . . . . . . . . . . . . . . . . . . . . .
Plots of Later Generations . . . . . . . . . . . . . . . . . . . . . . . . . .
Stopping Conditions for the Algorithm . . . . . . . . . . . . . . . .
Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reproduction Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mutation and Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-21
5-22
5-23
5-25
5-26
5-29
5-29
5-30
Mixed Integer Optimization . . . . . . . . . . . . . . . . . . . . . . . .
Solving Mixed Integer Optimization Problems . . . . . . . . . .
Characteristics of the Integer ga Solver . . . . . . . . . . . . . . .
Integer ga Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-32
5-32
5-34
5-40
Solving a Mixed Integer Engineering Design Problem
Using the Genetic Algorithm . . . . . . . . . . . . . . . . . . . . .
5-42
Nonlinear Constraint Solver Algorithm . . . . . . . . . . . . . .
5-52
Create Custom Plot Function . . . . . . . . . . . . . . . . . . . . . . .
About Custom Plot Functions . . . . . . . . . . . . . . . . . . . . . . . .
Creating the Custom Plot Function . . . . . . . . . . . . . . . . . . .
Using the Plot Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How the Plot Function Works . . . . . . . . . . . . . . . . . . . . . . .
5-54
5-54
5-54
5-55
5-56
Reproduce Results in Optimization App . . . . . . . . . . . . .
5-58
Resume ga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Resuming ga From the Final Population . . . . . . . . . . . . . . .
Resuming ga From a Previous Run . . . . . . . . . . . . . . . . . . .
5-59
5-59
5-64
Options and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Running ga with the Default Options . . . . . . . . . . . . . . . . .
Setting Options at the Command Line . . . . . . . . . . . . . . . .
Additional Output Arguments . . . . . . . . . . . . . . . . . . . . . . .
5-66
5-66
5-67
5-69
Use Exported Options and Problems . . . . . . . . . . . . . . . .
5-70
Reproduce Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-71
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Run ga from a File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-73
Population Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Importance of Population Diversity . . . . . . . . . . . . . . . . . . .
Setting the Initial Range . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linearly Constrained Population and Custom Plot
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setting the Population Size . . . . . . . . . . . . . . . . . . . . . . . . .
5-76
5-76
5-76
Fitness Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scaling the Fitness Scores . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparing Rank and Top Scaling . . . . . . . . . . . . . . . . . . . .
5-87
5-87
5-89
Vary Mutation and Crossover . . . . . . . . . . . . . . . . . . . . . . .
Setting the Amount of Mutation . . . . . . . . . . . . . . . . . . . . .
Setting the Crossover Fraction . . . . . . . . . . . . . . . . . . . . . . .
Comparing Results for Varying Crossover Fractions . . . . .
5-91
5-91
5-93
5-98
5-80
5-85
Global vs. Local Minima Using ga . . . . . . . . . . . . . . . . . . . 5-100
Searching for a Global Minimum . . . . . . . . . . . . . . . . . . . . . 5-100
Running the Genetic Algorithm on the Example . . . . . . . . 5-102
Include a Hybrid Function . . . . . . . . . . . . . . . . . . . . . . . . . 5-107
Set Maximum Number of Generations . . . . . . . . . . . . . . . 5-111
Vectorize the Fitness Function . . . . . . . . . . . . . . . . . . . . . . 5-114
Vectorize for Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-114
Vectorized Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-115
Constrained Minimization Using ga . . . . . . . . . . . . . . . . . 5-116
Using Simulated Annealing
6
What Is Simulated Annealing? . . . . . . . . . . . . . . . . . . . . . .
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6-2
xiii
Optimize Using Simulated Annealing . . . . . . . . . . . . . . . .
Calling simulannealbnd at the Command Line . . . . . . . . .
Using the Optimization App . . . . . . . . . . . . . . . . . . . . . . . . .
6-3
6-3
6-4
Minimize Function with Many Local Minima . . . . . . . . .
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Minimize at the Command Line . . . . . . . . . . . . . . . . . . . . . .
Minimize Using the Optimization App . . . . . . . . . . . . . . . .
6-6
6-6
6-7
6-7
Simulated Annealing Terminology . . . . . . . . . . . . . . . . . .
Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Annealing Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reannealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-9
6-9
6-9
6-10
6-10
How Simulated Annealing Works . . . . . . . . . . . . . . . . . . .
Outline of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stopping Conditions for the Algorithm . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-11
6-11
6-13
6-13
Command Line Simulated Annealing Optimization . . .
Run simulannealbnd With the Default Options . . . . . . . . .
Set Options for simulannealbnd at the Command Line . . .
Reproduce Your Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-15
6-15
6-16
6-18
Minimization Using Simulated Annealing Algorithm . .
6-20
Multiobjective Optimization
7
xiv
Contents
What Is Multiobjective Optimization? . . . . . . . . . . . . . . .
7-2
Use gamultiobj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Use gamultiobj with the Optimization app . . . . . . . . . . . . .
Multiobjective Optimization with Two Objectives . . . . . . .
Options and Syntax: Differences from ga . . . . . . . . . . . . . .
7-5
7-5
7-6
7-6
7-13
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-14
Parallel Processing
8
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parallel Processing Types in Global Optimization
Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How Toolbox Functions Distribute Processes . . . . . . . . . . .
8-2
How to Use Parallel Processing . . . . . . . . . . . . . . . . . . . . .
Multicore Processors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Processor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parallel Search Functions or Hybrid Functions . . . . . . . . .
8-12
8-12
8-13
8-15
Minimizing an Expensive Optimization Problem Using
Parallel Computing Toolbox™ . . . . . . . . . . . . . . . . . . . .
8-19
8-2
8-3
Options Reference
9
GlobalSearch and MultiStart Properties (Options) . . . .
How to Set Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of Both Objects . . . . . . . . . . . . . . . . . . . . . . . . . .
GlobalSearch Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MultiStart Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-2
9-2
9-2
9-7
9-8
Pattern Search Options . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization App vs. Command Line . . . . . . . . . . . . . . . . .
Plot Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Poll Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Search Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mesh Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constraint Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cache Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-9
9-9
9-10
9-12
9-14
9-19
9-20
9-21
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xv
Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output Function Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display to Command Window Options . . . . . . . . . . . . . . . .
Vectorize and Parallel Options (User Function
Evaluation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Options Table for Pattern Search Algorithms . . . . . . . . . . .
9-21
9-22
9-24
Genetic Algorithm Options . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization App vs. Command Line . . . . . . . . . . . . . . . . .
Plot Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Population Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitness Scaling Options . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Selection Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reproduction Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mutation Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Crossover Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Migration Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constraint Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiobjective Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hybrid Function Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stopping Criteria Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output Function Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display to Command Window Options . . . . . . . . . . . . . . . .
Vectorize and Parallel Options (User Function
Evaluation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-31
9-31
9-32
9-36
9-39
9-41
9-43
9-43
9-46
9-50
9-51
9-51
9-52
9-53
9-54
9-55
Simulated Annealing Options . . . . . . . . . . . . . . . . . . . . . . .
saoptimset At The Command Line . . . . . . . . . . . . . . . . . . . .
Plot Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Algorithm Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hybrid Function Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stopping Criteria Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output Function Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9-58
9-58
9-58
9-60
9-61
9-63
9-64
9-64
9-66
9-25
9-27
9-56
Functions — Alphabetical List
10
xvi
Contents
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‫‪Getting Started‬‬
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1
Introducing Global
Optimization Toolbox
Functions
• “Global Optimization Toolbox Product Description” on page 1-2
• “Solution Quality” on page 1-4
• “Comparison of Four Solvers” on page 1-5
• “What Is Global Optimization?” on page 1-13
• “Optimization Workflow” on page 1-18
• “Choose a Solver” on page 1-19
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1
Introducing Global Optimization Toolbox Functions
Global Optimization Toolbox Product Description
Solve multiple maxima, multiple minima, and nonsmooth
optimization problems
Global Optimization Toolbox provides methods that search for global
solutions to problems that contain multiple maxima or minima. It includes
global search, multistart, pattern search, genetic algorithm, and simulated
annealing solvers. You can use these solvers to solve optimization problems
where the objective or constraint function is continuous, discontinuous,
stochastic, does not possess derivatives, or includes simulations or black-box
functions with undefined values for some parameter settings.
Genetic algorithm and pattern search solvers support algorithmic
customization. You can create a custom genetic algorithm variant by
modifying initial population and fitness scaling options or by defining parent
selection, crossover, and mutation functions. You can customize pattern
search by defining polling, searching, and other functions.
Key Features
• Interactive tools for defining and solving optimization problems and
monitoring solution progress
• Global search and multistart solvers for finding single or multiple global
optima
• Genetic algorithm solver that supports linear, nonlinear, and bound
constraints
• Multiobjective genetic algorithm with Pareto-front identification, including
linear and bound constraints
• Pattern search solver that supports linear, nonlinear, and bound
constraints
• Simulated annealing tools that implement a random search method,
with options for defining annealing process, temperature schedule, and
acceptance criteria
• Parallel computing support in multistart, genetic algorithm, and pattern
search solver
1-2
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Global Optimization Toolbox Product Description
• Custom data type support in genetic algorithm, multiobjective genetic
algorithm, and simulated annealing solvers
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1-3
1
Introducing Global Optimization Toolbox Functions
Solution Quality
Global Optimization Toolbox solvers repeatedly attempt to locate a global
solution. However, no solver employs an algorithm that can certify a solution
as global.
You can extend the capabilities of Global Optimization Toolbox functions by
writing your own files, by using them in combination with other toolboxes, or
with the MATLAB® or Simulink® environments.
1-4
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Comparison of Four Solvers
Comparison of Four Solvers
In this section...
“Function to Optimize” on page 1-5
“Four Solution Methods” on page 1-6
“Compare Syntax and Solutions” on page 1-11
Function to Optimize
This example shows how to minimize Rastrigin’s function with four solvers.
Each solver has its own characteristics. The characteristics lead to different
solutions and run times. The results, examined in “Compare Syntax and
Solutions” on page 1-11, can help you choose an appropriate solver for your
own problems.
Rastrigin’s function has many local minima, with a global minimum at (0,0):
Ras( x) = 20 + x12 + x22 − 10 ( cos 2πx1 + cos 2πx2 ) .
Usually you don’t know the location of the global minimum of your objective
function. To show how the solvers look for a global solution, this example
starts all the solvers around the point [20,30], which is far from the global
minimum.
The rastriginsfcn.m file implements Rastrigin’s function. This file comes
with Global Optimization Toolbox software. This example employs a
scaled version of Rastrigin’s function with larger basins of attraction. For
information, see “Basins of Attraction” on page 1-14.
rf2 = @(x)rastriginsfcn(x/10);
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1-5
1
Introducing Global Optimization Toolbox Functions
This example minimizes rf2 using the default settings of fminunc (an
Optimization Toolbox™ solver), patternsearch, and GlobalSearch. It also
uses ga with a nondefault setting, to obtain an initial population around
the point [20,30].
Four Solution Methods
• “fminunc” on page 1-7
• “patternsearch” on page 1-8
• “ga” on page 1-8
1-6
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Comparison of Four Solvers
• “GlobalSearch” on page 1-10
fminunc
To solve the optimization problem using the fminunc Optimization Toolbox
solver, enter:
rf2 = @(x)rastriginsfcn(x/10); % objective
x0 = [20,30]; % start point away from the minimum
[xf ff flf of] = fminunc(rf2,x0)
fminunc returns
Local minimum found.
Optimization completed because the size of the gradient is
less than the default value of the function tolerance.
xf =
19.8991
29.8486
ff =
12.9344
flf =
1
of =
iterations: 3
funcCount: 15
stepsize: 1
firstorderopt: 5.9907e-009
algorithm: 'medium-scale: Quasi-Newton line search'
message: 'Local minimum found.
Optimization completed because ...'
• xf is the minimizing point.
• ff is the value of the objective, rf2, at xf.
• flf is the exit flag. An exit flag of 1 indicates xf is a local minimum.
• of is the output structure, which describes the fminunc calculations
leading to the solution.
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1-7
1
Introducing Global Optimization Toolbox Functions
patternsearch
To solve the optimization problem using the patternsearch Global
Optimization Toolbox solver, enter:
rf2 = @(x)rastriginsfcn(x/10); % objective
x0 = [20,30]; % start point away from the minimum
[xp fp flp op] = patternsearch(rf2,x0)
patternsearch returns
Optimization terminated: mesh size less than options.TolMesh.
xp =
19.8991
-9.9496
fp =
4.9748
flp =
1
op =
function: @(x)rastriginsfcn(x/10)
problemtype: 'unconstrained'
pollmethod: 'gpspositivebasis2n'
searchmethod: []
iterations: 48
funccount: 174
meshsize: 9.5367e-07
rngstate: [1x1 struct]
message: 'Optimization terminated: mesh size
less than options.T...'
• xp is the minimizing point.
• fp is the value of the objective, rf2, at xp.
• flp is the exit flag. An exit flag of 1 indicates xp is a local minimum.
• op is the output structure, which describes the patternsearch calculations
leading to the solution.
ga
To solve the optimization problem using the ga Global Optimization Toolbox
solver, enter:
1-8
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Comparison of Four Solvers
rng('default') % for reproducibility
rf2 = @(x)rastriginsfcn(x/10); % objective
x0 = [20,30]; % start point away from the minimum
initpop = 10*randn(20,2) + repmat([10 30],20,1);
opts = gaoptimset('InitialPopulation',initpop);
[xga fga flga oga] = ga(rf2,2,[],[],[],[],[],[],[],opts)
initpop is a 20-by-2 matrix. Each row of initpop has mean [10,30], and
each element is normally distributed with standard deviation 10. The rows of
initpop form an initial population matrix for the ga solver.
opts is an optimization structure setting initpop as the initial population.
The final line calls ga, using the optimization structure.
ga uses random numbers, and produces a random result. In this case ga
returns:
Optimization terminated: average change in the fitness value
less than options.TolFun.
xga =
0.1191
0.0089
fga =
0.0283
flga =
1
oga =
problemtype: 'unconstrained'
rngstate: [1x1 struct]
generations: 107
funccount: 5400
message: 'Optimization terminated: average
change in the fitness ...'
• xga is the minimizing point.
• fga is the value of the objective, rf2, at xga.
• flga is the exit flag. An exit flag of 1 indicates xga is a local minimum.
• oga is the output structure, which describes the ga calculations leading
to the solution.
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1-9
1
Introducing Global Optimization Toolbox Functions
GlobalSearch
To solve the optimization problem using the GlobalSearch solver, enter:
rf2 = @(x)rastriginsfcn(x/10); % objective
x0 = [20,30]; % start point away from the minimum
problem = createOptimProblem('fmincon','objective',rf2,...
'x0',x0);
gs = GlobalSearch;
[xg fg flg og] = run(gs,problem)
problem is an optimization problem structure. problem specifies the fmincon
solver, the rf2 objective function, and x0=[20,30]. For more information on
using createOptimProblem, see “Create Problem Structure” on page 3-7.
Note You must specify fmincon as the solver for GlobalSearch, even for
unconstrained problems.
gs is a default GlobalSearch object. The object contains options for solving
the problem. Calling run(gs,problem) runs problem from multiple start
points. The start points are random, so the following result is also random.
In this case, the run returns:
GlobalSearch stopped because it analyzed all the trial points.
All 8 local solver runs converged with a positive local solver exit flag.
xg =
1.0e-07 *
-0.1405
-0.1405
fg =
0
flg =
1
og =
1-10
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Comparison of Four Solvers
funcCount: 2257
localSolverTotal: 8
localSolverSuccess: 8
localSolverIncomplete: 0
localSolverNoSolution: 0
message: 'GlobalSearch stopped because it analyzed all ...'
• xg is the minimizing point.
• fg is the value of the objective, rf2, at xg.
• flg is the exit flag. An exit flag of 1 indicates all fmincon runs converged
properly.
• og is the output structure, which describes the GlobalSearch calculations
leading to the solution.
Compare Syntax and Solutions
One solution is better than another if its objective function value is smaller
than the other. The following table summarizes the results, accurate to one
decimal.
Results
fminunc
patternsearch
ga
GlobalSearch
solution
[19.9 29.9]
[19.9 -9.9]
[0.1 0]
[0 0]
objective
12.9
5
0.03
0
# Fevals
15
174
5400
2257
These results are typical:
• fminunc quickly reaches the local solution within its starting basin, but
does not explore outside this basin at all. fminunc has simple calling
syntax.
• patternsearch takes more function evaluations than fminunc, and
searches through several basins, arriving at a better solution than fminunc.
patternsearch calling syntax is the same as that of fminunc.
• ga takes many more function evaluations than patternsearch. By chance
it arrived at a better solution. ga is stochastic, so its results change with
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1-11
1
Introducing Global Optimization Toolbox Functions
every run. ga has simple calling syntax, but there are extra steps to have
an initial population near [20,30].
• GlobalSearch run takes the same order of magnitude of function
evaluations as ga, searches many basins, and arrives at an even better
solution. In this case, GlobalSearch found the global optimum. Setting
up GlobalSearch is more involved than setting up the other solvers. As
the example shows, before calling GlobalSearch, you must create both
a GlobalSearch object (gs in the example), and a problem structure
(problem). Then, call the run method with gs and problem. For more
details on how to run GlobalSearch, see “Optimization Workflow” on page
3-4.
1-12
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What Is Global Optimization?
What Is Global Optimization?
In this section...
“Local vs. Global Optima” on page 1-13
“Basins of Attraction” on page 1-14
Local vs. Global Optima
Optimization is the process of finding the point that minimizes a function.
More specifically:
• A local minimum of a function is a point where the function value is
smaller than or equal to the value at nearby points, but possibly greater
than at a distant point.
• A global minimum is a point where the function value is smaller than or
equal to the value at all other feasible points.
Global minimum
Local minimum
Generally, Optimization Toolbox solvers find a local optimum. (This local
optimum can be a global optimum.) They find the optimum in the basin
of attraction of the starting point. For more information, see “Basins of
Attraction” on page 1-14.
In contrast, Global Optimization Toolbox solvers are designed to search
through more than one basin of attraction. They search in various ways:
• GlobalSearch and MultiStart generate a number of starting points. They
then use a local solver to find the optima in the basins of attraction of the
starting points.
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1-13
1
Introducing Global Optimization Toolbox Functions
• ga uses a set of starting points (called the population) and iteratively
generates better points from the population. As long as the initial
population covers several basins, ga can examine several basins.
• simulannealbnd performs a random search. Generally, simulannealbnd
accepts a point if it is better than the previous point. simulannealbnd
occasionally accepts a worse point, in order to reach a different basin.
• patternsearch looks at a number of neighboring points before accepting
one of them. If some neighboring points belong to different basins,
patternsearch in essence looks in a number of basins at once.
Basins of Attraction
If an objective function f(x) is smooth, the vector –∇f(x) points in the direction
where f(x) decreases most quickly. The equation of steepest descent, namely
d
x(t) = −∇f ( x(t)),
dt
yields a path x(t) that goes to a local minimum as t gets large. Generally,
initial values x(0) that are close to each other give steepest descent paths that
tend to the same minimum point. The basin of attraction for steepest descent
is the set of initial values leading to the same local minimum.
The following figure shows two one-dimensional minima. The figure shows
different basins of attraction with different line styles, and it shows directions
of steepest descent with arrows. For this and subsequent figures, black dots
represent local minima. Every steepest descent path, starting at a point x(0),
goes to the black dot in the basin containing x(0).
f(x)
x
1-14
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What Is Global Optimization?
The following figure shows how steepest descent paths can be more
complicated in more dimensions.
The following figure shows even more complicated paths and basins of
attraction.
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1-15
1
Introducing Global Optimization Toolbox Functions
Constraints can break up one basin of attraction into several pieces. For
example, consider minimizing y subject to:
• y ≥ |x|
• y ≥ 5 – 4(x–2)2.
The figure shows the two basins of attraction with the final points.
1-16
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What Is Global Optimization?
The steepest descent paths are straight lines down to the constraint
boundaries. From the constraint boundaries, the steepest descent paths
travel down along the boundaries. The final point is either (0,0) or (11/4,11/4),
depending on whether the initial x-value is above or below 2.
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1-17
1
Introducing Global Optimization Toolbox Functions
Optimization Workflow
To solve an optimization problem:
1 Decide what type of problem you have, and whether you want a local or global
solution (see “Local vs. Global Optima” on page 1-13). Choose a solver per the
recommendations in “Choose a Solver” on page 1-19.
2 Write your objective function and, if applicable, constraint functions per the
syntax in “Compute Objective Functions” on page 2-2 and “Write Constraints”
on page 2-7.
3 Set appropriate options with psoptimset, gaoptimset, or saoptimset, or
prepare a GlobalSearch or MultiStart problem as described in “Optimization
Workflow” on page 3-4. For details, see “Pattern Search Options” on page 9-9,
“Genetic Algorithm Options” on page 9-31, or “Simulated Annealing Options”
on page 9-58.
4 Run the solver.
5 Examine the result. For information on the result, see “Solver Outputs and
Iterative Display” in the Optimization Toolbox documentation or “Examine
Results” for GlobalSearch or MultiStart.
6 If the result is unsatisfactory, change options or start points or otherwise
update your optimization and rerun it. For information, see “Improve
Results”, or see “When the Solver Fails”, “When the Solver Might Have
Succeeded”, or “When the Solver Succeeds” in the Optimization Toolbox
documentation.
1-18
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Choose a Solver
Choose a Solver
In this section...
“Table for Choosing a Solver” on page 1-19
“Solver Characteristics” on page 1-24
“Why Are Some Solvers Objects?” on page 1-26
Table for Choosing a Solver
There are six Global Optimization Toolbox solvers:
• ga (Genetic Algorithm)
• GlobalSearch
• MultiStart
• patternsearch, also called direct search
• simulannealbnd (Simulated Annealing)
• gamultiobj, which is not a minimizer; see “Multiobjective Optimization”
Choose an optimizer based on problem characteristics and on the type of
solution you want. “Solver Characteristics” on page 1-24 contains more
information that can help you decide which solver is likely to be most suitable.
Desired Solution
Smooth Objective and
Constraints
Nonsmooth Objective or
Constraints
“Explanation of “Desired
Solution”” on page 1-20
“Choosing Between Solvers for
Smooth Problems” on page 1-22
“Choosing Between Solvers for
Nonsmooth Problems” on page
1-23
Single local solution
Optimization Toolbox functions;
see “Optimization Decision
Table” in the Optimization
Toolbox documentation
fminbnd, patternsearch,
fminsearch, ga, simulannealbnd
Multiple local solutions
GlobalSearch, MultiStart
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1-19
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Introducing Global Optimization Toolbox Functions
Desired Solution
Smooth Objective and
Constraints
Nonsmooth Objective or
Constraints
Single global solution
GlobalSearch, MultiStart,
patternsearch, ga,
simulannealbnd
patternsearch, ga,
simulannealbnd
Single local solution
using parallel processing
MultiStart, Optimization
Toolbox functions
patternsearch, ga
Multiple local solutions
using parallel processing
MultiStart
Single global solution
using parallel processing
MultiStart
patternsearch, ga
Explanation of “Desired Solution”
To understand the meaning of the terms in “Desired Solution,” consider the
example
f(x)=100x2(1–x)2–x,
which has local minima x1 near 0 and x2 near 1:
1-20
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Choose a Solver
The minima are located at:
x1 = fminsearch(@(x)(100*x^2*(x - 1)^2 - x),0)
x1 =
0.0051
x2 = fminsearch(@(x)(100*x^2*(x - 1)^2 - x),1)
x2 =
1.0049
Description of the Terms
Term
Meaning
Single local solution
Find one local solution, a point x where the
objective function f(x) is a local minimum. For
more details, see “Local vs. Global Optima” on
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Introducing Global Optimization Toolbox Functions
Description of the Terms (Continued)
Term
Meaning
page 1-13. In the example, both x1 and x2 are
local solutions.
Multiple local solutions
Find a set of local solutions. In the example, the
complete set of local solutions is {x1,x2}.
Single global solution
Find the point x where the objective function f(x)
is a global minimum. In the example, the global
solution is x2.
Choosing Between Solvers for Smooth Problems
Single Global Solution.
1 Try GlobalSearch first. It is most focused on finding a global solution, and
has an efficient local solver, fmincon.
2 Try MultiStart second. It has efficient local solvers, and can search a
wide variety of start points.
3 Try patternsearch third. It is less efficient, since it does not use gradients.
However, patternsearch is robust and is more efficient than the remaining
local solvers.
4 Try ga fourth. It can handle all types of constraints, and is usually more
efficient than simulannealbnd.
5 Try simulannealbnd last. It can handle problems with no constraints or
bound constraints. simulannealbnd is usually the least efficient solver.
However, given a slow enough cooling schedule, it can find a global solution.
Multiple Local Solutions. GlobalSearch and MultiStart both provide
multiple local solutions. For the syntax to obtain multiple solutions, see
“Multiple Solutions” on page 3-30. GlobalSearch and MultiStart differ in
the following characteristics:
1-22
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Choose a Solver
• MultiStart can find more local minima. This is because GlobalSearch
rejects many generated start points (initial points for local solution).
Essentially, GlobalSearch accepts a start point only when it determines
that the point has a good chance of obtaining a global minimum. In
contrast, MultiStart passes all generated start points to a local solver. For
more information, see “GlobalSearch Algorithm” on page 3-54.
• MultiStart offers a choice of local solver: fmincon, fminunc, lsqcurvefit,
or lsqnonlin. The GlobalSearch solver uses only fmincon as its local
solver.
• GlobalSearch uses a scatter-search algorithm for generating start points.
In contrast, MultiStart generates points uniformly at random within
bounds, or allows you to provide your own points.
• MultiStart can run in parallel. See “How to Use Parallel Processing”
on page 8-12.
Choosing Between Solvers for Nonsmooth Problems
Choose the applicable solver with the lowest number. For problems with
integer constraints, use ga.
1 Use fminbnd first on one-dimensional bounded problems only. fminbnd
provably converges quickly in one dimension.
2 Use patternsearch on any other type of problem. patternsearch provably
converges, and handles all types of constraints.
3 Try fminsearch next for low-dimensional unbounded problems.
fminsearch is not as general as patternsearch and can fail to converge.
For low-dimensional problems, fminsearch is simple to use, since it has
few tuning options.
4 Try ga next. ga has little supporting theory and is often less efficient than
patternsearch. It handles all types of constraints. ga is the only solver
that handles integer constraints.
5 Try simulannealbnd last for unbounded problems, or for problems with
bounds. simulannealbnd provably converges only for a logarithmic cooling
schedule, which is extremely slow. simulannealbnd takes only bound
constraints, and is often less efficient than ga.
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1-23
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Introducing Global Optimization Toolbox Functions
Solver Characteristics
Solver
Convergence
Characteristics
GlobalSearch
Fast convergence to local
optima for smooth problems.
Deterministic iterates
Gradient-based
Automatic stochastic start points
Removes many start points
heuristically
MultiStart
Fast convergence to local
optima for smooth problems.
Deterministic iterates
Can run in parallel; see “How to Use
Parallel Processing” on page 8-12
Gradient-based
Stochastic or deterministic start
points, or combination of both
Automatic stochastic start points
Runs all start points
Choice of local solver: fmincon,
fminunc, lsqcurvefit, or
lsqnonlin
patternsearch
Proven convergence to
local optimum, slower than
gradient-based solvers.
Deterministic iterates
Can run in parallel; see “How to Use
Parallel Processing” on page 8-12
No gradients
User-supplied start point
1-24
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Choose a Solver
Solver
Convergence
Characteristics
ga
No convergence proof.
Stochastic iterates
Can run in parallel; see “How to Use
Parallel Processing” on page 8-12
Population-based
No gradients
Allows integer constraints; see
“Mixed Integer Optimization” on
page 5-32
Automatic start population, or
user-supplied population, or
combination of both
simulannealbnd
Proven to converge to global
optimum for bounded problems
with very slow cooling
schedule.
Stochastic iterates
No gradients
User-supplied start point
Only bound constraints
Explanation of some characteristics:
• Convergence — Solvers can fail to converge to any solution when
started far from a local minimum. When started near a local minimum,
gradient-based solvers converge to a local minimum quickly for smooth
problems. patternsearch provably converges for a wide range of problems,
but the convergence is slower than gradient-based solvers. Both ga and
simulannealbnd can fail to converge in a reasonable amount of time for
some problems, although they are often effective.
• Iterates — Solvers iterate to find solutions. The steps in the iteration are
iterates. Some solvers have deterministic iterates. Others use random
numbers and have stochastic iterates.
• Gradients — Some solvers use estimated or user-supplied derivatives in
calculating the iterates. Other solvers do not use or estimate derivatives,
but use only objective and constraint function values.
• Start points — Most solvers require you to provide a starting point for
the optimization. One reason they require a start point is to obtain the
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Introducing Global Optimization Toolbox Functions
dimension of the decision variables. ga does not require any starting points,
because it takes the dimension of the decision variables as an input. ga can
generate its start population automatically.
Compare the characteristics of Global Optimization Toolbox solvers to
Optimization Toolbox solvers.
Solver
Convergence
Characteristics
fmincon, fminunc,
fseminf, lsqcurvefit,
lsqnonlin
Proven quadratic convergence
to local optima for smooth
problems
Deterministic iterates
fminsearch
No convergence proof —
counterexamples exist.
Gradient-based
User-supplied starting point
Deterministic iterates
No gradients
User-supplied start point
No constraints
fminbnd
Proven convergence to local
optima for smooth problems,
slower than quadratic.
Deterministic iterates
No gradients
User-supplied start point
Only one-dimensional problems
All these Optimization Toolbox solvers:
• Have deterministic iterates
• Start from one user-supplied point
• Search just one basin of attraction
Why Are Some Solvers Objects?
GlobalSearch and MultiStart are objects. What does this mean for you?
• You create a GlobalSearch or MultiStart object before running your
problem.
• You can reuse the object for running multiple problems.
1-26
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Choose a Solver
• GlobalSearch and MultiStart objects are containers for algorithms and
global options. You use these objects to run a local solver multiple times.
The local solver has its own options.
For more information, see the “Object-Oriented Programming” documentation.
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1-27
1
1-28
Introducing Global Optimization Toolbox Functions
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2
Write Files for Optimization
Functions
• “Compute Objective Functions” on page 2-2
• “Write Constraints” on page 2-7
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2
Write Files for Optimization Functions
Compute Objective Functions
In this section...
“Objective (Fitness) Functions” on page 2-2
“Write a Function File” on page 2-2
“Write a Vectorized Function” on page 2-3
“Gradients and Hessians” on page 2-5
“Maximizing vs. Minimizing” on page 2-5
Objective (Fitness) Functions
To use Global Optimization Toolbox functions, first write a file (or an
anonymous function) that computes the function you want to optimize. This is
called an objective function for most solvers, or fitness function for ga. The
function should accept a vector, whose length is the number of independent
variables, and return a scalar. For gamultiobj, the function should return a
row vector of objective function values. For vectorized solvers, the function
should accept a matrix, where each row represents one input vector, and
return a vector of objective function values. This section shows how to write
the file.
Write a Function File
This example shows how to write a file for the function you want to optimize.
Suppose that you want to minimize the function
f ( x) = x12 − 2 x1 x2 + 6 x1 + 4 x22 − 3 x2 .
The file that computes this function must accept a vector x of length 2,
corresponding to the variables x1 and x2, and return a scalar equal to the
value of the function at x.
1 Select New > Script (Ctrl+N) from the MATLAB File menu. A new file
opens in the editor.
2 Enter the following two lines of code:
2-2
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Compute Objective Functions
function z = my_fun(x)
z = x(1)^2 - 2*x(1)*x(2) + 6*x(1) + 4*x(2)^2 - 3*x(2);
3 Save the file in a folder on the MATLAB path.
Check that the file returns the correct value.
my_fun([2 3])
ans =
31
For gamultiobj, suppose you have three objectives. Your objective function
returns a three-element vector consisting of the three objective function
values:
function z = my_fun(x)
z = zeros(1,3); % allocate output
z(1) = x(1)^2 - 2*x(1)*x(2) + 6*x(1) + 4*x(2)^2 - 3*x(2);
z(2) = x(1)*x(2) + cos(3*x(2)/(2+x(1)));
z(3) = tanh(x(1) + x(2));
Write a Vectorized Function
The ga, gamultiobj, and patternsearch solvers optionally compute the
objective functions of a collection of vectors in one function call. This method
can take less time than computing the objective functions of the vectors
serially. This method is called a vectorized function call.
To compute in vectorized fashion:
• Write your objective function to:
-
Accept a matrix with an arbitrary number of rows.
Return the vector of function values of each row.
For gamultiobj, return a matrix, where each row contains the objective
function values of the corresponding input matrix row.
• If you have a nonlinear constraint, be sure to write the constraint in a
vectorized fashion. For details, see “Vectorized Constraints” on page 2-8.
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2-3
2
Write Files for Optimization Functions
• Set the Vectorized option to 'on' with gaoptimset or psoptimset, or
set User function evaluation > Evaluate objective/fitness and
constraint functions to vectorized in the Optimization app. For
patternsearch, also set CompletePoll to 'on'. Be sure to pass the options
structure to the solver.
For example, to write the objective function of “Write a Function File” on page
2-2 in a vectorized fashion,
function z = my_fun(x)
z = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ...
4*x(:,2).^2 - 3*x(:,2);
To use my_fun as a vectorized objective function for patternsearch:
options = psoptimset('CompletePoll','on','Vectorized','on');
[x fval] = patternsearch(@my_fun,[1 1],[],[],[],[],[],[],...
[],options);
To use my_fun as a vectorized objective function for ga:
options = gaoptimset('Vectorized','on');
[x fval] = ga(@my_fun,2,[],[],[],[],[],[],[],options);
For gamultiobj,
function z = my_fun(x)
z = zeros(size(x,1),3); % allocate output
z(:,1) = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ...
4*x(:,2).^2 - 3*x(:,2);
z(:,2) = x(:,1).*x(:,2) + cos(3*x(:,2)./(2+x(:,1)));
z(:,3) = tanh(x(:,1) + x(:,2));
To use my_fun as a vectorized objective function for gamultiobj:
options = gaoptimset('Vectorized','on');
[x fval] = gamultiobj(@my_fun,2,[],[],[],[],[],[],options);
For more information on writing vectorized functions for patternsearch, see
“Vectorize the Objective and Constraint Functions” on page 4-85. For more
information on writing vectorized functions for ga, see “Vectorize the Fitness
Function” on page 5-114.
2-4
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Compute Objective Functions
Gradients and Hessians
If you use GlobalSearch or MultiStart, your objective function can return
derivatives (gradient, Jacobian, or Hessian). For details on how to include
this syntax in your objective function, see “Writing Objective Functions” in
the Optimization Toolbox documentation. Use optimoptions to set options so
that your solver uses the derivative information:
Local Solver = fmincon, fminunc
Condition
Option Setting
Objective function contains gradient
'GradObj' = 'on'
Objective function contains Hessian
'Hessian' = 'on'
Constraint function contains
gradient
'GradConstr' = 'on'
Calculate Hessians of Lagrangian in
an extra function
'Hessian' = 'on', 'HessFcn' =
function handle
For more information about Hessians for fmincon, see “Hessian”.
Local Solver = lsqcurvefit, lsqnonlin
Condition
Option Setting
Objective function contains Jacobian
'Jacobian' = 'on'
Maximizing vs. Minimizing
Global Optimization Toolbox optimization functions minimize the objective or
fitness function. That is, they solve problems of the form
min f ( x).
x
If you want to maximize f(x), minimize –f(x), because the point at which the
minimum of –f(x) occurs is the same as the point at which the maximum
of f(x) occurs.
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2-5
2
Write Files for Optimization Functions
For example, suppose you want to maximize the function
f ( x) = x12 − 2 x1 x2 + 6 x1 + 4 x22 − 3 x2 .
Write your function file to compute
g( x) = − f ( x) = − x12 + 2 x1 x2 − 6 x1 − 4 x22 + 3 x2 ,
and minimize g(x).
2-6
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Write Constraints
Write Constraints
In this section...
“Consult Optimization Toolbox Documentation” on page 2-7
“Set Bounds” on page 2-7
“Ensure ga Options Maintain Feasibility” on page 2-8
“Gradients and Hessians” on page 2-8
“Vectorized Constraints” on page 2-8
Consult Optimization Toolbox Documentation
Many Global Optimization Toolbox functions accept bounds, linear
constraints, or nonlinear constraints. To see how to include these constraints
in your problem, see “Writing Constraints” in the Optimization Toolbox
documentation. Try consulting these pertinent links to sections:
• “Bound Constraints”
• “Linear Inequality Constraints” (linear equality constraints have the same
form)
• “Nonlinear Constraints”
Set Bounds
It is more important to set bounds for global solvers than for local solvers.
Global solvers use bounds in a variety of ways:
• GlobalSearch requires bounds for its scatter-search point generation. If
you do not provide bounds, GlobalSearch bounds each component below
by -9999 and above by 10001. However, these bounds can easily be
inappropriate.
• If you do not provide bounds and do not provide custom start points,
MultiStart bounds each component below by -1000 and above by 1000.
However, these bounds can easily be inappropriate.
• ga uses bounds and linear constraints for its initial population generation.
For unbounded problems, ga uses a default of 0 as the lower bound and 1
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2-7
2
Write Files for Optimization Functions
as the upper bound for each dimension for initial point generation. For
bounded problems, and problems with linear constraints, ga uses the
bounds and constraints to make the initial population.
• simulannealbnd and patternsearch do not require bounds, although they
can use bounds.
Ensure ga Options Maintain Feasibility
The ga solver generally maintains strict feasibility with respect to bounds
and linear constraints. This means that, at every iteration, all members of a
population satisfy the bounds and linear constraints.
However, you can set options that cause this feasibility to fail. For example
if you set MutationFcn to @mutationgaussian or @mutationuniform, the
mutation function does not respect constraints, and your population can
become infeasible. Similarly, some crossover functions can cause infeasible
populations, although the default gacreationlinearfeasible does respect
bounds and linear constraints. Also, ga can have infeasible points when using
custom mutation or crossover functions.
To ensure feasibility, use the default crossover and mutation functions for ga.
Be especially careful that any custom functions maintain feasibility with
respect to bounds and linear constraints.
Gradients and Hessians
If you use GlobalSearch or MultiStart with fmincon, your nonlinear
constraint functions can return derivatives (gradient or Hessian). For details,
see “Gradients and Hessians” on page 2-5.
Vectorized Constraints
The ga and patternsearch solvers optionally compute the nonlinear
constraint functions of a collection of vectors in one function call. This method
can take less time than computing the objective functions of the vectors
serially. This method is called a vectorized function call.
For the solver to compute in a vectorized manner, you must vectorize both
your objective (fitness) function and nonlinear constraint function. For
details, see “Vectorize the Objective and Constraint Functions” on page 4-85.
2-8
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Write Constraints
As an example, suppose your nonlinear constraints for a three-dimensional
problem are
x12 x22 x32
+
+
≤6
4
9 25
x3 ≥ cosh ( x1 + x2 )
x1 x2 x3 = 2.
The following code gives these nonlinear constraints in a vectorized fashion,
assuming that the rows of your input matrix x are your population or input
vectors:
function [c ceq] = nlinconst(x)
c(:,1) = x(:,1).^2/4 + x(:,2).^2/9 + x(:,3).^2/25 - 6;
c(:,2) = cosh(x(:,1) + x(:,2)) - x(:,3);
ceq = x(:,1).*x(:,2).*x(:,3) - 2;
For example, minimize the vectorized quadratic function
function y = vfun(x)
y = -x(:,1).^2 - x(:,2).^2 - x(:,3).^2;
over the region with constraints nlinconst using patternsearch:
options = psoptimset('CompletePoll','on','Vectorized','on');
[x fval] = patternsearch(@vfun,[1,1,2],[],[],[],[],[],[],...
@nlinconst,options)
Optimization terminated: mesh size less than options.TolMesh
and constraint violation is less than options.TolCon.
x =
0.2191
0.7500
12.1712
fval =
-148.7480
Using ga:
options = gaoptimset('Vectorized','on');
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2-9
2
Write Files for Optimization Functions
[x fval] = ga(@vfun,3,[],[],[],[],[],[],@nlinconst,options)
Optimization terminated: maximum number of generations exceeded.
x =
-1.4098
-0.1216
11.6664
fval =
-138.1066
For this problem patternsearch computes the solution far more quickly
and accurately.
2-10
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3
Using GlobalSearch and
MultiStart
• “Problems That GlobalSearch and MultiStart Can Solve” on page 3-3
• “Optimization Workflow” on page 3-4
• “Inputs for Problem Structure” on page 3-6
• “Create Problem Structure” on page 3-7
• “Create Solver Object” on page 3-17
• “Set Start Points for MultiStart” on page 3-20
• “Run the Solver” on page 3-24
• “Single Solution” on page 3-28
• “Multiple Solutions” on page 3-30
• “Iterative Display” on page 3-35
• “Global Output Structures” on page 3-39
• “Visualize the Basins of Attraction” on page 3-40
• “Output Functions for GlobalSearch and MultiStart” on page 3-43
• “Plot Functions for GlobalSearch and MultiStart” on page 3-47
• “How GlobalSearch and MultiStart Work” on page 3-52
• “Can You Certify a Solution Is Global?” on page 3-62
• “Refine Start Points” on page 3-66
• “Change Options” on page 3-75
• “Reproduce Results” on page 3-79
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3
Using GlobalSearch and MultiStart
• “Find Global or Multiple Local Minima” on page 3-83
• “Optimize Using Only Feasible Start Points” on page 3-91
• “MultiStart Using lsqcurvefit or lsqnonlin” on page 3-96
• “Parallel MultiStart” on page 3-100
• “Isolated Global Minimum” on page 3-104
3-2
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Problems That GlobalSearch and MultiStart Can Solve
Problems That GlobalSearch and MultiStart Can Solve
The GlobalSearch and MultiStart solvers apply to problems with smooth
objective and constraint functions. The solvers search for a global minimum,
or for a set of local minima. For more information on which solver to use,
see “Choose a Solver” on page 1-19.
GlobalSearch and MultiStart work by starting a local solver, such as
fmincon, from a variety of start points. Generally the start points are random.
However, for MultiStart you can provide a set of start points. For more
information, see “How GlobalSearch and MultiStart Work” on page 3-52.
To find out how to use these solvers, see “Optimization Workflow” on page 3-4.
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3-3
3
Using GlobalSearch and MultiStart
Optimization Workflow
To find a global or multiple local solutions:
1 “Create Problem Structure” on page 3-7
2 “Create Solver Object” on page 3-17
3 (Optional, MultiStart only) “Set Start Points for MultiStart” on page 3-20
4 “Run the Solver” on page 3-24
The following figure illustrates these steps.
3-4
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Optimization Workflow
Information
you have
Local Solver
X0
Objective Constraints
Global options
Local Options
Command
to use
Resulting
Object or
Structure
createOptimProblem
Problem Structure
Optional,
MultiStart only
Start Points
GlobalSearch
or
MultiStart
RandomStartPointSet
or
CustomStartPointSet
Solver Object
Start Points Object
run
Results
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3-5
3
Using GlobalSearch and MultiStart
Inputs for Problem Structure
A problem structure defines a local optimization problem using a local solver,
such as fmincon. Therefore, most of the documentation on choosing a solver
or preparing the inputs is in the Optimization Toolbox documentation.
Required Inputs
Input
More Information
Local solver
“Optimization Decision Table” in the Optimization
Toolbox documentation.
Objective function
“Compute Objective Functions” on page 2-2
Start point x0
Gives the dimension of points for the objective
function.
Optional Inputs
3-6
Input
More Information
Constraint functions
“Write Constraints” on page 2-7
Local options
structure
“Set Options”
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Create Problem Structure
Create Problem Structure
In this section...
“About Problem Structures” on page 3-7
“Using the createOptimProblem Function” on page 3-7
“Exporting from the Optimization app” on page 3-10
About Problem Structures
To use the GlobalSearch or MultiStart solvers, you must first create a
problem structure. There are two ways to create a problem structure: using
the createOptimProblem function and exporting from the Optimization app.
For information on creating inputs for the problem structure, see “Inputs
for Problem Structure” on page 3-6.
Using the createOptimProblem Function
Follow these steps to create a problem structure using the
createOptimProblem function.
1 Define your objective function as a file or anonymous function. For
details, see “Compute Objective Functions” on page 2-2. If your solver is
lsqcurvefit or lsqnonlin, ensure the objective function returns a vector,
not scalar.
2 If relevant, create your constraints, such as bounds and nonlinear
constraint functions. For details, see “Write Constraints” on page 2-7.
3 Create a start point. For example, to create a three-dimensional random
start point xstart:
xstart = randn(3,1);
4 (Optional) Create an options structure using optimoptions. For example,
options = optimoptions(@fmincon,'Algorithm','interior-point');
5 Enter
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3-7
3
Using GlobalSearch and MultiStart
problem = createOptimProblem(solver,
where solver is the name of your local solver:
• For GlobalSearch: 'fmincon'
• For MultiStart the choices are:
– 'fmincon'
– 'fminunc'
– 'lsqcurvefit'
– 'lsqnonlin'
For help choosing, see “Optimization Decision Table”.
6 Set an initial point using the 'x0' parameter. If your initial point is
xstart, and your solver is fmincon, your entry is now
problem = createOptimProblem('fmincon','x0',xstart,
7 Include the function handle for your objective function in objective:
problem = createOptimProblem('fmincon','x0',xstart, ...
'objective',@objfun,
8 Set bounds and other constraints as applicable.
3-8
Constraint
Name
lower bounds
'lb'
upper bounds
'ub'
matrix Aineq for linear inequalities
Aineq x ≤ bineq
'Aineq'
vector bineq for linear inequalities
Aineq x ≤ bineq
'bineq'
matrix Aeq for linear equalities Aeq x = beq
'Aeq'
vector beq for linear equalities Aeq x = beq
'beq'
nonlinear constraint function
'nonlcon'
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Create Problem Structure
9 If using the lsqcurvefit local solver, include vectors of input data and
response data, named 'xdata' and 'ydata' respectively.
10 Best practice: validate the problem structure by running your solver on the
structure. For example, if your local solver is fmincon:
[x fval eflag output] = fmincon(problem);
Note Specify fmincon as the solver for GlobalSearch, even if you have
no constraints. However, you cannot run fmincon on a problem without
constraints. Add an artificial constraint to the problem to validate the
structure:
problem.lb = -Inf(size(x0));
Example: Creating a Problem Structure with
createOptimProblem
This example minimizes the function from “Run the Solver” on page 3-24,
subject to the constraint x1 + 2x2 ≥ 4. The objective is
sixmin = 4x2 – 2.1x4 + x6/3 + xy – 4y2 + 4y4.
Use the interior-point algorithm of fmincon, and set the start point to
[2;3].
1 Write a function handle for the objective function.
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
2 Write the linear constraint matrices. Change the constraint to “less than”
form:
A = [-1,-2];
b = -4;
3 Create the local options structure to use the interior-point algorithm:
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3-9
3
Using GlobalSearch and MultiStart
opts = optimoptions(@fmincon,'Algorithm','interior-point');
4 Create the problem structure with createOptimProblem:
problem = createOptimProblem('fmincon', ...
'x0',[2;3],'objective',sixmin, ...
'Aineq',A,'bineq',b,'options',opts)
5 The resulting structure:
problem =
objective:
x0:
Aineq:
bineq:
Aeq:
beq:
lb:
ub:
nonlcon:
solver:
options:
@(x)(4*x(1)^2-2.1*x(1)^4+x(1)^6/3+x(1)*x(2)-4*x(2)^2+4*x(2)^
[2x1 double]
[-1 -2]
-4
[]
[]
[]
[]
[]
'fmincon'
[1x1 optim.options.Fmincon]
6 Best practice: validate the problem structure by running your solver on the
structure:
[x fval eflag output] = fmincon(problem);
Note Specify fmincon as the solver for GlobalSearch, even if you have
no constraints. However, you cannot run fmincon on a problem without
constraints. Add an artificial constraint to the problem to validate the
structure:
problem.lb = -Inf(size(x0));
Exporting from the Optimization app
Follow these steps to create a problem structure using the Optimization app.
3-10
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Create Problem Structure
1 Define your objective function as a file or anonymous function. For
details, see “Compute Objective Functions” on page 2-2. If your solver is
lsqcurvefit or lsqnonlin, ensure the objective function returns a vector,
not scalar.
2 If relevant, create nonlinear constraint functions. For details, see
“Nonlinear Constraints”.
3 Create a start point. For example, to create a three-dimensional random
start point xstart:
xstart = randn(3,1);
4 Open the Optimization app by entering optimtool at the command line, or
by choosing the Optimization app from the Apps tab.
5 Choose the local Solver.
• For GlobalSearch: fmincon (default).
• For MultiStart:
– fmincon (default)
– fminunc
– lsqcurvefit
– lsqnonlin
For help choosing, see “Optimization Decision Table”.
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3-11
3
Using GlobalSearch and MultiStart
6 Choose an appropriate Algorithm. For help choosing, see “Choosing the
Algorithm”.
7 Set an initial point (Start point).
8 Include the function handle for your objective function in Objective
function, and, if applicable, include your Nonlinear constraint
function. For example,
9 Set bounds, linear constraints, or local Options. For details on constraints,
see “Writing Constraints”.
10 Best practice: run the problem to verify the setup.
3-12
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Create Problem Structure
Note You must specify fmincon as the solver for GlobalSearch, even if
you have no constraints. However, you cannot run fmincon on a problem
without constraints. Add an artificial constraint to the problem to verify
the setup.
11 Choose File > Export to Workspace and select Export problem and
options to a MATLAB structure named
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3-13
3
Using GlobalSearch and MultiStart
Example: Creating a Problem Structure with the Optimization
App
This example minimizes the function from “Run the Solver” on page 3-24,
subject to the constraint x1 + 2x2 ≥ 4. The objective is
sixmin = 4x2 – 2.1x4 + x6/3 + xy – 4y2 + 4y4.
Use the interior-point algorithm of fmincon, and set the start point to
[2;3].
1 Write a function handle for the objective function.
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
2 Write the linear constraint matrices. Change the constraint to “less than”
form:
A = [-1,-2];
b = -4;
3 Launch the Optimization app by entering optimtool at the MATLAB
command line.
4 Set the solver, algorithm, objective, start point, and constraints.
3-14
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Create Problem Structure
5 Best practice: run the problem to verify the setup.
The problem runs successfully.
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3-15
3
Using GlobalSearch and MultiStart
6 Choose File > Export to Workspace and select Export problem and
options to a MATLAB structure named
3-16
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Create Solver Object
Create Solver Object
In this section...
“What Is a Solver Object?” on page 3-17
“Properties (Global Options) of Solver Objects” on page 3-17
“Creating a Nondefault GlobalSearch Object” on page 3-19
“Creating a Nondefault MultiStart Object” on page 3-19
What Is a Solver Object?
A solver object contains your preferences for the global portion of the
optimization.
You do not need to set any preferences. Create a GlobalSearch object named
gs with default settings as follows:
gs = GlobalSearch;
Similarly, create a MultiStart object named ms with default settings as
follows:
ms = MultiStart;
Properties (Global Options) of Solver Objects
Global options are properties of a GlobalSearch or MultiStart object.
Properties for both GlobalSearch and MultiStart
Property Name
Meaning
Display
Detail level of iterative display. Set to 'off' for no
display, 'final' (default) for a report at the end of the
run, or 'iter' for reports as the solver progresses.
For more information and examples, see “Iterative
Display” on page 3-35.
TolFun
Solvers consider objective function values within
TolFun of each other to be identical (not distinct).
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3-17
3
Using GlobalSearch and MultiStart
Properties for both GlobalSearch and MultiStart (Continued)
Property Name
Meaning
Default: 1e-6. Solvers group solutions when the
solutions satisfy both TolFun and TolX tolerances.
TolX
Solvers consider solutions within TolX distance of each
other to be identical (not distinct). Default: 1e-6.
Solvers group solutions when the solutions satisfy both
TolFun and TolX tolerances.
MaxTime
Solvers halt if the run exceeds MaxTime seconds, as
measured by a clock (not processor seconds). Default:
Inf
StartPointsToRun Choose whether to run 'all' (default) start points,
only those points that satisfy 'bounds', or only those
points that are feasible with respect to bounds and
inequality constraints with 'bounds-ineqs'. For an
example, see “Optimize Using Only Feasible Start
Points” on page 3-91.
OutputFcns
Functions to run after each local solver run. See
“Output Functions for GlobalSearch and MultiStart”
on page 3-43. Default: []
PlotFcns
Plot functions to run after each local solver run. See
“Plot Functions for GlobalSearch and MultiStart” on
page 3-47. Default: []
Properties for GlobalSearch
Property Name
Meaning
NumTrialPoints
Number of trial points to examine.
Default: 1000
BasinRadiusFactor
See “Properties” on page 10-38 for detailed
descriptions of these properties.
DistanceThresholdFactor
MaxWaitCycle
NumStageOnePoints
3-18
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Create Solver Object
Properties for GlobalSearch (Continued)
Property Name
Meaning
PenaltyThresholdFactor
Properties for MultiStart
Property Name
Meaning
UseParallel
When true, MultiStart attempts to distribute start
points to multiple processors for the local solver.
Disable by setting to false (default). For details, see
“How to Use Parallel Processing” on page 8-12. For
an example, see “Parallel MultiStart” on page 3-100.
Creating a Nondefault GlobalSearch Object
Suppose you want to solve a problem and:
• Consider local solutions identical if they are within 0.01 of each other and
the function values are within the default TolFun tolerance.
• Spend no more than 2000 seconds on the computation.
To solve the problem, create a GlobalSearch object gs as follows:
gs = GlobalSearch('TolX',0.01,'MaxTime',2000);
Creating a Nondefault MultiStart Object
Suppose you want to solve a problem such that:
• You consider local solutions identical if they are within 0.01 of each other
and the function values are within the default TolFun tolerance.
• You spend no more than 2000 seconds on the computation.
To solve the problem, create a MultiStart object ms as follows:
ms = MultiStart('TolX',0.01,'MaxTime',2000);
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3-19
3
Using GlobalSearch and MultiStart
Set Start Points for MultiStart
In this section...
“Four Ways to Set Start Points” on page 3-20
“Positive Integer for Start Points” on page 3-20
“RandomStartPointSet Object for Start Points” on page 3-21
“CustomStartPointSet Object for Start Points” on page 3-21
“Cell Array of Objects for Start Points” on page 3-22
Four Ways to Set Start Points
There are four ways you tell MultiStart which start points to use for the
local solver:
• Pass a positive integer k. MultiStart generates k - 1 start points
as if using a RandomStartPointSet object and the problem structure.
MultiStart also uses the x0 start point from the problem structure, for a
total of k start points.
• Pass a RandomStartPointSet object.
• Pass a CustomStartPointSet object.
• Pass a cell array of RandomStartPointSet and CustomStartPointSet
objects. Pass a cell array if you have some specific points you want to run,
but also want MultiStart to use other random start points.
Note You can control whether MultiStart uses all start points, or only
those points that satisfy bounds or other inequality constraints. For more
information, see “Filter Start Points (Optional)” on page 3-59.
Positive Integer for Start Points
The syntax for running MultiStart for k start points is
[xmin,fmin,flag,outpt,allmins] = run(ms,problem,k);
3-20
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Set Start Points for MultiStart
The positive integer k specifies the number of start points MultiStart uses.
MultiStart generates random start points using the dimension of the problem
and bounds from the problem structure. MultiStart generates k - 1 random
start points, and also uses the x0 start point from the problem structure.
RandomStartPointSet Object for Start Points
Create a RandomStartPointSet object as follows:
stpoints = RandomStartPointSet;
By default a RandomStartPointSet object generates 10 start points. Control
the number of start points with the NumStartPoints property. For example,
to generate 40 start points:
stpoints = RandomStartPointSet('NumStartPoints',40);
You can set an ArtificialBound for a RandomStartPointSet. This
ArtificialBound works in conjunction with the bounds from the problem
structure:
• If a component has no bounds, RandomStartPointSet uses a lower bound
of -ArtificialBound, and an upper bound of ArtificialBound.
• If a component has a lower bound lb but no upper bound,
RandomStartPointSet uses an upper bound of lb + 2*ArtificialBound.
• Similarly, if a component has an upper bound ub but no lower bound,
RandomStartPointSet uses a lower bound of ub - 2*ArtificialBound.
For example, to generate 100 start points with an ArtificialBound of 50:
stpoints = RandomStartPointSet('NumStartPoints',100, ...
'ArtificialBound',50);
A RandomStartPointSet object generates start points with the same
dimension as the x0 point in the problem structure; see list.
CustomStartPointSet Object for Start Points
To use a specific set of starting points, package them in a
CustomStartPointSet as follows:
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3-21
3
Using GlobalSearch and MultiStart
1 Place the starting points in a matrix. Each row of the matrix represents one
starting point. MultiStart runs all the rows of the matrix, subject to filtering
with the StartPointsToRun property. For more information, see “MultiStart
Algorithm” on page 3-59.
2 Create a CustomStartPointSet object from the matrix:
tpoints = CustomStartPointSet(ptmatrix);
For example, create a set of 40 five-dimensional points, with each component
of a point equal to 10 plus an exponentially distributed variable with mean 25:
pts = -25*log(rand(40,5)) + 10;
tpoints = CustomStartPointSet(pts);
To get the original matrix of points from a CustomStartPointSet object,
use the list method:
pts = list(tpoints); % Assumes tpoints is a CustomStartPointSet
A CustomStartPointSet has two properties: DimStartPoints
and NumStartPoints. You can use these properties to query a
CustomStartPointSet object. For example, the tpoints object in the example
has the following properties:
tpoints.DimStartPoints
ans =
5
tpoints.NumStartPoints
ans =
40
Cell Array of Objects for Start Points
To use a specific set of starting points along with some randomly generated
points, pass a cell array of RandomStartPointSet or CustomStartPointSet
objects.
For example, to use both the 40 specific five-dimensional points of
“CustomStartPointSet Object for Start Points” on page 3-21 and 40 additional
five-dimensional points from RandomStartPointSet:
3-22
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Set Start Points for MultiStart
pts = -25*log(rand(40,5)) + 10;
tpoints = CustomStartPointSet(pts);
rpts = RandomStartPointSet('NumStartPoints',40);
allpts = {tpoints,rpts};
Run MultiStart with the allpts cell array:
% Assume ms and problem exist
[xmin,fmin,flag,outpt,allmins] = run(ms,problem,allpts);
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3-23
3
Using GlobalSearch and MultiStart
Run the Solver
In this section...
“Optimize by Calling run” on page 3-24
“Example of Run with GlobalSearch” on page 3-25
“Example of Run with MultiStart” on page 3-26
Optimize by Calling run
Running a solver is nearly identical for GlobalSearch and MultiStart. The
only difference in syntax is MultiStart takes an additional input describing
the start points.
For example, suppose you want to find several local minima of the sixmin
function
sixmin = 4x2 – 2.1x4 + x6/3 + xy – 4y2 + 4y4.
3-24
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Run the Solver
This function is also called the six-hump camel back function [3]. All the local
minima lie in the region –3 ≤ x,y ≤ 3.
Example of Run with GlobalSearch
To find several local minima of the sixmin function using GlobalSearch,
enter:
% % Set the random stream to get exactly the same output
% rng(14,'twister')
gs = GlobalSearch;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3],...
'options',opts);
[xming,fming,flagg,outptg,manyminsg] = run(gs,problem);
The output of the run (which varies, based on the random seed):
xming,fming,flagg,outptg,manyminsg
xming =
0.0898
-0.7127
fming =
-1.0316
flagg =
1
outptg =
funcCount:
localSolverTotal:
localSolverSuccess:
localSolverIncomplete:
localSolverNoSolution:
message:
2131
3
3
0
0
'GlobalSearch stopped because it analyzed all ..
manyminsg =
1x2 GlobalOptimSolution array with properties:
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3-25
3
Using GlobalSearch and MultiStart
X
Fval
Exitflag
Output
X0
Example of Run with MultiStart
To find several local minima of the sixmin function using 50 runs of fmincon
with MultiStart, enter:
% % Set the random stream to get exactly the same output
% rng(14,'twister')
ms = MultiStart;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3],...
'options',opts);
[xminm,fminm,flagm,outptm,manyminsm] = run(ms,problem,50);
The output of the run (which varies based on the random seed):
xminm,fminm,flagm,outptm,manyminsm
xminm =
0.0898
-0.7127
fminm =
-1.0316
flagm =
1
outptm =
funcCount:
localSolverTotal:
localSolverSuccess:
localSolverIncomplete:
3-26
2034
50
50
0
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Run the Solver
localSolverNoSolution: 0
message: 'MultiStart completed the runs from all start ..
manyminsm =
1x6 GlobalOptimSolution array with properties:
X
Fval
Exitflag
Output
X0
In this case, MultiStart located all six local minima, while GlobalSearch
located two. For pictures of the MultiStart solutions, see “Visualize the
Basins of Attraction” on page 3-40.
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3-27
3
Using GlobalSearch and MultiStart
Single Solution
You obtain the single best solution found during the run by calling run with
the syntax
[x fval eflag output] = run(...);
• x is the location of the local minimum with smallest objective function
value.
• fval is the objective function value evaluated at x.
• eflag is an exit flag for the global solver. Values:
Global Solver Exit Flags
2
At least one local minimum found. Some runs of the local solver
converged (had positive exit flag).
1
At least one local minimum found. All runs of the local solver
converged (had positive exit flag).
0
No local minimum found. Local solver called at least once, and
at least one local solver exceeded the MaxIter or MaxFunEvals
tolerances.
-1
Solver stopped by output function or plot function.
-2
No feasible local minimum found.
-5
MaxTime limit exceeded.
-8
No solution found. All runs had local solver exit flag -1 or
smaller.
-10
Failures encountered in user-provided functions.
• output is a structure with details about the multiple runs of the local
solver. For more information, see “Global Output Structures” on page 3-39.
The list of outputs is for the case eflag > 0. If eflag <= 0, then x is the
following:
3-28
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Single Solution
• If some local solutions are feasible, x represents the location of the lowest
objective function value. “Feasible” means the constraint violations are
smaller than problem.options.TolCon.
• If no solutions are feasible, x is the solution with lowest infeasibility.
• If no solutions exist, x, fval, and output are empty ([]).
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3-29
3
Using GlobalSearch and MultiStart
Multiple Solutions
In this section...
“About Multiple Solutions” on page 3-30
“Change the Definition of Distinct Solutions” on page 3-32
About Multiple Solutions
You obtain multiple solutions in an object by calling run with the syntax
[x fval eflag output manymins] = run(...);
manymins is a vector of solution objects; see GlobalOptimSolution. The
manymins vector is in order of objective function value, from lowest (best) to
highest (worst). Each solution object contains the following properties (fields):
• X — a local minimum
• Fval — the value of the objective function at X
• Exitflag — the exit flag for the local solver (described in the local solver
function reference page: fmincon, fminunc, lsqcurvefit, or lsqnonlin)
• Output — an output structure for the local solver (described in the local
solver function reference page: fmincon, fminunc, lsqcurvefit, or
lsqnonlin)
• X0 — a cell array of start points that led to the solution point X
There are several ways to examine the vector of solution objects:
• In the MATLAB Workspace Browser. Double-click the solution object, and
then double-click the resulting display in the Variables editor.
3-30
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Multiple Solutions
• Using dot notation. GlobalOptimSolution properties are capitalized. Use
proper capitalization to access the properties.
For example, to find the vector of function values, enter:
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3-31
3
Using GlobalSearch and MultiStart
fcnvals = [manymins.Fval]
fcnvals =
-1.0316
-0.2155
0
To get a cell array of all the start points that led to the lowest function
value (the first element of manymins), enter:
smallX0 = manymins(1).X0
• Plot some field values. For example, to see the range of resulting Fval,
enter:
hist([manymins.Fval])
This results in a histogram of the computed function values. (The figure
shows a histogram from a different example than the previous few figures.)
Change the Definition of Distinct Solutions
You might find out, after obtaining multiple local solutions, that your
tolerances were not appropriate. You can have many more local solutions
3-32
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Multiple Solutions
than you want, spaced too closely together. Or you can have fewer solutions
than you want, with GlobalSearch or MultiStart clumping together too
many solutions.
To deal with this situation, run the solver again with different tolerances. The
TolX and TolFun tolerances determine how the solvers group their outputs
into the GlobalOptimSolution vector. These tolerances are properties of the
GlobalSearch or MultiStart object.
For example, suppose you want to use the active-set algorithm in fmincon
to solve the problem in “Example of Run with MultiStart” on page 3-26.
Further suppose that you want to have tolerances of 0.01 for both TolX and
TolFun. The run method groups local solutions whose objective function
values are within TolFun of each other, and which are also less than TolX
apart from each other. To obtain the solution:
% % Set the random stream to get exactly the same output
% rng(14,'twister')
ms = MultiStart('TolFun',0.01,'TolX',0.01);
opts = optimoptions(@fmincon,'Algorithm','active-set');
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3],...
'options',opts);
[xminm,fminm,flagm,outptm,someminsm] = run(ms,problem,50);
MultiStart completed the runs from all start points.
All 50 local solver runs converged with a
positive local solver exit flag.
someminsm
someminsm =
1x5 GlobalOptimSolution
Properties:
X
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3-33
3
Using GlobalSearch and MultiStart
Fval
Exitflag
Output
X0
In this case, MultiStart generated five distinct solutions. Here “distinct”
means that the solutions are more than 0.01 apart in either objective function
value or location.
3-34
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Iterative Display
Iterative Display
In this section...
“Types of Iterative Display” on page 3-35
“Examine Types of Iterative Display” on page 3-36
Types of Iterative Display
Iterative display gives you information about the progress of solvers during
their runs.
There are two types of iterative display:
• Global solver display
• Local solver display
Both types appear at the command line, depending on global and local options.
Obtain local solver iterative display by setting the Display option in
the problem.options structure to 'iter' or 'iter-detailed' with
optimoptions. For more information, see “Iterative Display” in the
Optimization Toolbox documentation.
Obtain global solver iterative display by setting the Display property in the
GlobalSearch or MultiStart object to 'iter'.
Global solvers set the default Display option of the local solver to 'off',
unless the problem structure has a value for this option. Global solvers do
not override any setting you make for local options.
Note Setting the local solver Display option to anything other than 'off'
can produce a great deal of output. The default Display option created by
optimoptions(@solver) is 'final'.
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3-35
3
Using GlobalSearch and MultiStart
Examine Types of Iterative Display
Run the example described in “Run the Solver” on page 3-24 using
GlobalSearch with GlobalSearch iterative display:
% % Set the random stream to get exactly the same output
% rng(14,'twister')
gs = GlobalSearch('Display','iter');
opts = optimoptions(@fmincon,'Algorithm','interior-point');
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3],...
'options',opts);
[xming,fming,flagg,outptg,manyminsg] = run(gs,problem);
Best
Current
Threshold
Local
Local
Analyzed
Num Pts
F-count
f(x)
Penalty
Penalty
f(x)
exitflag
Procedure
0
34
-1.032
-1.032
1
Initial Point
200
1291
-1.032
-0.2155
1
300
1393
-1.032
248.7
-0.2137
400
1493
-1.032
278
1.134
446
1577
-1.032
1.6
2.073
500
1631
-1.032
9.055
0.3214
Stage 2 Search
600
1731
-1.032
-0.7299
-0.7686
Stage 2 Search
700
1831
-1.032
0.3191
-0.7431
Stage 2 Search
800
1931
-1.032
296.4
0.4577
Stage 2 Search
900
2031
-1.032
10.68
0.5116
Stage 2 Search
1000
2131
-1.032
-0.9207
-0.9254
Stage 2 Search
Stage 1 Local
Stage 2 Search
Stage 2 Search
-0.2155
1
Stage 2 Local
GlobalSearch stopped because it analyzed all the trial points.
All 3 local solver runs converged with a positive local solver exit flag.
Run the same example without GlobalSearch iterative display, but with
fmincon iterative display:
gs.Display = 'final';
problem.options.Display = 'iter';
[xming,fming,flagg,outptg,manyminsg] = run(gs,problem);
3-36
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Iterative Display
First-order
Norm of
f(x)
Feasibility
optimality
step
3
4.823333e+001
0.000e+000
1.088e+002
1
7
2.020476e+000
0.000e+000
2.176e+000
2
10
6.525252e-001
0.000e+000
1.937e+000
1.886e+000
3
13
-8.776121e-001
0.000e+000
9.076e-001
8.539e-001
4
16
-9.121907e-001
0.000e+000
9.076e-001
1.655e-001
5
19
-1.009367e+000
0.000e+000
7.326e-001
8.558e-002
6
22
-1.030423e+000
0.000e+000
2.172e-001
6.670e-002
7
25
-1.031578e+000
0.000e+000
4.278e-002
1.444e-002
8
28
-1.031628e+000
0.000e+000
8.777e-003
2.306e-003
9
31
-1.031628e+000
0.000e+000
8.845e-005
2.750e-004
10
34
-1.031628e+000
0.000e+000
8.744e-007
1.354e-006
Iter F-count
0
2.488e+000
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the selected value of the function tolerance,
and constraints were satisfied to within the selected value of the constraint tolerance.
<stopping criteria details>
Iter F-count
0
3
First-order
Norm of
f(x)
Feasibility
optimality
step
-1.980435e-02
0.000e+00
1.996e+00
0.000e+00
8.742e-07
... MANY ITERATIONS DELETED ...
8
33
-1.031628e+00
2.287e-07
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the selected value of the function tolerance,
and constraints were satisfied to within the selected value of the constraint tolerance.
<stopping criteria details>
GlobalSearch stopped because it analyzed all the trial points.
All 4 local solver runs converged with a positive local solver exit flag.
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3-37
3
Using GlobalSearch and MultiStart
Setting GlobalSearch iterative display, as well as fmincon iterative display,
yields both displays intermingled.
For an example of iterative display in a parallel environment, see “Parallel
MultiStart” on page 3-100.
3-38
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Global Output Structures
Global Output Structures
run can produce two types of output structures:
• A global output structure. This structure contains information about the
overall run from multiple starting points. Details follow.
• Local solver output structures. The vector of GlobalOptimSolution objects
contains one such structure in each element of the vector. For a description
of this structure, see “Output Structures” in the Optimization Toolbox
documentation, or the function reference pages for the local solvers:
fmincon, fminunc, lsqcurvefit, or lsqnonlin.
Global Output Structure
Field
Meaning
funcCount
Total number of calls to user-supplied functions (objective or
nonlinear constraint)
localSolverTotal
Number of local solver runs started
localSolverSuccess
Number of local solver runs that finished with a positive exit flag
localSolverIncomplete
Number of local solver runs that finished with a 0 exit flag
localSolverNoSolution
Number of local solver runs that finished with a negative exit
flag
message
GlobalSearch or MultiStart exit message
A positive exit flag from a local solver generally indicates a successful run. A
negative exit flag indicates a failure. A 0 exit flag indicates that the solver
stopped by exceeding the iteration or function evaluation limit. For more
information, see “Exit Flags and Exit Messages” or “Tolerances and Stopping
Criteria” in the Optimization Toolbox documentation.
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3-39
3
Using GlobalSearch and MultiStart
Visualize the Basins of Attraction
Which start points lead to which basin? For a steepest descent solver, nearby
points generally lead to the same basin; see “Basins of Attraction” on page
1-14. However, for Optimization Toolbox solvers, basins are more complicated.
Plot the MultiStart start points from the example, “Example of Run with
MultiStart” on page 3-26, color-coded with the basin where they end.
% rng(14,'twister')
% Uncomment the previous line to get the same output
ms = MultiStart;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3],...
'options',opts);
[xminm,fminm,flagm,outptm,manyminsm] = run(ms,problem,50);
possColors = 'kbgcrm';
hold on
for i = 1:size(manyminsm,2)
% Color of this line
cIdx = rem(i-1, length(possColors)) + 1;
color = possColors(cIdx);
% Plot start points
u = manyminsm(i).X0;
x0ThisMin = reshape([u{:}], 2, length(u));
plot(x0ThisMin(1, :), x0ThisMin(2, :), '.', ...
'Color',color,'MarkerSize',25);
% Plot the basin with color i
plot(manyminsm(i).X(1), manyminsm(i).X(2), '*', ...
'Color', color, 'MarkerSize',25);
end % basin center marked with a *, start points with dots
hold off
3-40
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Visualize the Basins of Attraction
The figure shows the centers of the basins by colored * symbols. Start points
with the same color as the * symbol converge to the center of the * symbol.
Start points do not always converge to the closest basin. For example, the red
points are closer to the cyan basin center than to the red basin center. Also,
many black and blue start points are closer to the opposite basin centers.
The magenta and red basins are shallow, as you can see in the following
contour plot.
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3-41
3
3-42
Using GlobalSearch and MultiStart
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Output Functions for GlobalSearch and MultiStart
Output Functions for GlobalSearch and MultiStart
In this section...
“What Are Output Functions?” on page 3-43
“GlobalSearch Output Function” on page 3-43
“No Parallel Output Functions” on page 3-46
What Are Output Functions?
Output functions allow you to examine intermediate results in an
optimization. Additionally, they allow you to halt a solver programmatically.
There are two types of output functions, like the two types of output
structures:
• Global output functions run after each local solver run. They also run when
the global solver starts and ends.
• Local output functions run after each iteration of a local solver. See
“Output Functions” in the Optimization Toolbox documentation.
To use global output functions:
• Write output functions using the syntax described in “OutputFcns” on
page 9-3.
• Set the OutputFcns property of your GlobalSearch or MultiStart solver
to the function handle of your output function. You can use multiple output
functions by setting the OutputFcns property to a cell array of function
handles.
GlobalSearch Output Function
This output function stops GlobalSearch after it finds five distinct local
minima with positive exit flags, or after it finds a local minimum value less
than 0.5. The output function uses a persistent local variable, foundLocal, to
store the local results. foundLocal enables the output function to determine
whether a local solution is distinct from others, to within a tolerance of 1e-4.
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3-43
3
Using GlobalSearch and MultiStart
To store local results using nested functions instead of persistent variables,
see “Example of a Nested Output Function” in the MATLAB Mathematics
documentation.
1 Write the output function using the syntax described in “OutputFcns” on
page 9-3.
function stop = StopAfterFive(optimValues, state)
persistent foundLocal
stop = false;
switch state
case 'init'
foundLocal = []; % initialized as empty
case 'iter'
newf = optimValues.localsolution.Fval;
eflag = optimValues.localsolution.Exitflag;
% Now check if the exit flag is positive and
% the new value differs from all others by at least 1e-4
% If so, add the new value to the newf list
if eflag > 0 && all(abs(newf - foundLocal) > 1e-4)
foundLocal = [foundLocal;newf];
% Now check if the latest value added to foundLocal
% is less than 1/2
% Also check if there are 5 local minima in foundLocal
% If so, then stop
if foundLocal(end) < 0.5 || length(foundLocal) >= 5
stop = true;
end
end
end
2 Save StopAfterFive.m as a file in a folder on your MATLAB path.
3 Write the objective function and create an optimization problem structure as
in “Find Global or Multiple Local Minima” on page 3-83.
function f = sawtoothxy(x,y)
[t r] = cart2pol(x,y); % change to polar coordinates
h = cos(2*t - 1/2)/2 + cos(t) + 2;
g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ...
.*r.^2./(r+1);
3-44
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Output Functions for GlobalSearch and MultiStart
f = g.*h;
end
4 Save sawtoothxy.m as a file in a folder on your MATLAB path.
5 At the command line, create the problem structure:
problem = createOptimProblem('fmincon',...
'objective',@(x)sawtoothxy(x(1),x(2)),...
'x0',[100,-50],'options',...
optimoptions(@fmincon,'Algorithm','sqp'));
6 Create a GlobalSearch object with @StopAfterFive as the output function,
and set the iterative display property to 'iter'.
gs = GlobalSearch('OutputFcns',@StopAfterFive,'Display','iter');
7 (Optional) To get the same answer as this example, set the default random
number stream.
rng('default')
8 Run the problem.
[x fval] = run(gs,problem)
Best
Current
Threshold
Local
Local
Analyzed
Num Pts
F-count
f(x)
Penalty
Penalty
f(x)
exitflag
Procedure
0
200
555.7
555.7
0
Initial Point
200
1479
1.547e-15
1.547e-15
1
Stage 1 Local
GlobalSearch stopped by the output or plot function.
1 out of 2 local solver runs converged with a positive local solver exit flag.
x =
1.0e-07 *
0.0414
0.1298
fval =
1.5467e-15
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3-45
3
Using GlobalSearch and MultiStart
The run stopped early because GlobalSearch found a point with a function
value less than 0.5.
No Parallel Output Functions
While MultiStart can run in parallel, it does not support global output
functions and plot functions in parallel. Furthermore, while local output
functions and plot functions run on workers when MultiStart runs in
parallel, the effect differs from running serially. Local output and plot
functions do not create a display when running on workers. You do not see
any other effects of output and plot functions until the worker passes its
results to the client (the originator of the MultiStart parallel jobs).
For information on running MultiStart in parallel, see “Parallel Computing”.
3-46
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Plot Functions for GlobalSearch and MultiStart
Plot Functions for GlobalSearch and MultiStart
In this section...
“What Are Plot Functions?” on page 3-47
“MultiStart Plot Function” on page 3-48
“No Parallel Plot Functions” on page 3-51
What Are Plot Functions?
The PlotFcns field of the options structure specifies one or more functions
that an optimization function calls at each iteration. Plot functions plot
various measures of progress while the algorithm executes. Pass a function
handle or cell array of function handles. The structure of a plot function is
the same as the structure of an output function. For more information on this
structure, see “OutputFcns” on page 9-3.
Plot functions are specialized output functions (see “Output Functions for
GlobalSearch and MultiStart” on page 3-43). There are two predefined plot
functions:
• @gsplotbestf plots the best objective function value.
• @gsplotfunccount plots the number of function evaluations.
Plot function windows have Pause and Stop buttons. By default, all plots
appear in one window.
To use global plot functions:
• Write plot functions using the syntax described in “OutputFcns” on page
9-3.
• Set the PlotFcns property of your GlobalSearch or MultiStart object
to the function handle of your plot function. You can use multiple plot
functions by setting the PlotFcns property to a cell array of function
handles.
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3-47
3
Using GlobalSearch and MultiStart
Details of Built-In Plot Functions
The built-in plot functions have characteristics that can surprise you.
• @gsplotbestf can have plots that are not strictly decreasing. This is
because early values can result from local solver runs with negative exit
flags (such as infeasible solutions). A subsequent local solution with
positive exit flag is better even if its function value is higher. Once a
local solver returns a value with a positive exit flag, the plot is monotone
decreasing.
• @gsplotfunccount might not plot the total number of function evaluations.
This is because GlobalSearch can continue to perform function evaluations
after it calls the plot function for the last time. For more information, see
“GlobalSearch Algorithm” on page 3-54Properties for GlobalSearch.
MultiStart Plot Function
This example plots the number of local solver runs it takes to obtain a better
local minimum for MultiStart. The example also uses a built-in plot function
to show the current best function value.
The example problem is the same as in “Find Global or Multiple Local
Minima” on page 3-83, with additional bounds.
The example uses persistent variables to store previous best values. The
plot function examines the best function value after each local solver run,
available in the bestfval field of the optimValues structure. If the value is
not lower than the previous best, the plot function adds 1 to the number of
consecutive calls with no improvement and draws a bar chart. If the value is
lower than the previous best, the plot function starts a new bar in the chart
with value 1. Before plotting, the plot function takes a logarithm of the
number of consecutive calls. The logarithm helps keep the plot legible, since
some values can be much larger than others.
To store local results using nested functions instead of persistent variables,
see “Example of a Nested Output Function” in the MATLAB Mathematics
documentation.
1 Write the objective function:
function f = sawtoothxy(x,y)
3-48
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Plot Functions for GlobalSearch and MultiStart
[t r] = cart2pol(x,y); % change to polar coordinates
h = cos(2*t - 1/2)/2 + cos(t) + 2;
g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ...
.*r.^2./(r+1);
f = g.*h;
2 Save sawtoothxy.m as a file in a folder on your MATLAB path.
3 Write the plot function:
function stop = NumberToNextBest(optimValues, state)
persistent bestfv bestcounter
stop = false;
switch state
case 'init'
% Initialize variable to record best function value.
bestfv = [];
% Initialize counter to record number of
% local solver runs to find next best minimum.
bestcounter = 1;
% Create the histogram.
bar(log(bestcounter),'tag','NumberToNextBest');
xlabel('Number of New Best Fval Found');
ylabel('Log Number of Local Solver Runs');
title('Number of Local Solver Runs to Find Lower Minimum')
case 'iter'
% Find the axes containing the histogram.
NumToNext = ...
findobj(get(gca,'Children'),'Tag','NumberToNextBest');
% Update the counter that records number of local
% solver runs to find next best minimum.
if ~isequal(optimValues.bestfval, bestfv)
bestfv = optimValues.bestfval;
bestcounter = [bestcounter 1];
else
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3-49
3
Using GlobalSearch and MultiStart
bestcounter(end) = bestcounter(end) + 1;
end
% Update the histogram.
set(NumToNext,'Ydata',log(bestcounter))
end
4 Save NumberToNextBest.m as a file in a folder on your MATLAB path.
5 Create the problem structure and global solver. Set lower bounds of
[-3e3,-4e3], upper bounds of [4e3,3e3] and set the global solver to use
the plot functions:
problem = createOptimProblem('fmincon',...
'objective',@(x)sawtoothxy(x(1),x(2)),...
'x0',[100,-50],'lb',[-3e3 -4e3],...
'ub',[4e3,3e3],'options',...
optimoptions(@fmincon,'Algorithm','sqp'));
ms = MultiStart('PlotFcns',{@NumberToNextBest,@gsplotbestf});
6 Run the global solver for 100 local solver runs:
[x fv] = run(ms,problem,100);
7 The plot functions produce the following figure (your results can differ, since
the solution process is stochastic):
3-50
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Plot Functions for GlobalSearch and MultiStart
No Parallel Plot Functions
While MultiStart can run in parallel, it does not support global output
functions and plot functions in parallel. Furthermore, while local output
functions and plot functions run on workers when MultiStart runs in
parallel, the effect differs from running serially. Local output and plot
functions do not create a display when running on workers. You do not see
any other effects of output and plot functions until the worker passes its
results to the client (the originator of the MultiStart parallel jobs).
For information on running MultiStart in parallel, see “Parallel Computing”.
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3-51
3
Using GlobalSearch and MultiStart
How GlobalSearch and MultiStart Work
In this section...
“Multiple Runs of a Local Solver” on page 3-52
“Differences Between the Solver Objects” on page 3-52
“GlobalSearch Algorithm” on page 3-54
“MultiStart Algorithm” on page 3-59
“Bibliography” on page 3-61
Multiple Runs of a Local Solver
GlobalSearch and MultiStart have similar approaches to finding global or
multiple minima. Both algorithms start a local solver (such as fmincon) from
multiple start points. The algorithms use multiple start points to sample
multiple basins of attraction. For more information, see “Basins of Attraction”
on page 1-14.
Differences Between the Solver Objects
GlobalSearch and MultiStart Algorithm Overview on page 3-53 contains a
sketch of the GlobalSearch and MultiStart algorithms.
3-52
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How GlobalSearch and MultiStart Work
GlobalSearch Algorithm
MultiStart Algorithm
Run fmincon from x0
Generate start points
Generate trial points
(potential start points)
Run start points
Stage 1:
Run best start point among the first
NumStageOnePoints trial points
Create GlobalOptimSolutions vector
Stage 2:
Loop through remaining trial points,
run fmincon if point satisfies
basin, score, and constraint filters
Create GlobalOptimSolutions vector
GlobalSearch and MultiStart Algorithm Overview
The main differences between GlobalSearch and MultiStart are:
• GlobalSearch uses a scatter-search mechanism for generating start points.
MultiStart uses uniformly distributed start points within bounds, or
user-supplied start points.
• GlobalSearch analyzes start points and rejects those points that are
unlikely to improve the best local minimum found so far. MultiStart runs
all start points (or, optionally, all start points that are feasible with respect
to bounds or inequality constraints).
• MultiStart gives a choice of local solver: fmincon, fminunc, lsqcurvefit,
or lsqnonlin. The GlobalSearch algorithm uses fmincon.
• MultiStart can run in parallel, distributing start points to multiple
processors for local solution. To run MultiStart in parallel, see “How to
Use Parallel Processing” on page 8-12.
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3-53
3
Using GlobalSearch and MultiStart
Deciding Which Solver to Use
The differences between these solver objects boil down to the following
decision on which to use:
• Use GlobalSearch to find a single global minimum most efficiently on a
single processor.
• Use MultiStart to:
-
Find multiple local minima.
Run in parallel.
Use a solver other than fmincon.
Search thoroughly for a global minimum.
Explore your own start points.
GlobalSearch Algorithm
For a description of the algorithm, see Ugray et al. [1].
When you run a GlobalSearch object, the algorithm performs the following
steps:
1 “Run fmincon from x0” on page 3-55
2 “Generate Trial Points” on page 3-55
3 “Obtain Stage 1 Start Point, Run” on page 3-55
4 “Initialize Basins, Counters, Threshold” on page 3-55
5 “Begin Main Loop” on page 3-56
6 “Examine Stage 2 Trial Point to See if fmincon Runs” on page 3-56
7 “When fmincon Runs” on page 3-56
8 “When fmincon Does Not Run” on page 3-58
9 “Create GlobalOptimSolution” on page 3-58
3-54
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How GlobalSearch and MultiStart Work
Run fmincon from x0
GlobalSearch runs fmincon from the start point you give in the problem
structure. If this run converges, GlobalSearch records the start point
and end point for an initial estimate on the radius of a basin of attraction.
Furthermore, GlobalSearch records the final objective function value for use
in the score function (see “Obtain Stage 1 Start Point, Run” on page 3-55).
The score function is the sum of the objective function value at a point and a
multiple of the sum of the constraint violations. So a feasible point has score
equal to its objective function value. The multiple for constraint violations is
initially 1000. GlobalSearch updates the multiple during the run.
Generate Trial Points
GlobalSearch uses the scatter search algorithm to generate a set of
NumTrialPoints trial points. Trial points are potential start points. For a
description of the scatter search algorithm, see Glover [2]. GlobalSearch
generates trial points within any finite bounds you set (lb and ub). Unbounded
components have artificial bounds imposed: lb = -1e4 + 1, ub = 1e4 + 1.
This range is not symmetric about the origin so that the origin is not in the
scatter search. Components with one-sided bounds have artificial bounds
imposed on the unbounded side, shifted by the finite bounds to keep lb < ub.
Obtain Stage 1 Start Point, Run
GlobalSearch evaluates the score function of a set of NumStageOnePoints
trial points. It then takes the point with the best score and runs fmincon
from that point. GlobalSearch removes the set of NumStageOnePoints trial
points from its list of points to examine.
Initialize Basins, Counters, Threshold
The localSolverThreshold is initially the smaller of the two objective
function values at the solution points. The solution points are the fmincon
solutions starting from x0 and from the Stage 1 start point. If both of these
solution points do not exist or are infeasible, localSolverThreshold is
initially the penalty function value of the Stage 1 start point.
The GlobalSearch heuristic assumption is that basins of attraction are
spherical. The initial estimate of basins of attraction for the solution point
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3-55
3
Using GlobalSearch and MultiStart
from x0 and the solution point from Stage 1 are spheres centered at the
solution points. The radius of each sphere is the distance from the initial
point to the solution point. These estimated basins can overlap.
There are two sets of counters associated with the algorithm. Each counter is
the number of consecutive trial points that:
• Lie within a basin of attraction. There is one counter for each basin.
• Have score function greater than localSolverThreshold. For a definition
of the score, see “Run fmincon from x0” on page 3-55.
All counters are initially 0.
Begin Main Loop
GlobalSearch repeatedly examines a remaining trial point from the list, and
performs the following steps. It continually monitors the time, and stops the
search if elapsed time exceeds MaxTime seconds.
Examine Stage 2 Trial Point to See if fmincon Runs
Call the trial point p. Run fmincon from p if the following conditions hold:
• p is not in any existing basin. The criterion for every basin i is:
|p - center(i)| > DistanceThresholdFactor * radius(i).
DistanceThresholdFactor is an option (default value 0.75).
radius is an estimated radius that updates in Update Basin Radius and
Threshold and React to Large Counter Values.
• score(p) < localSolverThreshold.
• (optional) p satisfies bound and/or inequality constraints. This test occurs
if you set the StartPointsToRun property of the GlobalSearch object to
'bounds' or 'bounds-ineqs'.
When fmincon Runs
1 Reset Counters
3-56
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How GlobalSearch and MultiStart Work
Set the counters for basins and threshold to 0.
2 Update Solution Set
If fmincon runs starting from p, it can yield a positive exit flag, which
indicates convergence. In that case, GlobalSearch updates the vector of
GlobalOptimSolution objects. Call the solution point xp and the objective
function value fp. There are two cases:
• For every other solution point xq with objective function value fq,
|xq - xp| > TolX * max(1,|xp|)
or
|fq - fp| > TolFun * max(1,|fp|).
In this case, GlobalSearch creates a new element in the vector of
GlobalOptimSolution objects. For details of the information contained in
each object, see GlobalOptimSolution.
• For some other solution point xq with objective function value fq,
|xq - xp| <= TolX * max(1,|xp|)
and
|fq - fp| <= TolFun * max(1,|fp|).
In this case, GlobalSearch regards xp as equivalent to xq. The
GlobalSearch algorithm modifies the GlobalOptimSolution of xq by
adding p to the cell array of X0 points.
There is one minor tweak that can happen to this update. If the exit flag for
xq is greater than 1, and the exit flag for xp is 1, then xp replaces xq. This
replacement can lead to some points in the same basin being more than
a distance of TolX from xp.
3 Update Basin Radius and Threshold
If the exit flag of the current fmincon run is positive:
a Set threshold to the score value at start point p.
b Set basin radius for xp equal to the maximum of the existing radius (if any)
and the distance between p and xp.
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3-57
3
Using GlobalSearch and MultiStart
4 Report to Iterative Display
When the GlobalSearch Display property is 'iter', every point that
fmincon runs creates one line in the GlobalSearch iterative display.
When fmincon Does Not Run
1 Update Counters
Increment the counter for every basin containing p. Reset the counter of
every other basin to 0.
Increment the threshold counter if score(p) >= localSolverThreshold.
Otherwise, reset the counter to 0.
2 React to Large Counter Values
For each basin with counter equal to MaxWaitCycle, multiply the basin radius
by 1 – BasinRadiusFactor. Reset the counter to 0. (Both MaxWaitCycle and
BasinRadiusFactor are settable properties of the GlobalSearch object.)
If the threshold counter equals MaxWaitCycle, increase the threshold:
new threshold = threshold + PenaltyThresholdFactor*(1 +
abs(threshold)).
Reset the counter to 0.
3 Report to Iterative Display
Every 200th trial point creates one line in the GlobalSearch iterative display.
Create GlobalOptimSolution
After reaching MaxTime seconds or running out of trial points, GlobalSearch
creates a vector of GlobalOptimSolution objects. GlobalSearch orders the
vector by objective function value, from lowest (best) to highest (worst). This
concludes the algorithm.
3-58
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How GlobalSearch and MultiStart Work
MultiStart Algorithm
When you run a MultiStart object, the algorithm performs the following
steps:
• “Generate Start Points” on page 3-59
• “Filter Start Points (Optional)” on page 3-59
• “Run Local Solver” on page 3-60
• “Check Stopping Conditions” on page 3-60
• “Create GlobalOptimSolution Object” on page 3-60
Generate Start Points
If you call MultiStart with the syntax
[x fval] = run(ms,problem,k)
for an integer k, MultiStart generates k - 1 start points exactly as if you
used a RandomStartPointSet object. The algorithm also uses the x0 start
point from the problem structure, for a total of k start points.
A RandomStartPointSet object does not have any points stored inside
the object. Instead, MultiStart calls the list method, which generates
random points within the bounds given by the problem structure. If an
unbounded component exists, list uses an artificial bound given by the
ArtificialBound property of the RandomStartPointSet object.
If you provide a CustomStartPointSet object, MultiStart does not generate
start points, but uses the points in the object.
Filter Start Points (Optional)
If you set the StartPointsToRun property of the MultiStart object to
'bounds' or 'bounds-ineqs', MultiStart does not run the local solver from
infeasible start points. In this context, “infeasible” means start points that
do not satisfy bounds, or start points that do not satisfy both bounds and
inequality constraints.
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3-59
3
Using GlobalSearch and MultiStart
The default setting of StartPointsToRun is 'all'. In this case, MultiStart
does not discard infeasible start points.
Run Local Solver
MultiStart runs the local solver specified in problem.solver, starting at the
points that pass the StartPointsToRun filter. If MultiStart is running in
parallel, it sends start points to worker processors one at a time, and the
worker processors run the local solver.
The local solver checks whether MaxTime seconds have elapsed at each of its
iterations. If so, it exits that iteration without reporting a solution.
When the local solver stops, MultiStart stores the results and continues
to the next step.
Report to Iterative Display. When the MultiStart Display property
is 'iter', every point that the local solver runs creates one line in the
MultiStart iterative display.
Check Stopping Conditions
MultiStart stops when it runs out of start points. It also stops when it
exceeds a total run time of MaxTime seconds.
Create GlobalOptimSolution Object
After MultiStart reaches a stopping condition, the algorithm creates a vector
of GlobalOptimSolution objects as follows:
1 Sort the local solutions by objective function value (Fval) from lowest to
highest. For the lsqnonlin and lsqcurvefit local solvers, the objective
function is the norm of the residual.
2 Loop over the local solutions j beginning with the lowest (best) Fval.
3 Find all the solutions k satisfying both:
|Fval(k) - Fval(j)| <= TolFun*max(1,|Fval(j)|)
|x(k) - x(j)| <= TolX*max(1,|x(j)|)
3-60
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How GlobalSearch and MultiStart Work
4 Record j, Fval(j), the local solver output structure for j, and a cell
array of the start points for j and all the k. Remove those points k
from the list of local solutions. This point is one entry in the vector of
GlobalOptimSolution objects.
The resulting vector of GlobalOptimSolution objects is in order by Fval,
from lowest (best) to highest (worst).
Report to Iterative Display. After examining all the local solutions,
MultiStart gives a summary to the iterative display. This summary includes
the number of local solver runs that converged, the number that failed to
converge, and the number that had errors.
Bibliography
[1] Ugray, Zsolt, Leon Lasdon, John C. Plummer, Fred Glover, James Kelly,
and Rafael Martí. Scatter Search and Local NLP Solvers: A Multistart
Framework for Global Optimization. INFORMS Journal on Computing, Vol.
19, No. 3, 2007, pp. 328–340.
[2] Glover, F. “A template for scatter search and path relinking.” Artificial
Evolution (J.-K. Hao, E.Lutton, E.Ronald, M.Schoenauer, D.Snyers, eds.).
Lecture Notes in Computer Science, 1363, Springer, Berlin/Heidelberg, 1998,
pp. 13–54.
[3] Dixon, L. and G. P. Szegö. “The Global Optimization Problem: an
Introduction.” Towards Global Optimisation 2 (Dixon, L. C. W. and G. P.
Szegö, eds.). Amsterdam, The Netherlands: North Holland, 1978.
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3-61
3
Using GlobalSearch and MultiStart
Can You Certify a Solution Is Global?
In this section...
“No Guarantees” on page 3-62
“Check if a Solution Is a Local Solution with patternsearch” on page 3-62
“Identify a Bounded Region That Contains a Global Solution” on page 3-63
“Use MultiStart with More Start Points” on page 3-64
No Guarantees
How can you tell if you have located the global minimum of your objective
function? The short answer is that you cannot; you have no guarantee that
the result of a Global Optimization Toolbox solver is a global optimum.
However, you can use the strategies in this section for investigating solutions.
Check if a Solution Is a Local Solution with
patternsearch
Before you can determine if a purported solution is a global minimum, first
check that it is a local minimum. To do so, run patternsearch on the problem.
To convert the problem to use patternsearch instead of fmincon or fminunc,
enter
problem.solver = 'patternsearch';
Also, change the start point to the solution you just found, and clear the
options:
problem.x0 = x;
problem.options = [];
For example, Check Nearby Points (in the Optimization Toolbox
documentation) shows the following:
options = optimoptions(@fmincon,'Algorithm','active-set');
ffun = @(x)(x(1)-(x(1)-x(2))^2);
3-62
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Can You Certify a Solution Is Global?
problem = createOptimProblem('fmincon', ...
'objective',ffun,'x0',[1/2 1/3], ...
'lb',[0 -1],'ub',[1 1],'options',options);
[x fval exitflag] = fmincon(problem)
x =
1.0e-007 *
0
0.1614
fval =
-2.6059e-016
exitflag =
1
However, checking this purported solution with patternsearch shows that
there is a better solution. Start patternsearch from the reported solution x:
% set the candidate solution x as the start point
problem.x0 = x;
problem.solver = 'patternsearch';
problem.options = [];
[xp fvalp exitflagp] = patternsearch(problem)
Optimization terminated: mesh size less than options.TolMesh.
xp =
1.0000
-1.0000
fvalp =
-3.0000
exitflagp =
1
Identify a Bounded Region That Contains a Global
Solution
Suppose you have a smooth objective function in a bounded region. Given
enough time and start points, MultiStart eventually locates a global solution.
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3-63
3
Using GlobalSearch and MultiStart
Therefore, if you can bound the region where a global solution can exist, you
can obtain some degree of assurance that MultiStart locates the global
solution.
For example, consider the function
⎛ x2 ⎞
f = x6 + y6 + sin( x + y) ( x2 + y2 ) − cos ⎜
⎟ ( 2 + x 4 + x 2 y2 + y4 ) .
⎜ 1 + y2 ⎟
⎝
⎠
The initial summands x6 + y6 force the function to become large and positive
for large values of |x| or |y|. The components of the global minimum of the
function must be within the bounds
–10 ≤ x,y ≤ 10,
since 106 is much larger than all the multiples of 104 that occur in the other
summands of the function.
You can identify smaller bounds for this problem; for example, the global
minimum is between –2 and 2. It is more important to identify reasonable
bounds than it is to identify the best bounds.
Use MultiStart with More Start Points
To check whether there is a better solution to your problem, run MultiStart
with additional start points. Use MultiStart instead of GlobalSearch for this
task because GlobalSearch does not run the local solver from all start points.
For example, see “Example: Searching for a Better Solution” on page 3-70.
Updating Unconstrained Problem from GlobalSearch
If you use GlobalSearch on an unconstrained problem, change your problem
structure before using MultiStart. You have two choices in updating a
problem structure for an unconstrained problem using MultiStart:
• Change the solver field to 'fminunc':
problem.solver = 'fminunc';
3-64
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Can You Certify a Solution Is Global?
To avoid a warning if your objective function does not compute a gradient,
change the local options to have Algorithm set to 'quasi-newton':
problem.options.Algorithm = 'quasi-newton';
• Add an artificial constraint, retaining fmincon as the local solver:
problem.lb = -Inf(size(x0));
To search a larger region than the default, see “Refine Start Points” on page
3-66.
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3-65
3
Using GlobalSearch and MultiStart
Refine Start Points
In this section...
“About Refining Start Points” on page 3-66
“Methods of Generating Start Points” on page 3-67
“Example: Searching for a Better Solution” on page 3-70
About Refining Start Points
If some components of your problem are unconstrained, GlobalSearch and
MultiStart use artificial bounds to generate random start points uniformly
in each component. However, if your problem has far-flung minima, you need
widely dispersed start points to find these minima.
Use these methods to obtain widely dispersed start points:
• Give widely separated bounds in your problem structure.
• Use a RandomStartPointSet object with the MultiStart algorithm. Set a
large value of the ArtificialBound property in the RandomStartPointSet
object.
• Use a CustomStartPointSet object with the MultiStart algorithm. Use
widely dispersed start points.
There are advantages and disadvantages of each method.
3-66
Method
Advantages
Disadvantages
Give bounds in problem
Automatic point generation
Makes a more complex Hessian
Can use with GlobalSearch
Unclear how large to set the
bounds
Easy to do
Changes problem
Bounds can be asymmetric
Only uniform points
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Refine Start Points
Method
Advantages
Disadvantages
Large ArtificialBound in
Automatic point generation
MultiStart only
RandomStartPointSet
Does not change problem
Only symmetric, uniform points
Easy to do
Unclear how large to set
ArtificialBound
CustomStartPointSet
Customizable
MultiStart only
Does not change problem
Requires programming for
generating points
Methods of Generating Start Points
• “Uniform Grid” on page 3-67
• “Perturbed Grid” on page 3-68
• “Widely Dispersed Points for Unconstrained Components” on page 3-69
Uniform Grid
To generate a uniform grid of start points:
1 Generate multidimensional arrays with ndgrid. Give the lower bound,
spacing, and upper bound for each component.
For example, to generate a set of three-dimensional arrays with
• First component from –2 through 0, spacing 0.5
• Second component from 0 through 2, spacing 0.25
• Third component from –10 through 5, spacing 1
[X,Y,Z] = ndgrid(-2:.5:0,0:.25:2,-10:5);
2 Place the arrays into a single matrix, with each row representing one start
point. For example:
W = [X(:),Y(:),Z(:)];
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3-67
3
Using GlobalSearch and MultiStart
In this example, W is a 720-by-3 matrix.
3 Put the matrix into a CustomStartPointSet object. For example:
custpts = CustomStartPointSet(W);
Call MultiStart run with the CustomStartPointSet object as the third
input. For example,
% Assume problem structure and ms MultiStart object exist
[x fval flag outpt manymins] = run(ms,problem,custpts);
Perturbed Grid
Integer start points can yield less robust solutions than slightly perturbed
start points.
To obtain a perturbed set of start points:
1 Generate a matrix of start points as in steps 1–2 of “Uniform Grid” on page
3-67.
2 Perturb the start points by adding a random normal matrix with 0 mean
and relatively small variance.
For the example in “Uniform Grid” on page 3-67, after making the W matrix,
add a perturbation:
[X,Y,Z] = ndgrid(-2:.5:0,0:.25:2,-10:5);
W = [X(:),Y(:),Z(:)];
W = W + 0.01*randn(size(W));
3 Put the matrix into a CustomStartPointSet object. For example:
custpts = CustomStartPointSet(W);
Call MultiStart run with the CustomStartPointSet object as the third
input. For example,
% Assume problem structure and ms MultiStart object exist
[x fval flag outpt manymins] = run(ms,problem,custpts);
3-68
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Refine Start Points
Widely Dispersed Points for Unconstrained Components
Some components of your problem can lack upper or lower bounds. For
example:
• Although no explicit bounds exist, there are levels that the components
cannot attain. For example, if one component represents the weight of a
single diamond, there is an implicit upper bound of 1 kg (the Hope Diamond
is under 10 g). In such a case, give the implicit bound as an upper bound.
• There truly is no upper bound. For example, the size of a computer file in
bytes has no effective upper bound. The largest size can be in gigabytes
or terabytes today, but in 10 years, who knows?
For truly unbounded components, you can use the following methods
of sampling. To generate approximately 1/n points in each region
(exp(n),exp(n+1)), use the following formula. If u is random and uniformly
distributed from 0 through 1, then r = 2u – 1 is uniformly distributed between
–1 and 1. Take
y = sgn(r) ( exp (1 / r ) − e ) .
y is symmetric and random. For a variable bounded below by lb, take
y = lb + ( exp (1 / u ) − e ) .
Similarly, for a variable bounded above by ub, take
y = ub − ( exp (1 / u ) − e ) .
For example, suppose you have a three-dimensional problem with
• x(1) > 0
• x(2) < 100
• x(3) unconstrained
To make 150 start points satisfying these constraints:
u = rand(150,3);
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3-69
3
Using GlobalSearch and MultiStart
r1 = 1./u(:,1);
r1 = exp(r1) - exp(1);
r2 = 1./u(:,2);
r2 = -exp(r2) + exp(1) + 100;
r3 = 1./(2*u(:,3)-1);
r3 = sign(r3).*(exp(abs(r3)) - exp(1));
custpts = CustomStartPointSet([r1,r2,r3]);
The following is a variant of this algorithm. Generate a number between 0
and infinity by the method for lower bounds. Use this number as the radius
of a point. Generate the other components of the point by taking random
numbers for each component and multiply by the radius. You can normalize
the random numbers, before multiplying by the radius, so their norm is 1. For
a worked example of this method, see “MultiStart Without Bounds, Widely
Dispersed Start Points” on page 3-108.
Example: Searching for a Better Solution
MultiStart fails to find the global minimum in “Multiple Local Minima Via
MultiStart” on page 3-87. There are two simple ways to search for a better
solution:
• Use more start points
• Give tighter bounds on the search space
Set up the problem structure and MultiStart object:
problem = createOptimProblem('fminunc',...
'objective',@(x)sawtoothxy(x(1),x(2)),...
'x0',[100,-50],'options',...
optimoptions(@fminunc,'Algorithm','quasi-newton'));
ms = MultiStart;
Use More Start Points
Run MultiStart on the problem for 200 start points instead of 50:
rng(14,'twister') % for reproducibility
[x fval eflag output manymins] = run(ms,problem,200)
3-70
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Refine Start Points
MultiStart completed some of the runs from the start points.
51 out of 200 local solver runs converged with a positive
local solver exit flag.
x =
1.0e-06 *
-0.2284
-0.5567
fval =
2.1382e-12
eflag =
2
output =
funcCount:
localSolverTotal:
localSolverSuccess:
localSolverIncomplete:
localSolverNoSolution:
message:
32760
200
51
149
0
'MultiStart completed some of the runs from th..
manymins =
1x51 GlobalOptimSolution
Properties:
X
Fval
Exitflag
Output
X0
This time MultiStart found the global minimum, and found 51 local minima.
To see the range of local solutions, enter hist([manymins.Fval]).
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3-71
3
Using GlobalSearch and MultiStart
Tighter Bound on the Start Points
Suppose you believe that the interesting local solutions have absolute
values of all components less than 100. The default value of the bound
on start points is 1000. To use a different value of the bound, generate a
RandomStartPointSet with the ArtificialBound property set to 100:
startpts = RandomStartPointSet('ArtificialBound',100,...
'NumStartPoints',50);
[x fval eflag output manymins] = run(ms,problem,startpts)
MultiStart completed some of the runs from the start points.
27 out of 50 local solver runs converged with a
positive local solver exit flag.
x =
1.0e-08 *
0.9725
-0.6198
fval =
1.4955e-15
3-72
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Refine Start Points
eflag =
2
output =
funcCount:
localSolverTotal:
localSolverSuccess:
localSolverIncomplete:
localSolverNoSolution:
message:
7482
50
27
23
0
'MultiStart completed some of the runs from th..
manymins =
1x22 GlobalOptimSolution
Properties:
X
Fval
Exitflag
Output
X0
MultiStart found the global minimum, and found 22 distinct local solutions.
To see the range of local solutions, enter hist([manymins.Fval]).
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3-73
3
Using GlobalSearch and MultiStart
Compared to the minima found in “Use More Start Points” on page 3-70, this
run found better (smaller) minima, and had a higher percentage of successful
runs.
3-74
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Change Options
Change Options
In this section...
“How to Determine Which Options to Change” on page 3-75
“Changing Local Solver Options” on page 3-76
“Changing Global Options” on page 3-77
How to Determine Which Options to Change
After you run a global solver, you might want to change some global or local
options. To determine which options to change, the guiding principle is:
• To affect the local solver, set local solver options.
• To affect the start points or solution set, change the problem structure, or
set the global solver object properties.
For example, to obtain:
• More local minima — Set global solver object properties.
• Faster local solver iterations — Set local solver options.
• Different tolerances for considering local solutions identical (to obtain more
or fewer local solutions) — Set global solver object properties.
• Different information displayed at the command line — Decide if you want
iterative display from the local solver (set local solver options) or global
information (set global solver object properties).
• Different bounds, to examine different regions — Set the bounds in the
problem structure.
Examples of Choosing Problem Options
• To start your local solver at points only satisfying inequality constraints,
set the StartPointsToRun property in the global solver object to
'bounds-ineqs'. This setting can speed your solution, since local solvers
do not have to attempt to find points satisfying these constraints. However,
the setting can result in many fewer local solver runs, since the global
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3-75
3
Using GlobalSearch and MultiStart
solver can reject many start points. For an example, see “Optimize Using
Only Feasible Start Points” on page 3-91.
• To use the fmincon interior-point algorithm, set the local solver
Algorithm option to 'interior-point'. For an example showing how to
do this, see “Examples of Updating Problem Options” on page 3-76.
• For your local solver to have different bounds, set the bounds in the
problem structure. Examine different regions by setting bounds.
• To see every solution that has positive local exit flag, set the TolX property
in the global solver object to 0. For an example showing how to do this, see
“Changing Global Options” on page 3-77.
Changing Local Solver Options
There are several ways to change values in a local options structure:
• Update the values using dot notation and optimoptions. The syntax is
problem.options =
optimoptions(problem.options,'Parameter',value,...);
You can also replace the local options entirely:
problem.options =
optimoptions(@solvername,'Parameter',value,...);
• Use dot notation on one local option. The syntax is
problem.options.Parameter = newvalue;
• Recreate the entire problem structure. For details, see “Create Problem
Structure” on page 3-7.
Examples of Updating Problem Options
1 Create a problem structure:
problem = createOptimProblem('fmincon','x0',[-1 2], ...
'objective',@rosenboth);
2 Set the problem to use the sqp algorithm in fmincon:
3-76
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Change Options
problem.options.Algorithm = 'sqp';
3 Update the problem to use the gradient in the objective function, have a
TolFun value of 1e-8, and a TolX value of 1e-7:
problem.options = optimoptions(problem.options,'GradObj','on', ...
'TolFun',1e-8,'TolX',1e-7);
Changing Global Options
There are several ways to change characteristics of a GlobalSearch or
MultiStart object:
• Use dot notation. For example, suppose you have a default MultiStart
object:
ms = MultiStart
MultiStart with properties:
UseParallel: 0
Display: 'final'
TolFun: 1.0000e-06
TolX: 1.0000e-06
MaxTime: Inf
StartPointsToRun: 'all'
OutputFcns: []
PlotFcns: []
To change ms to have its TolX value equal to 1e-3, update the TolX field:
ms.TolX = 1e-3
MultiStart with properties:
UseParallel: 0
Display: 'final'
TolFun: 1.0000e-06
TolX: 1.0000e-03
MaxTime: Inf
StartPointsToRun: 'all'
OutputFcns: []
PlotFcns: []
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3-77
3
Using GlobalSearch and MultiStart
• Reconstruct the object starting from the current settings. For example, to
set the TolFun field in ms to 1e-3, retaining the nondefault value for TolX:
ms = MultiStart(ms,'TolFun',1e-3)
MultiStart with properties:
UseParallel: 0
Display: 'final'
TolFun: 1.0000e-03
TolX: 1.0000e-03
MaxTime: Inf
StartPointsToRun: 'all'
OutputFcns: []
PlotFcns: []
• Convert a GlobalSearch object to a MultiStart object, or vice-versa.
For example, with the ms object from the previous example, create a
GlobalSearch object with the same values of TolX and TolFun:
gs = GlobalSearch(ms)
GlobalSearch with properties:
NumTrialPoints: 1000
BasinRadiusFactor: 0.2000
DistanceThresholdFactor: 0.7500
MaxWaitCycle: 20
NumStageOnePoints: 200
PenaltyThresholdFactor: 0.2000
Display: 'final'
TolFun: 1.0000e-03
TolX: 1.0000e-03
MaxTime: Inf
StartPointsToRun: 'all'
OutputFcns: []
PlotFcns: []
3-78
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Reproduce Results
Reproduce Results
In this section...
“Identical Answers with Pseudorandom Numbers” on page 3-79
“Steps to Take in Reproducing Results” on page 3-79
“Example: Reproducing a GlobalSearch or MultiStart Result” on page 3-80
“Parallel Processing and Random Number Streams” on page 3-82
Identical Answers with Pseudorandom Numbers
GlobalSearch and MultiStart use pseudorandom numbers in choosing start
points. Use the same pseudorandom number stream again to:
• Compare various algorithm settings.
• Have an example run repeatably.
• Extend a run, with known initial segment of a previous run.
Both GlobalSearch and MultiStart use the default random number stream.
Steps to Take in Reproducing Results
1 Before running your problem, store the current state of the default random
number stream:
stream = rng;
2 Run your GlobalSearch or MultiStart problem.
3 Restore the state of the random number stream:
rng(stream)
4 If you run your problem again, you get the same result.
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3-79
3
Using GlobalSearch and MultiStart
Example: Reproducing a GlobalSearch or MultiStart
Result
This example shows how to obtain reproducible results for “Find Global or
Multiple Local Minima” on page 3-83. The example follows the procedure in
“Steps to Take in Reproducing Results” on page 3-79.
1 Store the current state of the default random number stream:
stream = rng;
2 Create the sawtoothxy function file:
function f = sawtoothxy(x,y)
[t r] = cart2pol(x,y); % change to polar coordinates
h = cos(2*t - 1/2)/2 + cos(t) + 2;
g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ...
.*r.^2./(r+1);
f = g.*h;
end
3 Create the problem structure and GlobalSearch object:
problem = createOptimProblem('fmincon',...
'objective',@(x)sawtoothxy(x(1),x(2)),...
'x0',[100,-50],'options',...
optimoptions(@fmincon,'Algorithm','sqp'));
gs = GlobalSearch('Display','iter');
4 Run the problem:
[x fval] = run(gs,problem)
Num Pts
3-80
Best
Current
Threshold
Local
Local
Analyzed
F-count
f(x)
Penalty
Penalty
f(x)
exitflag
Procedure
0
465
422.9
422.9
2
Initial Point
200
1730
1.547e-015
1.547e-015
1
300
1830
1.547e-015
6.01e+004
400
1930
1.547e-015
1.47e+005
4.16
Stage 2 Search
500
2030
1.547e-015
2.63e+004
11.84
Stage 2 Search
600
2130
1.547e-015
1.341e+004
30.95
Stage 2 Search
700
2230
1.547e-015
2.562e+004
65.25
Stage 2 Search
1.074
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Stage 1 Local
Stage 2 Search
Reproduce Results
800
2330
1.547e-015
5.217e+004
163.8
900
2430
1.547e-015
7.704e+004
409.2
981
2587
1.547e-015
42.24
516.6
1000
2606
1.547e-015
3.299e+004
42.24
Stage 2 Search
Stage 2 Search
7.573
1
Stage 2 Local
Stage 2 Search
GlobalSearch stopped because it analyzed all the trial points.
All 3 local solver runs converged with a positive local solver exit flag.
x =
1.0e-007 *
0.0414
0.1298
fval =
1.5467e-015
You might obtain a different result when running this problem, since the
random stream was in an unknown state at the beginning of the run.
5 Restore the state of the random number stream:
rng(stream)
6 Run the problem again. You get the same output.
[x fval] = run(gs,problem)
Best
Current
Threshold
Local
Local
Analyzed
Num Pts
F-count
f(x)
Penalty
Penalty
f(x)
exitflag
Procedure
0
465
422.9
422.9
2
Initial Point
200
1730
1.547e-015
1.547e-015
1
Stage 1 Local
... Output deleted to save space ...
x =
1.0e-007 *
0.0414
0.1298
fval =
1.5467e-015
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3-81
3
Using GlobalSearch and MultiStart
Parallel Processing and Random Number Streams
You obtain reproducible results from MultiStart when you run the
algorithm in parallel the same way as you do for serial computation. Runs
are reproducible because MultiStart generates pseudorandom start points
locally, and then distributes the start points to parallel processors. Therefore,
the parallel processors do not use random numbers.
To reproduce a parallel MultiStart run, use the procedure described in “Steps
to Take in Reproducing Results” on page 3-79. For a description of how to run
MultiStart in parallel, see “How to Use Parallel Processing” on page 8-12.
3-82
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Find Global or Multiple Local Minima
Find Global or Multiple Local Minima
In this section...
“Function to Optimize” on page 3-83
“Single Global Minimum Via GlobalSearch” on page 3-85
“Multiple Local Minima Via MultiStart” on page 3-87
Function to Optimize
This example illustrates how GlobalSearch finds a global minimum
efficiently, and how MultiStart finds many more local minima.
The objective function for this example has many local minima and a unique
global minimum. In polar coordinates, the function is
f(r,t) = g(r)h(t),
where
2
sin(2r) sin(3r) sin(4 r)
⎛
⎞ r
g(r) = ⎜ sin(r) −
+
−
+ 4⎟
2
3
4
⎝
⎠ r +1
1⎞
⎛
cos ⎜ 2t − ⎟
2⎠.
⎝
h(t) = 2 + cos(t) +
2
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3-83
3
Using GlobalSearch and MultiStart
The global minimum is at r = 0, with objective function 0. The function g(r)
grows approximately linearly in r, with a repeating sawtooth shape. The
function h(t) has two local minima, one of which is global.
3-84
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Find Global or Multiple Local Minima
Single Global Minimum Via GlobalSearch
1 Write a function file to compute the objective:
function f = sawtoothxy(x,y)
[t r] = cart2pol(x,y); % change to polar coordinates
h = cos(2*t - 1/2)/2 + cos(t) + 2;
g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ...
.*r.^2./(r+1);
f = g.*h;
end
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3-85
3
Using GlobalSearch and MultiStart
2 Create the problem structure. Use the 'sqp' algorithm for fmincon:
problem = createOptimProblem('fmincon',...
'objective',@(x)sawtoothxy(x(1),x(2)),...
'x0',[100,-50],'options',...
optimoptions(@fmincon,'Algorithm','sqp','Display','off'));
The start point is [100,-50] instead of [0,0], so GlobalSearch does not
start at the global solution.
3 Add an artificial bound, and validate the problem structure by running
fmincon:
problem.lb = -Inf([2,1]);
[x fval] = fmincon(problem)
x =
45.6965 -107.6645
fval =
555.6941
4 Create the GlobalSearch object, and set iterative display:
gs = GlobalSearch('Display','iter');
5 Run the solver:
rng(14,'twister') % for reproducibility
[x fval] = run(gs,problem)
Num Pts
3-86
Best
Current
Threshold
Local
Local
Analyzed
F-count
f(x)
Penalty
Penalty
f(x)
exitflag
Procedure
0
200
555.7
555.7
0
Initial Point
200
1479
1.547e-15
1.547e-15
1
Stage 1 Local
300
1580
1.547e-15
5.858e+04
1.074
Stage 2 Search
400
1680
1.547e-15
1.84e+05
4.16
Stage 2 Search
500
1780
1.547e-15
2.683e+04
11.84
Stage 2 Search
600
1880
1.547e-15
1.122e+04
30.95
Stage 2 Search
700
1980
1.547e-15
1.353e+04
65.25
Stage 2 Search
800
2080
1.547e-15
6.249e+04
163.8
Stage 2 Search
900
2180
1.547e-15
4.119e+04
409.2
Stage 2 Search
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Find Global or Multiple Local Minima
950
2372
1.547e-15
477
589.7
387
2
952
2436
1.547e-15
368.4
477
250.7
2
1000
2484
1.547e-15
4.031e+04
530.9
Stage 2 Local
Stage 2 Local
Stage 2 Search
GlobalSearch stopped because it analyzed all the trial points.
3 out of 4 local solver runs converged with a positive local solver exit flag.
x =
1.0e-07 *
0.0414
0.1298
fval =
1.5467e-15
You can get different results, since GlobalSearch is stochastic.
The solver found three local minima, and it found the global minimum near
[0,0].
Multiple Local Minima Via MultiStart
1 Write a function file to compute the objective:
function f = sawtoothxy(x,y)
[t r] = cart2pol(x,y); % change to polar coordinates
h = cos(2*t - 1/2)/2 + cos(t) + 2;
g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ...
.*r.^2./(r+1);
f = g.*h;
end
2 Create the problem structure. Use the fminunc solver with the Algorithm
option set to 'quasi-newton'. The reasons for these choices are:
• The problem is unconstrained. Therefore, fminunc is the appropriate
solver; see “Optimization Decision Table” in the Optimization Toolbox
documentation.
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3-87
3
Using GlobalSearch and MultiStart
• The default fminunc algorithm requires a gradient; see “Choosing the
Algorithm” in the Optimization Toolbox documentation. Therefore, set
Algorithm to 'quasi-newton'.
problem = createOptimProblem('fminunc',...
'objective',@(x)sawtoothxy(x(1),x(2)),...
'x0',[100,-50],'options',...
optimoptions(@fminunc,'Algorithm','quasi-newton','Display','off'));
3 Validate the problem structure by running it:
[x fval] = fminunc(problem)
x =
8.4420 -110.2602
fval =
435.2573
4 Create a default MultiStart object:
ms = MultiStart;
5 Run the solver for 50 iterations, recording the local minima:
[x fval eflag output manymins] = run(ms,problem,50)
MultiStart completed some of the runs from the start points.
16 out of 50 local solver runs converged with a positive
local solver exit flag.
x =
-379.3434
559.6154
fval =
1.7590e+003
eflag =
2
3-88
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Find Global or Multiple Local Minima
output =
funcCount:
localSolverTotal:
localSolverSuccess:
localSolverIncomplete:
localSolverNoSolution:
message:
manymins =
1x16 GlobalOptimSolution
7803
50
16
34
0
'MultiStart completed some of the runs from
Properties:
X
Fval
Exitflag
Output
X0
You can get different results, since MultiStart is stochastic.
The solver did not find the global minimum near [0,0]. It found 16 distinct
local minima.
6 Plot the function values at the local minima:
hist([manymins.Fval])
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3-89
3
Using GlobalSearch and MultiStart
Plot the function values at the three best points:
bestf = [manymins.Fval];
hist(bestf(1:3))
MultiStart started fminunc from start points with components uniformly
distributed between –1000 and 1000. fminunc often got stuck in one of
the many local minima. fminunc exceeded its iteration limit or function
evaluation limit 34 times.
3-90
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Optimize Using Only Feasible Start Points
Optimize Using Only Feasible Start Points
You can set the StartPointsToRun option so that MultiStart and
GlobalSearch use only start points that satisfy inequality constraints. This
option can speed your optimization, since the local solver does not have to
search for a feasible region. However, the option can cause the solvers to
miss some basins of attraction.
There are three settings for the StartPointsToRun option:
• all — Accepts all start points
• bounds — Rejects start points that do not satisfy bounds
• bounds-ineqs — Rejects start points that do not satisfy bounds or
inequality constraints
For example, suppose your objective function is
function y = tiltcircle(x)
vx = x(:)-[4;4]; % ensure vx is in column form
y = vx'*[1;1] + sqrt(16 - vx'*vx); % complex if norm(x-[4;4])>4
tiltcircle returns complex values for norm(x - [4 4]) > 4.
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3-91
3
Using GlobalSearch and MultiStart
Write a constraint function that is positive on the set where norm(x - [4
4]) > 4
function [c ceq] = myconstraint(x)
ceq = [];
cx = x(:) - [4;4]; % ensure x is a column vector
c = cx'*cx - 16; % negative where tiltcircle(x) is real
Set GlobalSearch to use only start points satisfying inequality constraints:
gs = GlobalSearch('StartPointsToRun','bounds-ineqs');
To complete the example, create a problem structure and run the solver:
3-92
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Optimize Using Only Feasible Start Points
opts = optimoptions(@fmincon,'Algorithm','interior-point');
problem = createOptimProblem('fmincon',...
'x0',[4 4],'objective',@tiltcircle,...
'nonlcon',@myconstraint,'lb',[-10 -10],...
'ub',[10 10],'options',opts);
rng(7,'twister'); % for reproducibility
[x,fval,exitflag,output,solutionset] = run(gs,problem)
GlobalSearch stopped because it analyzed all the trial points.
All 5 local solver runs converged with a positive local solver exit flag.
x =
1.1716
1.1716
fval =
-5.6530
exitflag =
1
output =
funcCount:
localSolverTotal:
localSolverSuccess:
localSolverIncomplete:
localSolverNoSolution:
message:
3256
5
5
0
0
'GlobalSearch stopped because it analyzed all th
solutionset =
1x4 GlobalOptimSolution array with properties:
X
Fval
Exitflag
Output
X0
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3-93
3
Using GlobalSearch and MultiStart
tiltcircle With Local Minima
The tiltcircle function has just one local minimum. Yet GlobalSearch
(fmincon) stops at several points. Does this mean fmincon makes an error?
The reason that fmincon stops at several boundary points is subtle. The
tiltcircle function has an infinite gradient on the boundary, as you can see
from a one-dimensional calculation:
d
−x
= ±∞ at x = 4.
16 − x2 =
dx
16 − x2
3-94
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Optimize Using Only Feasible Start Points
So there is a huge gradient normal to the boundary. This gradient overwhelms
the small additional tilt from the linear term. As far as fmincon can tell,
boundary points are stationary points for the constrained problem.
This behavior can arise whenever you have a function that has a square root.
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3-95
3
Using GlobalSearch and MultiStart
MultiStart Using lsqcurvefit or lsqnonlin
This example shows how to fit a function to data using lsqcurvefit together
with MultiStart.
Many fitting problems have multiple local solutions. MultiStart can help
find the global solution, meaning the best fit. While you can use lsqnonlin
as the local solver, this example uses lsqcurvefit simply because it has
a convenient syntax.
The model is
y  a  bx1 sin  cx2  d  ,
where the input data is x = (x1x2), and the parameters a, b, c, and d are the
unknown model coefficients.
Step 1. Create the objective function.
Write an anonymous function that takes a data matrix xdata with N rows
and two columns, and returns a response vector with N rows. It also takes a
coefficient matrix p, corresponding to the coefficient vector (a,b,c,d).
fitfcn = @(p,xdata)p(1) + p(2)*xdata(:,1).*sin(p(3)*xdata(:,2)+p(4));
Step 2. Create the training data.
Create 200 data points and responses. Use the values a = –3, b = 1/4, c = 1/2,
and d = 1. Include random noise in the response.
rng default % for reproducibility
N = 200; % number of data points
preal = [-3,1/4,1/2,1]; % real coefficients
xdata = 5*rand(N,2); % data points
ydata = fitfcn(preal,xdata) + 0.1*randn(N,1); % response data with noise
Step 3. Set bounds and initial point.
3-96
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MultiStart Using lsqcurvefit or lsqnonlin
Set bounds for lsqcurvefit. There is no reason for d to exceed π in absolute
value, because the sine function takes values in its full range over any
interval of width 2π. Assume that the coefficient c must be smaller than 20 in
absolute value, because allowing a very high frequency can cause unstable
responses or spurious convergence.
lb = [-Inf,-Inf,-20,-pi];
ub = [Inf,Inf,20,pi];
Set the initial point arbitrarily to (5,5,5,0).
p0 = 5*ones(1,4); % Arbitrary initial point
p0(4) = 0; % so the initial point satisfies the bounds
Step 4. Find the best local fit.
Fit the parameters to the data, starting at p0.
[xfitted,errorfitted] = lsqcurvefit(fitfcn,p0,xdata,ydata,lb,ub)
Local minimum possible.
lsqcurvefit stopped because the final change in the sum of squares relative
its initial value is less than the default value of the function tolerance.
xfitted =
-2.6149
-0.0238
6.0191
-1.6998
errorfitted =
28.2524
lsqcurvefit found a local solution that is not particularly close to the model
parameter values (–3,1/4,1/2,1).
Step 5. Set up the problem for MultiStart.
Create a problem structure so MultiStart can solve the same problem.
problem = createOptimProblem('lsqcurvefit','x0',p0,'objective',fitfcn,...
'lb',lb,'ub',ub,'xdata',xdata,'ydata',ydata);
Step 6. Find a global solution.
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3-97
3
Using GlobalSearch and MultiStart
Solve the fitting problem using MultiStart with 50 iterations. Plot the
smallest error as the number of MultiStart iterations.
ms = MultiStart('PlotFcns',@gsplotbestf);
[xmulti,errormulti] = run(ms,problem,50)
MultiStart completed the runs from all start points.
All 50 local solver runs converged with a positive local solver exit flag.
xmulti =
-2.9852
-0.2472
-0.4968
-1.0438
errormulti =
1.6464
3-98
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MultiStart Using lsqcurvefit or lsqnonlin
MultiStart found a global solution near the parameter values
(–3,–1/4,–1/2,–1). (This is equivalent to a solution near preal = (–3,1/4,1/2,1),
because changing the sign of all the coefficients except the first gives the same
numerical values of fitfcn.) The norm of the residual error decreased from
about 28 to about 1.6, a decrease of more than a factor of 10.
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3-99
3
Using GlobalSearch and MultiStart
Parallel MultiStart
In this section...
“Steps for Parallel MultiStart” on page 3-100
“Speedup with Parallel Computing” on page 3-102
Steps for Parallel MultiStart
If you have a multicore processor or access to a processor network, you can
use Parallel Computing Toolbox™ functions with MultiStart. This example
shows how to find multiple minima in parallel for a problem, using a processor
with two cores. The problem is the same as in “Multiple Local Minima Via
MultiStart” on page 3-87.
1 Write a function file to compute the objective:
function f = sawtoothxy(x,y)
[t r] = cart2pol(x,y); % change to polar coordinates
h = cos(2*t - 1/2)/2 + cos(t) + 2;
g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ...
.*r.^2./(r+1);
f = g.*h;
end
2 Create the problem structure:
problem = createOptimProblem('fminunc',...
'objective',@(x)sawtoothxy(x(1),x(2)),...
'x0',[100,-50],'options',...
optimoptions(@fminunc,'Algorithm','quasi-newton'));
3 Validate the problem structure by running it:
[x fval] = fminunc(problem)
x =
8.4420 -110.2602
fval =
3-100
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Parallel MultiStart
435.2573
4 Create a MultiStart object, and set the object to use parallel processing
and iterative display:
ms = MultiStart('UseParallel',true,'Display','iter');
5 Set up parallel processing:
parpool
Starting parpool using the 'local' profile ... connected to 4 workers.
ans =
Pool with properties:
Connected:
NumWorkers:
Cluster:
AttachedFiles:
IdleTimeout:
SpmdEnabled:
true
4
local
{}
30 minute(s) (30 minutes remaining)
true
6 Run the problem on 50 start points:
[x fval eflag output manymins] = run(ms,problem,50);
Running the local solvers in parallel.
Run
Index
17
16
34
33
Local
Local
Local
exitflag
f(x)
# iter
2
3953
4
0
1331
45
0
7271
54
2
8249
4
... Many iterations omitted ...
47
2
2740
5
35
0
8501
48
50
0
1225
40
Local
F-count
21
201
201
18
21
201
201
First-order
optimality
0.1626
65.02
520.9
2.968
0.0422
424.8
21.89
MultiStart completed some of the runs from the start points.
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3-101
3
Using GlobalSearch and MultiStart
17 out of 50 local solver runs converged with a positive
local solver exit flag.
Notice that the run indexes look random. Parallel MultiStart runs its start
points in an unpredictable order.
Notice that MultiStart confirms parallel processing in the first line of output,
which states: “Running the local solvers in parallel.”
7 When finished, shut down the parallel environment:
delete(gcp)
Parallel pool using the 'local' profile is shutting down.
For an example of how to obtain better solutions to this problem, see
“Example: Searching for a Better Solution” on page 3-70. You can use parallel
processing along with the techniques described in that example.
Speedup with Parallel Computing
The results of MultiStart runs are stochastic. The timing of runs is
stochastic, too. Nevertheless, some clear trends are apparent in the following
table. The data for the table came from one run at each number of start
points, on a machine with two cores.
Start Points
Parallel Seconds
Serial Seconds
50
3.6
3.4
100
4.9
5.7
200
8.3
10
500
16
23
1000
31
46
Parallel computing can be slower than serial when you use only a few start
points. As the number of start points increases, parallel computing becomes
increasingly more efficient than serial.
3-102
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Parallel MultiStart
There are many factors that affect speedup (or slowdown) with parallel
processing. For more information, see “Improving Performance with Parallel
Computing” in the Optimization Toolbox documentation.
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3-103
3
Using GlobalSearch and MultiStart
Isolated Global Minimum
In this section...
“Difficult-To-Locate Global Minimum” on page 3-104
“Default Settings Cannot Find the Global Minimum — Add Bounds” on
page 3-106
“GlobalSearch with Bounds and More Start Points” on page 3-106
“MultiStart with Bounds and Many Start Points” on page 3-107
“MultiStart Without Bounds, Widely Dispersed Start Points” on page 3-108
“MultiStart with a Regular Grid of Start Points” on page 3-109
“MultiStart with Regular Grid and Promising Start Points” on page 3-109
Difficult-To-Locate Global Minimum
Finding a start point in the basin of attraction of the global minimum can be
difficult when the basin is small or when you are unsure of the location of the
minimum. To solve this type of problem you can:
• Add sensible bounds
• Take a huge number of random start points
• Make a methodical grid of start points
• For an unconstrained problem, take widely dispersed random start points
This example shows these methods and some variants.
The function –sech(x) is nearly 0 for all |x| > 5, and –sech(0) = –1. The
example is a two-dimensional version of the sech function, with one minimum
at [1,1], the other at [1e5,-1e5]:
f(x,y) = –10sech(|x – (1,1)|) – 20sech(.0003(|x – (1e5,–1e5)|) – 1.
f has a global minimum of –21 at (1e5,–1e5), and a local minimum of –11
at (1,1).
3-104
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Isolated Global Minimum
The minimum at (1e5,–1e5) shows as a narrow spike. The minimum at (1,1)
does not show since it is too narrow.
The following sections show various methods of searching for the global
minimum. Some of the methods are not successful on this problem.
Nevertheless, you might find each method useful for different problems.
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3-105
3
Using GlobalSearch and MultiStart
Default Settings Cannot Find the Global Minimum
— Add Bounds
GlobalSearch and MultiStart cannot find the global minimum using default
global options, since the default start point components are in the range
(–9999,10001) for GlobalSearch and (–1000,1000) for MultiStart.
With additional bounds of –1e6 and 1e6 in problem, GlobalSearch usually
does not find the global minimum:
x1 = [1;1];x2 = [1e5;-1e5];
f = @(x)-10*sech(norm(x(:)-x1)) -20*sech((norm(x(:)-x2))*3e-4) -1;
opts = optimoptions(@fmincon,'Algorithm','active-set');
problem = createOptimProblem('fmincon','x0',[0,0],'objective',f,...
'lb',[-1e6;-1e6],'ub',[1e6;1e6],'options',opts);
gs = GlobalSearch;
rng(14,'twister') % for reproducibility
[xfinal fval] = run(gs,problem)
GlobalSearch stopped because it analyzed all the trial points.
All 32 local solver runs converged with a positive
local solver exit flag.
xfinal =
1.0000
1.0000
fval =
-11.0000
GlobalSearch with Bounds and More Start Points
To find the global minimum, you can search more points. This example uses
1e5 start points, and a MaxTime of 300 s:
gs.NumTrialPoints = 1e5;
gs.MaxTime = 300;
[xg fvalg] = run(gs,problem)
GlobalSearch stopped because maximum time is exceeded.
3-106
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Isolated Global Minimum
GlobalSearch called the local solver 2186 times before exceeding
the clock time limit (MaxTime = 300 seconds).
1943 local solver runs converged with a positive
local solver exit flag.
xg =
1.0e+04 *
10.0000 -10.0000
fvalg =
-21.0000
In this case, GlobalSearch found the global minimum.
MultiStart with Bounds and Many Start Points
Alternatively, you can search using MultiStart with many start points. This
example uses 1e5 start points, and a MaxTime of 300 s:
ms = MultiStart(gs);
[xm fvalm] = run(ms,problem,1e5)
MultiStart stopped because maximum time was exceeded.
MultiStart called the local solver 17266 times before exceeding
the clock time limit (MaxTime = 300 seconds).
17266 local solver runs converged with a positive
local solver exit flag.
xm =
1.0000
1.0000
fvalm =
-11.0000
In this case, MultiStart failed to find the global minimum.
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3-107
3
Using GlobalSearch and MultiStart
MultiStart Without Bounds, Widely Dispersed Start
Points
You can also use MultiStart to search an unbounded region to find the global
minimum. Again, you need many start points to have a good chance of finding
the global minimum.
The first five lines of code generate 10,000 widely dispersed random
start points using the method described in “Widely Dispersed Points for
Unconstrained Components” on page 3-69. newprob is a problem structure
using the fminunc local solver and no bounds:
rng(0,'twister') % for reproducibility
u = rand(1e4,1);
u = 1./u;
u = exp(u) - exp(1);
s = rand(1e4,1)*2*pi;
stpts = [u.*cos(s),u.*sin(s)];
startpts = CustomStartPointSet(stpts);
opts = optimoptions(@fminunc,'Algorithm','quasi-newton');
newprob = createOptimProblem('fminunc','x0',[0;0],'objective',f,...
'options',opts);
[xcust fcust] = run(ms,newprob,startpts)
MultiStart completed the runs from all start points.
All 10000 local solver runs converged with a positive
local solver exit flag.
xcust =
1.0e+05 *
1.0000
-1.0000
fcust =
-21.0000
In this case, MultiStart found the global minimum.
3-108
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Isolated Global Minimum
MultiStart with a Regular Grid of Start Points
You can also use a grid of start points instead of random start points. To
learn how to construct a regular grid for more dimensions, or one that has
small perturbations, see “Uniform Grid” on page 3-67 or “Perturbed Grid”
on page 3-68.
xx = -1e6:1e4:1e6;
[xxx yyy] = meshgrid(xx,xx);
z = [xxx(:),yyy(:)];
bigstart = CustomStartPointSet(z);
[xgrid fgrid] = run(ms,newprob,bigstart)
MultiStart completed the runs from all start points.
All 10000 local solver runs converged with a positive
local solver exit flag.
xgrid =
1.0e+004 *
10.0000
-10.0000
fgrid =
-21.0000
In this case, MultiStart found the global minimum.
MultiStart with Regular Grid and Promising Start
Points
Making a regular grid of start points, especially in high dimensions, can use
an inordinate amount of memory or time. You can filter the start points to
run only those with small objective function value.
To perform this filtering most efficiently, write your objective function in a
vectorized fashion. For information, see “Write a Vectorized Function” on
page 2-3 or “Vectorize the Objective and Constraint Functions” on page 4-85.
The following function handle computes a vector of objectives based on an
input matrix whose rows represent start points:
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3-109
3
Using GlobalSearch and MultiStart
x1 = [1;1];x2 = [1e5;-1e5];
g = @(x) -10*sech(sqrt((x(:,1)-x1(1)).^2 + (x(:,2)-x1(2)).^2)) ...
-20*sech(sqrt((x(:,1)-x2(1)).^2 + (x(:,2)-x2(2)).^2))-1;
Suppose you want to run the local solver only for points where the value is
less than –2. Start with a denser grid than in “MultiStart with a Regular
Grid of Start Points” on page 3-109, then filter out all the points with high
function value:
xx = -1e6:1e3:1e6;
[xxx yyy] = meshgrid(xx,xx);
z = [xxx(:),yyy(:)];
idx = g(z) < -2; % index of promising start points
zz = z(idx,:);
smallstartset = CustomStartPointSet(zz);
opts = optimoptions(@fminunc,'Algorithm','quasi-newton','Display','off');
newprobg = createOptimProblem('fminunc','x0',[0,0],...
'objective',g,'options',opts);
% row vector x0 since g expects rows
[xfew ffew] = run(ms,newprobg,smallstartset)
MultiStart completed the runs from all start points.
All 2 local solver runs converged with a positive
local solver exit flag.
xfew =
100000
-100000
ffew =
-21
In this case, MultiStart found the global minimum. There are only two start
points in smallstartset, one of which is the global minimum.
3-110
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4
Using Direct Search
• “What Is Direct Search?” on page 4-2
• “Optimize Using Pattern Search” on page 4-3
• “Optimize Using the GPS Algorithm” on page 4-8
• “Pattern Search Terminology” on page 4-12
• “How Pattern Search Polling Works” on page 4-15
• “Searching and Polling” on page 4-27
• “Setting Solver Tolerances” on page 4-32
• “Search and Poll” on page 4-33
• “Nonlinear Constraint Solver Algorithm” on page 4-39
• “Custom Plot Function” on page 4-42
• “Set Options” on page 4-48
• “Polling Types” on page 4-51
• “Set Mesh Options” on page 4-63
• “Constrained Minimization Using patternsearch” on page 4-74
• “Use Cache” on page 4-81
• “Vectorize the Objective and Constraint Functions” on page 4-85
• “Optimize an ODE in Parallel” on page 4-91
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4
Using Direct Search
What Is Direct Search?
Direct search is a method for solving optimization problems that does not
require any information about the gradient of the objective function. Unlike
more traditional optimization methods that use information about the
gradient or higher derivatives to search for an optimal point, a direct search
algorithm searches a set of points around the current point, looking for one
where the value of the objective function is lower than the value at the current
point. You can use direct search to solve problems for which the objective
function is not differentiable, or is not even continuous.
Global Optimization Toolbox functions include three direct search algorithms
called the generalized pattern search (GPS) algorithm, the generating set
search (GSS) algorithm, and the mesh adaptive search (MADS) algorithm.
All are pattern search algorithms that compute a sequence of points that
approach an optimal point. At each step, the algorithm searches a set of
points, called a mesh, around the current point—the point computed at the
previous step of the algorithm. The mesh is formed by adding the current
point to a scalar multiple of a set of vectors called a pattern. If the pattern
search algorithm finds a point in the mesh that improves the objective
function at the current point, the new point becomes the current point at the
next step of the algorithm.
The GPS algorithm uses fixed direction vectors. The GSS algorithm is
identical to the GPS algorithm, except when there are linear constraints, and
when the current point is near a linear constraint boundary. The MADS
algorithm uses a random selection of vectors to define the mesh. For details,
see “Patterns” on page 4-12.
4-2
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Optimize Using Pattern Search
Optimize Using Pattern Search
In this section...
“Call patternsearch at the Command Line” on page 4-3
“Pattern Search on Unconstrained Problems” on page 4-3
“Pattern Search on Constrained Problems” on page 4-4
“Additional Output Arguments” on page 4-5
“Use the Optimization App for Pattern Search” on page 4-5
Call patternsearch at the Command Line
To perform a pattern search on an unconstrained problem at the command
line, call the function patternsearch with the syntax
[x fval] = patternsearch(@objfun, x0)
where
• @objfun is a handle to the objective function.
• x0 is the starting point for the pattern search.
The results are:
• x — Point at which the final value is attained
• fval — Final value of the objective function
Pattern Search on Unconstrained Problems
For an unconstrained problem, call patternsearch with the syntax
[x fval] = patternsearch(@objectfun, x0)
The output arguments are
• x — The final point
• fval — The value of the objective function at x
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4-3
4
Using Direct Search
The required input arguments are
• @objectfun — A function handle to the objective function objectfun,
which you can write as a function file. See “Compute Objective Functions”
on page 2-2 to learn how to do this.
• x0 — The initial point for the pattern search algorithm.
As an example, you can run the example described in “Optimize Using the
GPS Algorithm” on page 4-8 from the command line by entering
[x fval] = patternsearch(@ps_example, [2.1 1.7])
This returns
Optimization terminated: mesh size less than options.TolMesh.
x =
-4.7124
-0.0000
fval =
-2.0000
Pattern Search on Constrained Problems
If your problem has constraints, use the syntax
[x fval] = patternsearch(@objfun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
where
• A is a matrix and b is vector that represent inequality constraints of the
form A·x ≤ b.
• Aeq is a matrix and beq is a vector that represent equality constraints of
the form Aeq·x = beq.
• lb and ub are vectors representing bound constraints of the form lb ≤ x and
x ≤ ub, respectively.
• nonlcon is a function that returns the nonlinear equality and inequality
vectors, c and ceq, respectively. The function is minimized such that
c(x) ≤ 0 and ceq(x) = 0.
4-4
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Optimize Using Pattern Search
You only need to pass in the constraints that are part of the problem. For
example, if there are no bound constraints or a nonlinear constraint function,
use the syntax
[x fval] = patternsearch(@objfun,x0,A,b,Aeq,beq)
Use empty brackets [] for constraint arguments that are not needed for the
problem. For example, if there are no inequality constraints or a nonlinear
constraint function, use the syntax
[x fval] = patternsearch(@objfun,x0,[],[],Aeq,beq,lb,ub)
Additional Output Arguments
To get more information about the performance of the pattern search, you can
call patternsearch with the syntax
[x fval exitflag output] = patternsearch(@objfun,x0)
Besides x and fval, this returns the following additional output arguments:
• exitflag — Integer indicating whether the algorithm was successful
• output — Structure containing information about the performance of the
solver
For more information about these arguments, see the patternsearch
reference page.
Use the Optimization App for Pattern Search
To open the Optimization app, enter
optimtool('patternsearch')
at the command line, or enter optimtool and then choose patternsearch
from the Solver menu.
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Using Direct Search
Set options
Expand or contract help
Choose solver
Enter problem
and constraints
Run solver
View results
See final point
You can also start the tool from the MATLAB Apps tab.
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Optimize Using Pattern Search
To use the Optimization app, first enter the following information:
• Objective function — The objective function you want to minimize. Enter
the objective function in the form @objfun, where objfun.m is a file that
computes the objective function. The @ sign creates a function handle to
objfun.
• Start point — The initial point at which the algorithm starts the
optimization.
In the Constraints pane, enter linear constraints, bounds, or a nonlinear
constraint function as a function handle for the problem. If the problem is
unconstrained, leave these fields blank.
Then, click Start. The tool displays the results of the optimization in the
Run solver and view results pane.
In the Options pane, set the options for the pattern search. To view the
options in a category, click the + sign next to it.
“Finding the Minimum of the Function” on page 4-8 gives an example of using
the Optimization app.
For more information, see “Optimization App” in the Optimization Toolbox
documentation.
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Using Direct Search
Optimize Using the GPS Algorithm
In this section...
“Objective Function” on page 4-8
“Finding the Minimum of the Function” on page 4-8
“Plotting the Objective Function Values and Mesh Sizes” on page 4-10
Objective Function
This example uses the objective function, ps_example, which is included
with Global Optimization Toolbox software. View the code for the function
by entering
type ps_example
The following figure shows a plot of the function.
Finding the Minimum of the Function
To find the minimum of ps_example, perform the following steps:
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Optimize Using the GPS Algorithm
1 Enter
optimtool
and then choose the patternsearch solver.
2 In the Objective function field of the Optimization app, enter
@ps_example.
3 In the Start point field, type [2.1 1.7].
Leave the fields in the Constraints pane blank because the problem is
unconstrained.
4 Click Start to run the pattern search.
The Run solver and view results pane displays the results of the pattern
search.
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Using Direct Search
The reason the optimization terminated is that the mesh size became smaller
than the acceptable tolerance value for the mesh size, defined by the Mesh
tolerance parameter in the Stopping criteria pane. The minimum function
value is approximately –2. The Final point pane displays the point at which
the minimum occurs.
Plotting the Objective Function Values and Mesh Sizes
To see the performance of the pattern search, display plots of the best function
value and mesh size at each iteration. First, select the following check boxes
in the Plot functions pane:
• Best function value
• Mesh size
Then click Start to run the pattern search. This displays the following plots.
4-10
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Optimize Using the GPS Algorithm
The upper plot shows the objective function value of the best point at each
iteration. Typically, the objective function values improve rapidly at the early
iterations and then level off as they approach the optimal value.
The lower plot shows the mesh size at each iteration. The mesh size increases
after each successful iteration and decreases after each unsuccessful one,
explained in “How Pattern Search Polling Works” on page 4-15.
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Using Direct Search
Pattern Search Terminology
In this section...
“Patterns” on page 4-12
“Meshes” on page 4-13
“Polling” on page 4-14
“Expanding and Contracting” on page 4-14
Patterns
A pattern is a set of vectors {vi} that the pattern search algorithm uses to
determine which points to search at each iteration. The set {vi} is defined by
the number of independent variables in the objective function, N, and the
positive basis set. Two commonly used positive basis sets in pattern search
algorithms are the maximal basis, with 2N vectors, and the minimal basis,
with N+1 vectors.
With GPS, the collection of vectors that form the pattern are fixed-direction
vectors. For example, if there are three independent variables in the
optimization problem, the default for a 2N positive basis consists of the
following pattern vectors:
v1 = [1 0 0] v2 = [0 1 0] v3 = [0 0 1]
v4 = [ −1 0 0] v5 = [0 −1 0] v6 = [0 0 −1]
An N+1 positive basis consists of the following default pattern vectors.
v1 = [1 0 0]
v2 = [0 1 0]
v3 = [0 0 1]
v4 = [ −1 −1 −1]
With GSS, the pattern is identical to the GPS pattern, except when there are
linear constraints and the current point is near a constraint boundary. For a
description of the way in which GSS forms a pattern with linear constraints,
see Kolda, Lewis, and Torczon [1]. The GSS algorithm is more efficient than
the GPS algorithm when you have linear constraints. For an example showing
the efficiency gain, see “Compare the Efficiency of Poll Options” on page 4-56.
4-12
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Pattern Search Terminology
With MADS, the collection of vectors that form the pattern are randomly
selected by the algorithm. Depending on the poll method choice, the number
of vectors selected will be 2N or N+1. As in GPS, 2N vectors consist of N
vectors and their N negatives, while N+1 vectors consist of N vectors and one
that is the negative of the sum of the others.
References
[1] Kolda, Tamara G., Robert Michael Lewis, and Virginia Torczon.
“A generating set direct search augmented Lagrangian algorithm for
optimization with a combination of general and linear constraints.” Technical
Report SAND2006-5315, Sandia National Laboratories, August 2006.
Meshes
At each step, patternsearch searches a set of points, called a mesh, for a
point that improves the objective function. patternsearch forms the mesh by
1 Generating a set of vectors {di} by multiplying each pattern vector vi by a
scalar Δm. Δm is called the mesh size.
2 Adding the { di } to the current point—the point with the best objective
function value found at the previous step.
For example, using the GPS algorithm. suppose that:
• The current point is [1.6 3.4].
• The pattern consists of the vectors
v1 = [1 0]
v2 = [0 1]
v3 = [ −1 0]
v4 = [0 −1]
• The current mesh size Δm is 4.
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Using Direct Search
The algorithm multiplies the pattern vectors by 4 and adds them to the
current point to obtain the following mesh.
[1.6
[1.6
[1.6
[1.6
3.4]
3.4]
3.4]
3.4]
+
+
+
+
4*[1 0] = [5.6 3.4]
4*[0 1] = [1.6 7.4]
4*[-1 0] = [-2.4 3.4]
4*[0 -1] = [1.6 -0.6]
The pattern vector that produces a mesh point is called its direction.
Polling
At each step, the algorithm polls the points in the current mesh by computing
their objective function values. When the Complete poll option has the
(default) setting Off, the algorithm stops polling the mesh points as soon as it
finds a point whose objective function value is less than that of the current
point. If this occurs, the poll is called successful and the point it finds becomes
the current point at the next iteration.
The algorithm only computes the mesh points and their objective function
values up to the point at which it stops the poll. If the algorithm fails to find a
point that improves the objective function, the poll is called unsuccessful and
the current point stays the same at the next iteration.
When the Complete poll option has the setting On, the algorithm computes
the objective function values at all mesh points. The algorithm then compares
the mesh point with the smallest objective function value to the current
point. If that mesh point has a smaller value than the current point, the
poll is successful.
Expanding and Contracting
After polling, the algorithm changes the value of the mesh size Δm. The
default is to multiply Δm by 2 after a successful poll, and by 0.5 after an
unsuccessful poll.
4-14
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How Pattern Search Polling Works
How Pattern Search Polling Works
In this section...
“Context” on page 4-15
“Successful Polls” on page 4-16
“An Unsuccessful Poll” on page 4-19
“Displaying the Results at Each Iteration” on page 4-20
“More Iterations” on page 4-21
“Poll Method” on page 4-22
“Complete Poll” on page 4-24
“Stopping Conditions for the Pattern Search” on page 4-24
“Robustness of Pattern Search” on page 4-26
Context
patternsearch finds a sequence of points, x0, x1, x2, ... , that approach an
optimal point. The value of the objective function either decreases or remains
the same from each point in the sequence to the next. This section explains
how pattern search works for the function described in “Optimize Using the
GPS Algorithm” on page 4-8.
To simplify the explanation, this section describes how the generalized
pattern search (GPS) works using a maximal positive basis of 2N, with Scale
set to Off in Mesh options.
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4
Using Direct Search
This section does not show how the patternsearch algorithm works
with bounds or linear constraints. For bounds and linear constraints,
patternsearch modifies poll points to be feasible, meaning to satisfy all
bounds and linear constraints.
This section does not encompass nonlinear constraints. To understand how
patternsearch works with nonlinear constraints, see “Nonlinear Constraint
Solver Algorithm” on page 4-39.
The problem setup:
Successful Polls
The pattern search begins at the initial point x0 that you provide. In this
example, x0 = [2.1 1.7].
4-16
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How Pattern Search Polling Works
Iteration 1
At the first iteration, the mesh size is 1 and the GPS algorithm adds the
pattern vectors to the initial point x0 = [2.1 1.7] to compute the following
mesh points:
[1 0] + x0 = [3.1 1.7]
[0 1] + x0 = [2.1 2.7]
[-1 0] + x0 = [1.1 1.7]
[0 -1] + x0 = [2.1 0.7]
The algorithm computes the objective function at the mesh points in the order
shown above. The following figure shows the value of ps_example at the
initial point and mesh points.
Objective Function Values at Initial Point and Mesh Points
3
Initial point x0
Mesh points
5.6347
2.5
2
4.5146
4.6347
4.782
1.5
1
3.6347
0.5
1
1.5
2
2.5
3
3.5
First polled point that improves the objective function
The algorithm polls the mesh points by computing their objective function
values until it finds one whose value is smaller than 4.6347, the value at x0.
In this case, the first such point it finds is [1.1 1.7], at which the value of
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Using Direct Search
the objective function is 4.5146, so the poll at iteration 1 is successful. The
algorithm sets the next point in the sequence equal to
x1 = [1.1 1.7]
Note By default, the GPS pattern search algorithm stops the current
iteration as soon as it finds a mesh point whose fitness value is smaller than
that of the current point. Consequently, the algorithm might not poll all the
mesh points. You can make the algorithm poll all the mesh points by setting
Complete poll to On.
Iteration 2
After a successful poll, the algorithm multiplies the current mesh size by 2,
the default value of Expansion factor in the Mesh options pane. Because
the initial mesh size is 1, at the second iteration the mesh size is 2. The mesh
at iteration 2 contains the following points:
2*[1 0] + x1 = [3.1 1.7]
2*[0 1] + x1 = [1.1 3.7]
2*[-1 0] + x1 = [-0.9 1.7]
2*[0 -1] + x1 = [1.1 -0.3]
The following figure shows the point x1 and the mesh points, together with
the corresponding values of ps_example.
4-18
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How Pattern Search Polling Works
Objective Function Values at x1 and Mesh Points
4
x1
Mesh points
6.5416
3.5
3
2.5
2
3.25
4.5146
4.7282
1.5
1
0.5
0
3.1146
−0.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
The algorithm polls the mesh points until it finds one whose value is smaller
than 4.5146, the value at x1. The first such point it finds is [-0.9 1.7], at
which the value of the objective function is 3.25, so the poll at iteration 2 is
again successful. The algorithm sets the second point in the sequence equal to
x2 = [-0.9 1.7]
Because the poll is successful, the algorithm multiplies the current mesh size
by 2 to get a mesh size of 4 at the third iteration.
An Unsuccessful Poll
By the fourth iteration, the current point is
x3 = [-4.9 1.7]
and the mesh size is 8, so the mesh consists of the points
8*[1 0] + x3 = [3.1 1.7]
8*[0 1] + x3 = [-4.9 9.7]
8*[-1 0] + x3 = [-12.9 1.7]
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4
Using Direct Search
8*[0 -1] + x3 = [-4.9 -1.3]
The following figure shows the mesh points and their objective function values.
Objective Function Values at x3 and Mesh Points
10
x3
Mesh points
7.7351
8
6
4
2
64.11
−0.2649
4.7282
0
−2
−4
−6
−8
4.3351
−10
−5
0
5
At this iteration, none of the mesh points has a smaller objective function
value than the value at x3, so the poll is unsuccessful. In this case, the
algorithm does not change the current point at the next iteration. That is,
x4 = x3;
At the next iteration, the algorithm multiplies the current mesh size by
0.5, the default value of Contraction factor in the Mesh options pane, so
that the mesh size at the next iteration is 4. The algorithm then polls with
a smaller mesh size.
Displaying the Results at Each Iteration
You can display the results of the pattern search at each iteration by setting
Level of display to Iterative in the Display to command window
options. This enables you to evaluate the progress of the pattern search and
to make changes to options if necessary.
4-20
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How Pattern Search Polling Works
With this setting, the pattern search displays information about each iteration
at the command line. The first four iterations are
Iter
0
1
2
3
4
f-count
1
4
7
10
14
f(x)
4.63474
4.51464
3.25
-0.264905
-0.264905
MeshSize
1
2
4
8
4
Method
Successful Poll
Successful Poll
Successful Poll
Refine Mesh
The entry Successful Poll below Method indicates that the current iteration
was successful. For example, the poll at iteration 2 is successful. As a result,
the objective function value of the point computed at iteration 2, displayed
below f(x), is less than the value at iteration 1.
At iteration 4, the entry Refine Mesh tells you that the poll is unsuccessful.
As a result, the function value at iteration 4 remains unchanged from
iteration 3.
By default, the pattern search doubles the mesh size after each successful poll
and halves it after each unsuccessful poll.
More Iterations
The pattern search performs 60 iterations before stopping. The following
plot shows the points in the sequence computed in the first 13 iterations of
the pattern search.
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Using Direct Search
Points at First 13 Iterations of Pattern Search
2
1.5
3
2
1
0
1
0.5
0
−0.5
−1
−6
10
13
6
−5
−4
−3
−2
−1
0
1
2
3
The numbers below the points indicate the first iteration at which
the algorithm finds the point. The plot only shows iteration numbers
corresponding to successful polls, because the best point doesn’t change after
an unsuccessful poll. For example, the best point at iterations 4 and 5 is
the same as at iteration 3.
Poll Method
At each iteration, the pattern search polls the points in the current
mesh—that is, it computes the objective function at the mesh points to see
if there is one whose function value is less than the function value at the
current point. “How Pattern Search Polling Works” on page 4-15 provides
an example of polling. You can specify the pattern that defines the mesh
by the Poll method option. The default pattern, GPS Positive basis 2N,
consists of the following 2N directions, where N is the number of independent
variables for the objective function.
4-22
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How Pattern Search Polling Works
[1 0 0...0]
[0 1 0...0]
...
[0 0 0...1]
[–1 0 0...0]
[0 –1 0...0]
[0 0 0...–1].
For example, if the objective function has three independent variables, the
GPS Positive basis 2N, consists of the following six vectors.
[1 0 0]
[0 1 0]
[0 0 1]
[–1 0 0]
[0 –1 0]
[0 0 –1].
Alternatively, you can set Poll method to GPS Positive basis NP1, the
pattern consisting of the following N + 1 directions.
[1 0 0...0]
[0 1 0...0]
...
[0 0 0...1]
[–1 –1 –1...–1].
For example, if objective function has three independent variables, the GPS
Positive basis Np1, consists of the following four vectors.
[1 0 0]
[0 1 0]
[0 0 1]
[–1 –1 –1].
A pattern search will sometimes run faster using GPS Positive basis Np1
rather than the GPS Positive basis 2N as the Poll method, because
the algorithm searches fewer points at each iteration. Although not being
addressed in this example, the same is true when using the MADS Positive
basis Np1 over the MADS Positive basis 2N, and similarly for GSS. For
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Using Direct Search
example, if you run a pattern search on the example described in “Linearly
Constrained Problem” on page 4-74, the algorithm performs 1588 function
evaluations with GPS Positive basis 2N, the default Poll method, but only
877 function evaluations using GPS Positive basis Np1. For more detail,
see “Compare the Efficiency of Poll Options” on page 4-56.
However, if the objective function has many local minima, using GPS
Positive basis 2N as the Poll method might avoid finding a local
minimum that is not the global minimum, because the search explores more
points around the current point at each iteration.
Complete Poll
By default, if the pattern search finds a mesh point that improves the value
of the objective function, it stops the poll and sets that point as the current
point for the next iteration. When this occurs, some mesh points might not
get polled. Some of these unpolled points might have an objective function
value that is even lower than the first one the pattern search finds.
For problems in which there are several local minima, it is sometimes
preferable to make the pattern search poll all the mesh points at each
iteration and choose the one with the best objective function value. This
enables the pattern search to explore more points at each iteration and
thereby potentially avoid a local minimum that is not the global minimum.
In the Optimization app you can make the pattern search poll the entire
mesh setting Complete poll to On in Poll options. At the command line, use
psoptimset to set the CompletePoll option to 'on'.
Stopping Conditions for the Pattern Search
The criteria for stopping the pattern search algorithm are listed in the
Stopping criteria section of the Optimization app:
4-24
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How Pattern Search Polling Works
The algorithm stops when any of the following conditions occurs:
• The mesh size is less than Mesh tolerance.
• The number of iterations performed by the algorithm reaches the value of
Max iteration.
• The total number of objective function evaluations performed by the
algorithm reaches the value of Max function evaluations.
• The time, in seconds, the algorithm runs until it reaches the value of
Time limit.
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Using Direct Search
• The distance between the point found in two consecutive iterations and the
mesh size are both less than X tolerance.
• The change in the objective function in two consecutive iterations and the
mesh size are both less than Function tolerance.
Nonlinear constraint tolerance is not used as stopping criterion. It
determines the feasibility with respect to nonlinear constraints.
The MADS algorithm uses an additional parameter called the poll parameter,
Δp, in the mesh size stopping criterion:
⎧⎪ N Δ m
Δp = ⎨
⎩⎪ Δ m
for positive basis N + 1 poll
for positive basis 2N poll,
where Δm is the mesh size. The MADS stopping criterion is:
Δp ≤ Mesh tolerance.
Robustness of Pattern Search
The pattern search algorithm is robust in relation to objective function
failures. This means patternsearch tolerates function evaluations resulting
in NaN, Inf, or complex values. When the objective function at the initial point
x0 is a real, finite value, patternsearch treats poll point failures as if the
objective function values are large, and ignores them.
For example, if all points in a poll evaluate to NaN, patternsearch considers
the poll unsuccessful, shrinks the mesh, and reevaluates. If even one point
in a poll evaluates to a smaller value than any seen yet, patternsearch
considers the poll successful, and expands the mesh.
4-26
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Searching and Polling
Searching and Polling
In this section...
“Definition of Search” on page 4-27
“How to Use a Search Method” on page 4-29
“Search Types” on page 4-30
“When to Use Search” on page 4-30
Definition of Search
In patternsearch, a search is an algorithm that runs before a poll. The
search attempts to locate a better point than the current point. (Better means
one with lower objective function value.) If the search finds a better point, the
better point becomes the current point, and no polling is done at that iteration.
If the search does not find a better point, patternsearch performs a poll.
By default, patternsearch does not use search. To search, see “How to Use a
Search Method” on page 4-29.
The figure patternsearch With a Search Method on page 4-28 contains a flow
chart of direct search including using a search method.
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4
Using Direct Search
Start
Stop
Yes
Update
current
point
Done?
No
Search
Yes
Search
enabled?
No
Yes
Success?
Expand mesh
No
Yes
Poll
Success?
No
patternsearch With a Search Method
4-28
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Refine mesh
Searching and Polling
Iteration limit applies to all built-in search methods except those that are poll
methods. If you select an iteration limit for the search method, the search
is enabled until the iteration limit is reached. Afterwards, patternsearch
stops searching and only polls.
How to Use a Search Method
To use search in patternsearch:
• In Optimization app, choose a Search method in the Search pane.
• At the command line, create an options structure with a search method
using psoptimset. For example, to use Latin hypercube search:
opts = psoptimset('SearchMethod',@searchlhs);
For more information, including a list of all built-in search methods,
consult the psoptimset function reference page, and the “Search Options”
on page 9-14 section of the options reference.
You can write your own search method. Use the syntax described in
“Structure of the Search Function” on page 9-17. To use your search method
in a pattern search, give its function handle as the Custom Function
(SearchMethod) option.
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Using Direct Search
Search Types
• Poll methods — You can use any poll method as a search algorithm.
patternsearch conducts one poll step as a search. For this type of search
to be beneficial, your search type should be different from your poll type.
(patternsearch does not search if the selected search method is the same
as the poll type.) Therefore, use a MADS search with a GSS or GPS poll,
or use a GSS or GPS search with a MADS poll.
• fminsearch, also called Nelder-Mead — fminsearch is for unconstrained
problems only. fminsearch runs to its natural stopping criteria; it does not
take just one step. Therefore, use fminsearch for just one iteration. This is
the default setting. To change settings, see “Search Options” on page 9-14.
• ga — ga runs to its natural stopping criteria; it does not take just one step.
Therefore, use ga for just one iteration. This is the default setting. To
change settings, see “Search Options” on page 9-14.
• Latin hypercube search — Described in “Search Options” on page 9-14.
By default, searches 15n points, where n is the number of variables, and
only searches during the first iteration. To change settings, see “Search
Options” on page 9-14.
When to Use Search
There are two main reasons to use a search method:
• To speed an optimization (see “Search Methods for Increased Speed” on
page 4-30)
• To obtain a better local solution, or to obtain a global solution (see “Search
Methods for Better Solutions” on page 4-31)
Search Methods for Increased Speed
Generally, you do not know beforehand whether a search method speeds an
optimization or not. So try a search method when:
• You are performing repeated optimizations on similar problems, or on the
same problem with different parameters.
• You can experiment with different search methods to find a lower solution
time.
4-30
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Searching and Polling
Search does not always speed an optimization. For one example where it does,
see “Search and Poll” on page 4-33.
Search Methods for Better Solutions
Since search methods run before poll methods, using search can be equivalent
to choosing a different starting point for your optimization. This comment
holds for the Nelder-Mead, ga, and Latin hypercube search methods, all of
which, by default, run once at the beginning of an optimization. ga and
Latin hypercube searches are stochastic, and can search through several
basins of attraction.
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Using Direct Search
Setting Solver Tolerances
Tolerance refers to how small a parameter, such a mesh size, can become
before the search is halted or changed in some way. You can specify the value
of the following tolerances:
• Mesh tolerance — When the current mesh size is less than the value of
Mesh tolerance, the algorithm halts.
• X tolerance — After a successful poll, if the distance from the previous
best point to the current best point is less than the value of X tolerance,
the algorithm halts.
• Function tolerance — After a successful poll, if the difference between
the function value at the previous best point and function value at the
current best point is less than the value of Function tolerance, the
algorithm halts.
• Nonlinear constraint tolerance — The algorithm treats a point to be
feasible if constraint violation is less than TolCon.
• Bind tolerance — Bind tolerance applies to linearly constrained problems.
It specifies how close a point must get to the boundary of the feasible
region before a linear constraint is considered to be active. When a linear
constraint is active, the pattern search polls points in directions parallel to
the linear constraint boundary as well as the mesh points.
Usually, you should set Bind tolerance to be at least as large as the
maximum of Mesh tolerance, X tolerance, and Function tolerance.
4-32
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Search and Poll
Search and Poll
In this section...
“Using a Search Method” on page 4-33
“Search Using a Different Solver” on page 4-37
Using a Search Method
In addition to polling the mesh points, the pattern search algorithm can
perform an optional step at every iteration, called search. At each iteration,
the search step applies another optimization method to the current point. If
this search does not improve the current point, the poll step is performed.
The following example illustrates the use of a search method on the problem
described in “Linearly Constrained Problem” on page 4-74. To set up the
example, enter the following commands at the MATLAB prompt to define the
initial point and constraints.
x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = 7;
Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
beq = [84 62 65 1];
Then enter the settings shown in the following figures in the Optimization
app.
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4-33
4
Using Direct Search
For comparison, click Start to run the example without a search method.
This displays the plots shown in the following figure.
4-34
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Search and Poll
To see the effect of using a search method, select MADS Positive Basis 2N
in the Search method field in Search options.
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4
Using Direct Search
This sets the search method to be a pattern search using the pattern for
MADS Positive Basis 2N. Then click Start to run the pattern search. This
displays the following plots.
Note that using the search method reduces the total function
evaluations—from 1206 to 1159—and reduces the number of iterations from
106 to 97.
4-36
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Search and Poll
Search Using a Different Solver
patternsearch takes a long time to minimize Rosenbrock’s function. The
(
function is f ( x) = 100 x2 − x12
)
2
+ (1 − x1 ) .
2
Rosenbrock’s function is described and plotted in “Solve a Constrained
Nonlinear Problem” in the Optimization Toolbox documentation. The
dejong2fcn.m file, which in included in the toolbox, calculates this function.
1 Set patternsearch options to MaxFunEvals = 5000 and MaxIter = 2000:
opts = psoptimset('MaxFunEvals',5000,'MaxIter',2000);
2 Run patternsearch starting from [-1.9 2]:
[x feval eflag output] = patternsearch(@dejong2fcn,...
[-1.9,2],[],[],[],[],[],[],[],opts);
Maximum number of function evaluations exceeded:
increase options.MaxFunEvals.
feval
feval =
0.8560
The optimization did not complete, and the result is not very close to the
optimal value of 0.
3 Set the options to use fminsearch as the search method:
opts = psoptimset(opts,'SearchMethod',@searchneldermead);
4 Rerun the optimization, the results are much better:
[x2 feval2 eflag2 output2] = patternsearch(@dejong2fcn,...
[-1.9,2],[],[],[],[],[],[],[],opts);
Optimization terminated: mesh size less than options.TolMesh.
feval2
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4
Using Direct Search
feval2 =
4.0686e-010
fminsearch is not as closely tied to coordinate directions as the default GPS
patternsearch poll method. Therefore, fminsearch is more efficient at
getting close to the minimum of Rosenbrock’s function. Adding the search
method in this case is effective.
4-38
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Nonlinear Constraint Solver Algorithm
Nonlinear Constraint Solver Algorithm
The pattern search algorithm uses the Augmented Lagrangian Pattern Search
(ALPS) algorithm to solve nonlinear constraint problems. The optimization
problem solved by the ALPS algorithm is
min f ( x)
x
such that
ci ( x) ≤ 0, i = 1 m
ceqi ( x) = 0, i = m + 1 mt
A⋅x ≤ b
Aeq ⋅ x = beq
lb ≤ x ≤ ub,
where c(x) represents the nonlinear inequality constraints, ceq(x) represents
the equality constraints, m is the number of nonlinear inequality constraints,
and mt is the total number of nonlinear constraints.
The ALPS algorithm attempts to solve a nonlinear optimization problem
with nonlinear constraints, linear constraints, and bounds. In this approach,
bounds and linear constraints are handled separately from nonlinear
constraints. A subproblem is formulated by combining the objective function
and nonlinear constraint function using the Lagrangian and the penalty
parameters. A sequence of such optimization problems are approximately
minimized using a pattern search algorithm such that the linear constraints
and bounds are satisfied.
Each subproblem solution represents one iteration. The number of function
evaluations per iteration is therefore much higher when using nonlinear
constraints than otherwise.
A subproblem formulation is defined as
m
( x,  , s,  )  f ( x)   i si log(si  ci ( x)) 
i1
mt

i m 1
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i ceqi ( x) 
 mt
 ceqi ( x)2 ,
2 i m1
4-39
4
Using Direct Search
where
• The components λi of the vector λ are nonnegative and are known as
Lagrange multiplier estimates
• The elements si of the vector s are nonnegative shifts
• ρ is the positive penalty parameter.
The algorithm begins by using an initial value for the penalty parameter
(InitialPenalty).
The pattern search minimizes a sequence of subproblems, each of which is
an approximation of the original problem. Each subproblem has a fixed
value of λ, s, and ρ. When the subproblem is minimized to a required
accuracy and satisfies feasibility conditions, the Lagrangian estimates
are updated. Otherwise, the penalty parameter is increased by a penalty
factor (PenaltyFactor). This results in a new subproblem formulation and
minimization problem. These steps are repeated until the stopping criteria
are met.
Each subproblem solution represents one iteration. The number of function
evaluations per iteration is therefore much higher when using nonlinear
constraints than otherwise.
For a complete description of the algorithm, see the following references:
References
[1] Kolda, Tamara G., Robert Michael Lewis, and Virginia Torczon.
“A generating set direct search augmented Lagrangian algorithm for
optimization with a combination of general and linear constraints.” Technical
Report SAND2006-5315, Sandia National Laboratories, August 2006.
[2] Conn, A. R., N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent
Augmented Lagrangian Algorithm for Optimization with General Constraints
and Simple Bounds,” SIAM Journal on Numerical Analysis, Volume 28,
Number 2, pages 545–572, 1991.
4-40
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Nonlinear Constraint Solver Algorithm
[3] Conn, A. R., N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent
Augmented Lagrangian Barrier Algorithm for Optimization with General
Inequality Constraints and Simple Bounds,” Mathematics of Computation,
Volume 66, Number 217, pages 261–288, 1997.
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4
Using Direct Search
Custom Plot Function
In this section...
“About Custom Plot Functions” on page 4-42
“Creating the Custom Plot Function” on page 4-42
“Setting Up the Problem” on page 4-43
“Using the Custom Plot Function” on page 4-44
“How the Plot Function Works” on page 4-46
About Custom Plot Functions
To use a plot function other than those included with the software, you can
write your own custom plot function that is called at each iteration of the
pattern search to create the plot. This example shows how to create a plot
function that displays the logarithmic change in the best objective function
value from the previous iteration to the current iteration. More plot function
details are available in “Plot Options” on page 9-32.
Creating the Custom Plot Function
To create the plot function for this example, copy and paste the following code
into a new function file in the MATLAB Editor:
function stop = psplotchange(optimvalues, flag)
% PSPLOTCHANGE Plots the change in the best objective function
% value from the previous iteration.
% Best objective function value in the previous iteration
persistent last_best
stop = false;
if(strcmp(flag,'init'))
set(gca,'Yscale','log'); %Set up the plot
hold on;
xlabel('Iteration');
ylabel('Log Change in Values');
title(['Change in Best Function Value']);
4-42
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Custom Plot Function
end
% Best objective function value in the current iteration
best = min(optimvalues.fval);
% Set last_best to best
if optimvalues.iteration == 0
last_best = best;
else
%Change in objective function value
change = last_best - best;
plot(optimvalues.iteration, change, '.r');
end
Then save the file as psplotchange.m in a folder on the MATLAB path.
Setting Up the Problem
The problem is the same as “Linearly Constrained Problem” on page 4-74.
To set up the problem:
1 Enter the following at the MATLAB command line:
x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = 7;
Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
beq = [84 62 65 1];
2 Enter optimtool to open the Optimization app.
3 Choose the patternsearch solver.
4 Set up the problem to match the following figure.
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Using Direct Search
5 Since this is a linearly constrained problem, set the Poll method to GSS
Positive basis 2N.
Using the Custom Plot Function
To use the custom plot function, select Custom function in the Plot
functions pane and enter @psplotchange in the field to the right. To
compare the custom plot with the best function value plot, also select Best
function value.
4-44
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Custom Plot Function
Now, when you run the example, the pattern search tool displays the plots
shown in the following figure.
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Using Direct Search
Note that because the scale of the y-axis in the lower custom plot is
logarithmic, the plot will only show changes that are greater than 0.
How the Plot Function Works
The plot function uses information contained in the following structures,
which the Optimization app passes to the function as input arguments:
• optimvalues — Structure containing the current state of the solver
• flag — String indicating the current status of the algorithm
The most important statements of the custom plot function, psplotchange.m,
are summarized in the following table.
Custom Plot Function Statements
4-46
Statement
Description
persistent last_best
Creates the persistent variable
last_best, the best objective
function value in the previous
generation. Persistent variables are
preserved over multiple calls to the
plot function.
set(gca,'Yscale','log')
Sets up the plot before the algorithm
starts.
best = min(optimvalues.fval)
Sets best equal to the minimum
objective function value. The field
optimvalues.fval contains the
objective function value in the
current iteration. The variable best
is the minimum objective function
value. For a complete description
of the fields of the structure
optimvalues, see “Structure of the
Plot Functions” on page 9-11.
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Custom Plot Function
Custom Plot Function Statements (Continued)
Statement
Description
change = last_best - best
Sets the variable change to the
best objective function value at
the previous iteration minus the
best objective function value in the
current iteration.
plot(optimvalues.iteration,
change, '.r')
Plots the variable change at the
current objective function value,
for the current iteration contained
inoptimvalues.iteration.
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4-47
4
Using Direct Search
Set Options
In this section...
“Set Options Using psoptimset” on page 4-48
“Create Options and Problems Using the Optimization App” on page 4-50
Set Options Using psoptimset
You can specify any available patternsearch options by passing an options
structure as an input argument to patternsearch using the syntax
[x fval] = patternsearch(@fitnessfun,nvars, ...
A,b,Aeq,beq,lb,ub,nonlcon,options)
Pass in empty brackets [] for any constraints that do not appear in the
problem.
You create the options structure using the function psoptimset.
options = psoptimset(@patternsearch)
This returns the options structure with the default values for its fields.
options =
TolMesh:
TolCon:
TolX:
TolFun:
TolBind:
MaxIter:
MaxFunEvals:
TimeLimit:
MeshContraction:
MeshExpansion:
MeshAccelerator:
MeshRotate:
InitialMeshSize:
ScaleMesh:
MaxMeshSize:
4-48
1.0000e-06
1.0000e-06
1.0000e-06
1.0000e-06
1.0000e-03
'100*numberofvariables'
'2000*numberofvariables'
Inf
0.5000
2
'off'
'on'
1
'on'
Inf
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Set Options
InitialPenalty:
PenaltyFactor:
PollMethod:
CompletePoll:
PollingOrder:
SearchMethod:
CompleteSearch:
Display:
OutputFcns:
PlotFcns:
PlotInterval:
Cache:
CacheSize:
CacheTol:
Vectorized:
UseParallel:
10
100
'gpspositivebasis2n'
'off'
'consecutive'
[]
'off'
'final'
[]
[]
1
'off'
10000
2.2204e-16
'off'
0
The patternsearch function uses these default values if you do not pass
in options as an input argument.
The value of each option is stored in a field of the options structure, such as
options.MeshExpansion. You can display any of these values by entering
options followed by the name of the field. For example, to display the mesh
expansion factor for the pattern search, enter
options.MeshExpansion
ans =
2
To create an options structure with a field value that is different from the
default, use psoptimset. For example, to change the mesh expansion factor
to 3 instead of its default value 2, enter
options = psoptimset('MeshExpansion',3);
This creates the options structure with all values set to empty except for
MeshExpansion, which is set to 3. (An empty field causes patternsearch to
use the default value.)
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4-49
4
Using Direct Search
If you now call patternsearch with the argument options, the pattern
search uses a mesh expansion factor of 3.
If you subsequently decide to change another field in the options structure,
such as setting PlotFcns to @psplotmeshsize, which plots the mesh size at
each iteration, call psoptimset with the syntax
options = psoptimset(options, 'PlotFcns', @psplotmeshsize)
This preserves the current values of all fields of options except for PlotFcns,
which is changed to @plotmeshsize. Note that if you omit the options input
argument, psoptimset resets MeshExpansion to its default value, which is 2.
You can also set both MeshExpansion and PlotFcns with the single command
options = psoptimset('MeshExpansion',3,'PlotFcns',@plotmeshsize)
Create Options and Problems Using the Optimization
App
As an alternative to creating the options structure using psoptimset, you can
set the values of options in the Optimization app and then export the options
to a structure in the MATLAB workspace, as described in the “Importing and
Exporting Your Work” section of the Optimization Toolbox documentation.
If you export the default options in the Optimization app, the resulting
options structure has the same settings as the default structure returned
by the command
options = psoptimset
except for the default value of 'Display', which is 'final' when created by
psoptimset, but 'none' when created in the Optimization app.
You can also export an entire problem from the Optimization app and run it
from the command line. Enter
patternsearch(problem)
where problem is the name of the exported problem.
4-50
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Polling Types
Polling Types
In this section...
“Using a Complete Poll in a Generalized Pattern Search” on page 4-51
“Compare the Efficiency of Poll Options” on page 4-56
Using a Complete Poll in a Generalized Pattern
Search
As an example, consider the following function.
⎧ x12 + x22 − 25
for x12 + x22 ≤ 25
⎪
⎪
2
2
f ( x1 , x2 ) = ⎨ x12 + ( x2 − 9 ) − 16 for x12 + ( x2 − 9 ) ≤ 16
⎪0
otherwise.
⎪⎩
The following figure shows a plot of the function.
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4-51
4
Using Direct Search
0
−5
−10
−15
10
−20
5
−25
15
0
10
5
0
−5
−5
−10
−10
Global minimum at (0,0)
Local minimum at (0,9)
The global minimum of the function occurs at (0, 0), where its value is -25.
However, the function also has a local minimum at (0, 9), where its value is
-16.
To create a file that computes the function, copy and paste the following code
into a new file in the MATLAB Editor.
function z = poll_example(x)
if x(1)^2 + x(2)^2 <= 25
z = x(1)^2 + x(2)^2 - 25;
elseif x(1)^2 + (x(2) - 9)^2 <= 16
z = x(1)^2 + (x(2) - 9)^2 - 16;
else z = 0;
end
Then save the file as poll_example.m in a folder on the MATLAB path.
4-52
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Polling Types
To run a pattern search on the function, enter the following in the
Optimization app:
• Set Solver to patternsearch.
• Set Objective function to @poll_example.
• Set Start point to [0 5].
• Set Level of display to Iterative in the Display to command window
options.
Click Start to run the pattern search with Complete poll set to Off, its
default value. The Optimization app displays the results in the Run solver
and view results pane, as shown in the following figure.
The pattern search returns the local minimum at (0, 9). At the initial point,
(0, 5), the objective function value is 0. At the first iteration, the search polls
the following mesh points.
f((0, 5) + (1, 0)) = f(1, 5) = 0
f((0, 5) + (0, 1)) = f(0, 6) = -7
As soon as the search polls the mesh point (0, 6), at which the objective
function value is less than at the initial point, it stops polling the current
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4-53
4
Using Direct Search
mesh and sets the current point at the next iteration to (0, 6). Consequently,
the search moves toward the local minimum at (0, 9) at the first iteration. You
see this by looking at the first two lines of the command line display.
Iter
0
1
f-count
1
3
f(x)
0
-7
MeshSize
1
2
Method
Successful Poll
Note that the pattern search performs only two evaluations of the objective
function at the first iteration, increasing the total function count from 1 to 3.
Next, set Complete poll to On and click Start. The Run solver and view
results pane displays the following results.
This time, the pattern search finds the global minimum at (0, 0). The
difference between this run and the previous one is that with Complete poll
set to On, at the first iteration the pattern search polls all four mesh points.
f((0, 5) + (1, 0)) = f(1, 5) = 0
f((0, 5) + (0, 1)) = f(0, 6) = -6
f((0, 5) + (-1, 0)) = f(-1, 5) = 0
4-54
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Polling Types
f((0, 5) + (0, -1)) = f(0, 4) = -9
Because the last mesh point has the lowest objective function value, the
pattern search selects it as the current point at the next iteration. The first
two lines of the command-line display show this.
Iter
f-count
1
5
0
1
f(x)
0
-9
MeshSize
1
2
Method
Successful Poll
In this case, the objective function is evaluated four times at the first iteration.
As a result, the pattern search moves toward the global minimum at (0, 0).
The following figure compares the sequence of points returned when
Complete poll is set to Off with the sequence when Complete poll is On.
14
Initial point
Complete poll off
Complete poll on
12
10
8
Local minimum
6
4
Global minimum
2
0
−2
−4
−6
−6
−4
−2
0
2
4
6
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8
10
12
14
4-55
4
Using Direct Search
Compare the Efficiency of Poll Options
This example shows how several poll options interact in terms of iterations
and total function evaluations. The main results are:
• GSS is more efficient than GPS or MADS for linearly constrained problems.
• Whether setting CompletePoll to 'on' increases efficiency or decreases
efficiency is unclear, although it affects the number of iterations.
• Similarly, whether having a 2N poll is more or less efficient than having
an Np1 poll is also unclear. The most efficient poll is GSS Positive Basis
Np1 with Complete poll set to on. The least efficient is MADS Positive
Basis Np1 with Complete poll set to on.
Note The efficiency of an algorithm depends on the problem. GSS is
efficient for linearly constrained problems. However, predicting the efficiency
implications of the other poll options is difficult, as is knowing which poll type
works best with other constraints.
Problem setup
The problem is the same as in “Performing a Pattern Search on the Example”
on page 4-75. This linearly constrained problem uses the lincontest7 file
that comes with the toolbox:
1 Enter the following into your MATLAB workspace:
x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = 7;
Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
beq = [84 62 65 1];
2 Open the Optimization app by entering optimtool at the command line.
3 Choose the patternsearch solver.
4 Enter the problem and constraints as pictured.
4-56
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Polling Types
5 Ensure that the Poll method is GPS Positive basis 2N.
Generate the Results
1 Run the optimization.
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4
Using Direct Search
2 Choose File > Export to Workspace.
3 Export the results to a structure named gps2noff.
4 Set Options > Poll > Complete poll to on.
4-58
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Polling Types
5 Run the optimization.
6 Export the result to a structure named gps2non.
7 Set Options > Poll > Poll method to GPS Positive basis Np1 and set
Complete poll to off.
8 Run the optimization.
9 Export the result to a structure named gpsnp1off.
10 Set Complete poll to on.
11 Run the optimization.
12 Export the result to a structure named gpsnp1on.
13 Continue in a like manner to create solution structures for the other poll
methods with Complete poll set on and off: gss2noff, gss2non, gssnp1off,
gssnp1on, mads2noff, mads2non, madsnp1off, and madsnp1on.
Examine the Results
You have the results of 12 optimization runs. The following table shows the
efficiency of the runs, measured in total function counts and in iterations.
Your MADS results could differ, since MADS is a stochastic algorithm.
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4-59
4
Using Direct Search
Algorithm
Function Count
Iterations
GPS2N, complete poll off
1462
136
GPS2N, complete poll on
1396
96
GPSNp1, complete poll off
864
118
GPSNp1, complete poll on
1007
104
GSS2N, complete poll off
758
84
GSS2N, complete poll on
889
74
GSSNp1, complete poll off
533
94
GSSNp1, complete poll on
491
70
MADS2N, complete poll off
922
162
MADS2N, complete poll on
2285
273
MADSNp1, complete poll off
1155
201
MADSNp1, complete poll on
1651
201
To obtain, say, the first row in the table, enter gps2noff.output.funccount
and gps2noff.output.iterations. You can also examine a structure in the
Variables editor by double-clicking the structure in the Workspace Browser,
and then double-clicking the output structure.
4-60
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Polling Types
The main results gleaned from the table are:
• Setting Complete poll to on generally lowers the number of iterations
for GPS and GSS, but the change in number of function evaluations is
unpredictable.
• Setting Complete poll to on does not necessarily change the number of
iterations for MADS, but substantially increases the number of function
evaluations.
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Using Direct Search
• The most efficient algorithm/options settings, with efficiency meaning
lowest function count:
1 GSS Positive basis Np1 with Complete poll set to on (function count
491)
2 GSS Positive basis Np1 with Complete poll set to off (function
count 533)
3 GSS Positive basis 2N with Complete poll set to off (function count
758)
4 GSS Positive basis 2N with Complete poll set to on (function count
889)
The other poll methods had function counts exceeding 900.
• For this problem, the most efficient poll is GSS Positive Basis Np1 with
Complete poll set to on, although the Complete poll setting makes only
a small difference. The least efficient poll is MADS Positive Basis 2N
with Complete poll set to on. In this case, the Complete poll setting
makes a substantial difference.
4-62
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Set Mesh Options
Set Mesh Options
In this section...
“Mesh Expansion and Contraction” on page 4-63
“Mesh Accelerator” on page 4-70
Mesh Expansion and Contraction
The Expansion factor and Contraction factor options, in Mesh options,
control how much the mesh size is expanded or contracted at each iteration.
With the default Expansion factor value of 2, the pattern search multiplies
the mesh size by 2 after each successful poll. With the default Contraction
factor value of 0.5, the pattern search multiplies the mesh size by 0.5 after
each unsuccessful poll.
You can view the expansion and contraction of the mesh size during the
pattern search by selecting Mesh size in the Plot functions pane. To also
display the values of the mesh size and objective function at the command
line, set Level of display to Iterative in the Display to command
window options.
For example, set up the problem described in “Linearly Constrained Problem”
on page 4-74 as follows:
1 Enter the following at the command line:
x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = 7;
Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
beq = [84 62 65 1];
2 Set up your problem in the Optimization app to match the following figures.
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4-63
‫‪Using Direct Search‬‬
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‫‪4‬‬
‫‪4-64‬‬
Set Mesh Options
3 Run the optimization.
The Optimization app displays the following plot.
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4-65
4
Using Direct Search
To see the changes in mesh size more clearly, change the y-axis to logarithmic
scaling as follows:
1 Select Axes Properties from the Edit menu in the plot window.
2 In the Properties Editor, select the Y Axis tab.
3 Set Scale to Log.
Updating these settings in the MATLAB Property Editor shows the plot in
the following figure.
4-66
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Set Mesh Options
The first 5 iterations result in successful polls, so the mesh sizes increase
steadily during this time. You can see that the first unsuccessful poll occurs
at iteration 6 by looking at the command-line display:
Iter
0
1
2
3
4
5
6
f-count
1
2
3
4
5
6
15
f(x)
2273.76
2251.69
2209.86
2135.43
2023.48
1947.23
1947.23
MeshSize
1
2
4
8
16
32
16
Method
Successful Poll
Successful Poll
Successful Poll
Successful Poll
Successful Poll
Refine Mesh
Note that at iteration 5, which is successful, the mesh size doubles for the
next iteration. But at iteration 6, which is unsuccessful, the mesh size is
multiplied 0.5.
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4-67
4
Using Direct Search
To see how Expansion factor and Contraction factor affect the pattern
search, make the following changes:
• Set Expansion factor to 3.0.
• Set Contraction factor to 2/3.
Then click Start. The Run solver and view results pane shows that
the final point is approximately the same as with the default settings of
Expansion factor and Contraction factor, but that the pattern search
takes longer to reach that point.
4-68
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Set Mesh Options
When you change the scaling of the y-axis to logarithmic, the mesh size plot
appears as shown in the following figure.
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4-69
4
Using Direct Search
Note that the mesh size increases faster with Expansion factor set to 3.0,
as compared with the default value of 2.0, and decreases more slowly with
Contraction factor set to 0.75, as compared with the default value of 0.5.
Mesh Accelerator
The mesh accelerator can make a pattern search converge faster to an
optimal point by reducing the number of iterations required to reach the mesh
tolerance. When the mesh size is below a certain value, the pattern search
contracts the mesh size by a factor smaller than the Contraction factor
factor. Mesh accelerator applies only to the GPS and GSS algorithms.
4-70
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Set Mesh Options
Note For best results, use the mesh accelerator for problems in which the
objective function is not too steep near the optimal point, or you might lose
some accuracy. For differentiable problems, this means that the absolute
value of the derivative is not too large near the solution.
To use the mesh accelerator, set Accelerator to On in the Mesh options. Or,
at the command line, set the MeshAccelerator option to 'on'.
For example, set up the problem described in “Linearly Constrained Problem”
on page 4-74 as follows:
1 Enter the following at the command line:
x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = 7;
Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
beq = [84 62 65 1];
2 Set up your problem in the Optimization app to match the following figures.
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4-71
4
Using Direct Search
3 Run the optimization.
The number of iterations required to reach the mesh tolerance is 78, as
compared with 84 when Accelerator is set to Off.
You can see the effect of the mesh accelerator by setting Level of display
to Iterative in Display to command window. Run the example with
Accelerator set to On, and then run it again with Accelerator set to Off.
The mesh sizes are the same until iteration 70, but differ at iteration 71. The
MATLAB Command Window displays the following lines for iterations 70
and 71 with Accelerator set to Off.
4-72
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Set Mesh Options
Iter
70
71
f-count
618
630
f(x)
1919.54
1919.54
MeshSize
6.104e-05
3.052e-05
Method
Refine Mesh
Refine Mesh
Note that the mesh size is multiplied by 0.5, the default value of Contraction
factor.
For comparison, the Command Window displays the following lines for the
same iteration numbers with Accelerator set to On.
Iter
70
71
f-count
618
630
f(x)
1919.54
1919.54
MeshSize
6.104e-05
1.526e-05
Method
Refine Mesh
Refine Mesh
In this case the mesh size is multiplied by 0.25.
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4-73
4
Using Direct Search
Constrained Minimization Using patternsearch
In this section...
“Linearly Constrained Problem” on page 4-74
“Nonlinearly Constrained Problem” on page 4-77
Linearly Constrained Problem
Problem Description
This section presents an example of performing a pattern search on a
constrained minimization problem. The example minimizes the function
F ( x) =
1 T
x Hx + f T x,
2
where
⎡36
⎢17
⎢
⎢19
H=⎢
⎢12
⎢8
⎢
⎢⎣15
17 19 12 8 15 ⎤
33 18 11 7 14 ⎥⎥
18 43 13 8 16 ⎥
⎥,
11 13 18 6 11 ⎥
7 8 6 9 8⎥
⎥
14 16 11 8 29 ⎥⎦
f = [20 15 21 18 29 24 ],
subject to the constraints
A ⋅ x ≤ b,
Aeq ⋅ x = beq,
where
4-74
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Constrained Minimization Using patternsearch
A = [ −8 7 3 −4 9 0],
b = 7,
⎡7 1 8 3 3 3⎤
⎢ 5 0 −5 1 −5 8 ⎥
⎥,
Aeq = ⎢
⎢ −2 −6 7 1 1 9 ⎥
⎢
⎥
⎣ 1 −1 2 −2 3 −3⎦
beq = [84 62 65 1].
Performing a Pattern Search on the Example
To perform a pattern search on the example, first enter
optimtool('patternsearch')
to open the Optimization app, or enter optimtool and then choose
patternsearch from the Solver menu. Then type the following function in
the Objective function field:
@lincontest7
lincontest7 is a file included in Global Optimization Toolbox software that
computes the objective function for the example. Because the matrices and
vectors defining the starting point and constraints are large, it is more
convenient to set their values as variables in the MATLAB workspace first
and then enter the variable names in the Optimization app. To do so, enter
x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = 7;
Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
beq = [84 62 65 1];
Then, enter the following in the Optimization app:
• Set Start point to x0.
• Set the following Linear inequalities:
-
Set A to Aineq.
Set b to bineq.
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4-75
4
Using Direct Search
-
Set Aeq to Aeq.
Set beq to beq.
The following figure shows these settings in the Optimization app.
Since this is a linearly constrained problem, set the Poll method to GSS
Positive basis 2N. For more information about the efficiency of the
GSS search methods for linearly constrained problems, see “Compare the
Efficiency of Poll Options” on page 4-56.
Then click Start to run the pattern search. When the search is finished, the
results are displayed in Run solver and view results pane, as shown in
the following figure.
4-76
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Constrained Minimization Using patternsearch
Nonlinearly Constrained Problem
Suppose you want to minimize the simple objective function of two variables
x1 and x2,
(
)
(
)
min f ( x) = 4 - 2.1 x12 − x14 / 3 x12 + x1 x2 + −4 + 4 x22 x22
x
subject to the following nonlinear inequality constraints and bounds
x1 x2 + x1 − x2 + 1.5 ≤ 0
10 − x1 x2 ≤ 0
0 ≤ x1 ≤ 1
0 ≤ x2 ≤ 13
(nonlinear constraint)
(nonlinear constraint)
(bound)
(bound)
Begin by creating the objective and constraint functions. First, create a file
named simple_objective.m as follows:
function y = simple_objective(x)
y = (4 - 2.1*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + (-4 + 4*x(2)^2)*x(2)^2;
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4-77
4
Using Direct Search
The pattern search solver assumes the objective function will take one input x
where x has as many elements as number of variables in the problem. The
objective function computes the value of the function and returns that scalar
value in its one return argument y.
Then create a file named simple_constraint.m containing the constraints:
function [c, ceq] = simple_constraint(x)
c = [1.5 + x(1)*x(2) + x(1) - x(2);
-x(1)*x(2) + 10];
ceq = [];
The pattern search solver assumes the constraint function will take one input
x, where x has as many elements as the number of variables in the problem.
The constraint function computes the values of all the inequality and equality
constraints and returns two vectors, c and ceq, respectively.
Next, to minimize the objective function using the patternsearch function,
you need to pass in a function handle to the objective function as well as
specifying a start point as the second argument. Lower and upper bounds
are provided as LB and UB respectively. In addition, you also need to pass a
function handle to the nonlinear constraint function.
ObjectiveFunction = @simple_objective;
X0 = [0 0];
% Starting point
LB = [0 0];
% Lower bound
UB = [1 13];
% Upper bound
ConstraintFunction = @simple_constraint;
[x,fval] = patternsearch(ObjectiveFunction,X0,[],[],[],[],...
LB,UB,ConstraintFunction)
Optimization terminated: mesh size less than options.TolMesh
and constraint violation is less than options.TolCon.
x =
0.8122
12.3122
fval =
9.1324e+004
4-78
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Constrained Minimization Using patternsearch
Next, plot the results. Create an options structure using psoptimset that
selects two plot functions. The first plot function psplotbestf plots the
best objective function value at every iteration. The second plot function
psplotmaxconstr plots the maximum constraint violation at every iteration.
Note You can also visualize the progress of the algorithm by displaying
information to the Command Window using the 'Display' option.
options = psoptimset('PlotFcns',{@psplotbestf,@psplotmaxconstr},'Display','iter');
[x,fval] = patternsearch(ObjectiveFunction,X0,[],[],[],[],LB,UB,ConstraintFunction,options)
max
Iter
f-count
0
f(x)
constraint
10
MeshSize
Method
1
0
0.8919
1
28
113580
0
0.001
Increase penalty
2
105
91324
1.782e-007
1e-005
Increase penalty
3
192
91324
1.188e-011
1e-007
Increase penalty
Optimization terminated: mesh size less than options.TolMesh
and constraint violation is less than options.TolCon.
x =
0.8122
12.3122
fval =
9.1324e+004
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4-79
4
Using Direct Search
Best Objective Function Value and Maximum Constraint Violation at Each
Iteration
4-80
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Use Cache
Use Cache
Typically, at any given iteration of a pattern search, some of the mesh points
might coincide with mesh points at previous iterations. By default, the
pattern search recomputes the objective function at these mesh points even
though it has already computed their values and found that they are not
optimal. If computing the objective function takes a long time—say, several
minutes—this can make the pattern search run significantly longer.
You can eliminate these redundant computations by using a cache, that is,
by storing a history of the points that the pattern search has already visited.
To do so, set Cache to On in Cache options. At each poll, the pattern search
checks to see whether the current mesh point is within a specified tolerance,
Tolerance, of a point in the cache. If so, the search does not compute the
objective function for that point, but uses the cached function value and
moves on to the next point.
Note When Cache is set to On, the pattern search might fail to identify a
point in the current mesh that improves the objective function because it is
within the specified tolerance of a point in the cache. As a result, the pattern
search might run for more iterations with Cache set to On than with Cache
set to Off. It is generally a good idea to keep the value of Tolerance very
small, especially for highly nonlinear objective functions.
For example, set up the problem described in “Linearly Constrained Problem”
on page 4-74 as follows:
1 Enter the following at the command line:
x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = 7;
Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3];
beq = [84 62 65 1];
2 Set up your problem in the Optimization app to match the following figures.
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4-81
‫‪Using Direct Search‬‬
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‫‪4‬‬
‫‪4-82‬‬
Use Cache
3 Run the optimization.
After the pattern search finishes, the plots appear as shown in the following
figure.
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4-83
4
Using Direct Search
Note that the total function count is 758.
Now, set Cache to On and run the example again. This time, the plots appear
as shown in the following figure.
This time, the total function count is reduced to 734.
4-84
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Vectorize the Objective and Constraint Functions
Vectorize the Objective and Constraint Functions
In this section...
“Vectorize for Speed” on page 4-85
“Vectorized Objective Function” on page 4-85
“Vectorized Constraint Functions” on page 4-88
“Example of Vectorized Objective and Constraints” on page 4-89
Vectorize for Speed
Direct search often runs faster if you vectorize the objective and nonlinear
constraint functions. This means your functions evaluate all the points in
a poll or search pattern at once, with one function call, without having to
loop through the points one at a time. Therefore, the option Vectorized
= 'on' works only when CompletePoll or CompleteSearch is also set to
'on'. However, when you set Vectorized = 'on', patternsearch checks
that the objective and any nonlinear constraint functions give outputs of
the correct shape for vectorized calculations, regardless of the setting of the
CompletePoll or CompleteSearch options.
If there are nonlinear constraints, the objective function and the nonlinear
constraints all need to be vectorized in order for the algorithm to compute in
a vectorized manner.
Note Write your vectorized objective function or nonlinear constraint
function to accept a matrix with an arbitrary number of points.
patternsearch sometimes evaluates a single point even during a vectorized
calculation.
Vectorized Objective Function
A vectorized objective function accepts a matrix as input and generates a
vector of function values, where each function value corresponds to one row
or column of the input matrix. patternsearch resolves the ambiguity in
whether the rows or columns of the matrix represent the points of a pattern
as follows. Suppose the input matrix has m rows and n columns:
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4-85
4
Using Direct Search
• If the initial point x0 is a column vector of size m, the objective function
takes each column of the matrix as a point in the pattern and returns a
row vector of size n.
• If the initial point x0 is a row vector of size n, the objective function takes
each row of the matrix as a point in the pattern and returns a column
vector of size m.
• If the initial point x0 is a scalar, patternsearch assumes that x0 is a
row vector. Therefore, the input matrix has one column (n = 1, the input
matrix is a vector), and each entry of the matrix represents one row for
the objective function to evaluate. The output of the objective function in
this case is a column vector of size m.
Pictorially, the matrix and calculation are represented by the following figure.
4-86
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Vectorize the Objective and Constraint Functions
Results
The matrix passed to
objective and constraint functions
If x0 is a
column
...
Each column represents
one search/poll point
Results
If x0 is a row
...
Each row represents
one search/poll point
The matrix passed to
objective and constraint functions
Structure of Vectorized Functions
For example, suppose the objective function is
f ( x) = x14 + x24 − 4 x12 − 2 x22 + 3 x1 − x2 / 2.
If the initial vector x0 is a column vector, such as [0;0], a function for
vectorized evaluation is
function f = vectorizedc(x)
f = x(1,:).^4+x(2,:).^4-4*x(1,:).^2-2*x(2,:).^2 ...
+3*x(1,:)-.5*x(2,:);
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4-87
4
Using Direct Search
If the initial vector x0 is a row vector, such as [0,0], a function for vectorized
evaluation is
function f = vectorizedr(x)
f = x(:,1).^4+x(:,2).^4-4*x(:,1).^2-2*x(:,2).^2 ...
+3*x(:,1)-.5*x(:,2);
Tip If you want to use the same objective (fitness) function for both pattern
search and genetic algorithm, write your function to have the points
represented by row vectors, and write x0 as a row vector. The genetic
algorithm always takes individuals as the rows of a matrix. This was a design
decision—the genetic algorithm does not require a user-supplied population,
so needs to have a default format.
To minimize vectorizedc, enter the following commands:
options=psoptimset('Vectorized','on','CompletePoll','on');
x0=[0;0];
[x fval]=patternsearch(@vectorizedc,x0,...
[],[],[],[],[],[],[],options)
MATLAB returns the following output:
Optimization terminated: mesh size less than options.TolMesh.
x =
-1.5737
1.0575
fval =
-10.0088
Vectorized Constraint Functions
Only nonlinear constraints need to be vectorized; bounds and linear
constraints are handled automatically. If there are nonlinear constraints, the
objective function and the nonlinear constraints all need to be vectorized in
order for the algorithm to compute in a vectorized manner.
4-88
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Vectorize the Objective and Constraint Functions
The same considerations hold for constraint functions as for objective
functions: the initial point x0 determines the type of points (row or column
vectors) in the poll or search. If the initial point is a row vector of size k, the
matrix x passed to the constraint function has k columns. Similarly, if the
initial point is a column vector of size k, the matrix of poll or search points
has k rows. The figure Structure of Vectorized Functions on page 4-87 may
make this clear. If the initial point is a scalar, patternsearch assumes that it
is a row vector.
Your nonlinear constraint function returns two matrices, one for inequality
constraints, and one for equality constraints. Suppose there are nc nonlinear
inequality constraints and nceq nonlinear equality constraints. For row vector
x0, the constraint matrices have nc and nceq columns respectively, and the
number of rows is the same as in the input matrix. Similarly, for a column
vector x0, the constraint matrices have nc and nceq rows respectively, and the
number of columns is the same as in the input matrix. In figure Structure of
Vectorized Functions on page 4-87, “Results” includes both nc and nceq.
Example of Vectorized Objective and Constraints
Suppose that the nonlinear constraints are
x12 x22
+
≤ 1 (the interior of an ellipse),
9
4
x2 ≥ cosh ( x1 ) − 1.
Write a function for these constraints for row-form x0 as follows:
function [c ceq] = ellipsecosh(x)
c(:,1)=x(:,1).^2/9+x(:,2).^2/4-1;
c(:,2)=cosh(x(:,1))-x(:,2)-1;
ceq=[];
Minimize vectorizedr (defined in “Vectorized Objective Function” on page
4-85) subject to the constraints ellipsecosh:
x0=[0,0];
options=psoptimset('Vectorized','on','CompletePoll','on');
[x fval]=patternsearch(@vectorizedr,x0,...
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4-89
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Using Direct Search
[],[],[],[],[],[],@ellipsecosh,options)
MATLAB returns the following output:
Optimization terminated: mesh size less than options.TolMesh
and constraint violation is less than options.TolCon.
x =
-1.3516
1.0612
fval =
-9.5394
4-90
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Optimize an ODE in Parallel
Optimize an ODE in Parallel
This example shows how to optimize parameters of an ODE.
It also shows how to avoid computing the objective and nonlinear constraint
function twice when the ODE solution returns both. The example compares
patternsearch and ga in terms of time to run the solver and the quality
of the solutions.
You need a Parallel Computing Toolbox license to use parallel computing.
Step 1. Define the problem.
The problem is to change the position and angle of a cannon to fire a projectile
as far as possible beyond a wall. The cannon has a muzzle velocity of 300 m/s.
The wall is 20 m high. If the cannon is too close to the wall, it has to fire at too
steep an angle, and the projectile does not travel far enough. If the cannon is
too far from the wall, the projectile does not travel far enough either.
Air resistance slows the projectile. The resisting force is proportional to the
square of the velocity, with proportionality constant 0.02. Gravity acts on the
projectile, accelerating it downward with constant 9.81 m/s2. Therefore, the
equations of motion for the trajectory x(t) are
d2 x(t)
dt2
 0.02 x(t) x(t)  (0, 9.81).
The initial position x0 and initial velocity xp0 are 2-D vectors. However,
the initial height x0(2) is 0, so the initial position depends only on the
scalar x0(1). And the initial velocity xp0 has magnitude 300 (the muzzle
velocity), so depends only on the initial angle, a scalar. For an initial angle th,
xp0 = 300*(cos(th),sin(th). Therefore, the optimization problem depends
only on two scalars, so it is a 2-D problem. Use the horizontal distance and
the angle as the decision variables.
Step 2. Formulate the ODE model.
ODE solvers require you to formulate your model as a first-order system.
Augment the trajectory vector (x1(t),x2(t)) with its time derivative (x’1(t),x’2(t))
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4-91
4
Using Direct Search
to form a 4-D trajectory vector. In terms of this augmented vector, the
differential equation becomes
x3 (t)




x4 (t)
d


.
x(t)  
.02  x3 (t), x4 (t)  x3 (t) 
dt


 .02  x3 (t), x4 (t)  x4 (t)  9.81


Write the differential equation as a function file, and save it on your MATLAB
path.
function f = cannonfodder(t,x)
f = [x(3);x(4);x(3);x(4)]; % initial, gets f(1) and f(2) correct
nrm = norm(x(3:4)) * .02; % norm of the velocity times constant
f(3) = -x(3)*nrm; % horizontal acceleration
f(4) = -x(4)*nrm - 9.81; % vertical acceleration
Visualize the solution of the ODE starting 30 m from the wall at an angle
of pi/3.
4-92
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Optimize an ODE in Parallel
Code for generating the figure
x0 = [-30;0;300*cos(pi/3);300*sin(pi/3)];
sol = ode45(@cannonfodder,[0,10],x0);
% Find the time when the projectile lands
zerofnd = fzero(@(r)deval(sol,r,2),[sol.x(2),sol.x(end)]);
t = linspace(0,zerofnd); % equal times for plot
xs = deval(sol,t,1); % interpolated x values
ys = deval(sol,t,2); % interpolated y values
plot(xs,ys)
hold on
plot([0,0],[0,20],'k') % Draw the wall
xlabel('Horizontal distance')
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4-93
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Using Direct Search
ylabel('Trajectory height')
legend('Trajectory','Wall','Location','NW')
ylim([0 120])
hold off
Step 3. Solve using patternsearch.
The problem is to find initial position x0(1) and initial angle x0(2) to
maximize the distance from the wall the projectile lands. Because this
is a maximization problem, minimize the negative of the distance (see
“Maximizing vs. Minimizing” on page 2-5).
To use patternsearch to solve this problem, you must provide the objective,
constraint, initial guess, and options.
These two files are the objective and constraint functions. Copy them to a
folder on your MATLAB path.
function f = cannonobjective(x)
x0 = [x(1);0;300*cos(x(2));300*sin(x(2))];
sol = ode45(@cannonfodder,[0,15],x0);
% Find the time t when y_2(t) = 0
zerofnd = fzero(@(r)deval(sol,r,2),[sol.x(2),sol.x(end)]);
% Then find the x-position at that time
f = deval(sol,zerofnd,1);
f = -f; % take negative of distance for maximization
function [c,ceq] = cannonconstraint(x)
ceq = [];
x0 = [x(1);0;300*cos(x(2));300*sin(x(2))];
sol = ode45(@cannonfodder,[0,15],x0);
if sol.y(1,end) <= 0 % projectile never reaches wall
4-94
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Optimize an ODE in Parallel
c = 20 - sol.y(2,end);
else
% Find when the projectile crosses x = 0
zerofnd = fzero(@(r)deval(sol,r,1),[sol.x(2),sol.x(end)]);
% Then find the height there, and subtract from 20
c = 20 - deval(sol,zerofnd,2);
end
Notice that the objective and constraint functions set their input variable x0
to a 4-D initial point for the ODE solver. The ODE solver does not stop if
the projectile hits the wall. Instead, the constraint function simply becomes
positive, indicating an infeasible initial value.
The initial position x0(1) cannot be above 0, and it is futile to have it be
below –200. (It should be near –20 because, with no air resistance, the longest
trajectory would start at –20 at an angle pi/4.) Similarly, the initial angle
x0(2) cannot be below 0, and cannot be above pi/2. Set bounds slightly away
from these initial values:
lb = [-200;0.05];
ub = [-1;pi/2-.05];
x0 = [-30,pi/3]; % initial guess
Set the CompletePoll option to 'on'. This gives a higher-quality solution,
and enables direct comparison with parallel processing, because computing in
parallel requires this setting.
opts = psoptimset('CompletePoll','on');
Call patternsearch to solve the problem.
tic % time the solution
[xsolution,distance,eflag,outpt] = patternsearch(@cannonobjective,x0,...
[],[],[],[],lb,ub,@cannonconstraint,opts)
toc
Optimization terminated: mesh size less than options.TolMesh
and constraint violation is less than options.TolCon.
xsolution =
-28.8123
0.6095
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4-95
4
Using Direct Search
distance =
-125.9880
eflag =
1
outpt =
function:
problemtype:
pollmethod:
searchmethod:
iterations:
funccount:
meshsize:
maxconstraint:
message:
@cannonobjective
'nonlinearconstr'
'gpspositivebasis2n'
[]
5
269
8.9125e-07
0
[1x115 char]
Elapsed time is 3.174088 seconds.
Starting the projectile about 29 m from the wall at an angle 0.6095 radian
results in the farthest distance, about 126 m. The reported distance is
negative because the objective function is the negative of the distance to the
wall.
Visualize the solution.
x0 = [xsolution(1);0;300*cos(xsolution(2));300*sin(xsolution(2))];
sol = ode45(@cannonfodder,[0,15],x0);
% Find the time when the projectile lands
zerofnd = fzero(@(r)deval(sol,r,2),[sol.x(2),sol.x(end)]);
t = linspace(0,zerofnd); % equal times for plot
xs = deval(sol,t,1); % interpolated x values
ys = deval(sol,t,2); % interpolated y values
plot(xs,ys)
hold on
plot([0,0],[0,20],'k') % Draw the wall
xlabel('Horizontal distance')
ylabel('Trajectory height')
4-96
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Optimize an ODE in Parallel
legend('Trajectory','Wall','Location','NW')
ylim([0 70])
hold off
Step 4. Avoid calling the expensive subroutine twice.
Both the objective and nonlinear constraint function call the ODE solver
to calculate their values. Use the technique in “Objective and Nonlinear
Constraints in the Same Function” to avoid calling the solver twice. The
runcannon file implements this technique. Copy this file to a folder on your
MATLAB path.
function [x,f,eflag,outpt] = runcannon(x0,opts)
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Using Direct Search
if nargin == 1 % No options supplied
opts = [];
end
xLast = []; % Last place ode solver was called
sol = []; % ODE solution structure
fun = @objfun; % the objective function, nested below
cfun = @constr; % the constraint function, nested below
lb = [-200;0.05];
ub = [-1;pi/2-.05];
% Call patternsearch
[x,f,eflag,outpt] = patternsearch(fun,x0,[],[],[],[],lb,ub,cfun,opts);
function y = objfun(x)
if ~isequal(x,xLast) % Check if computation is necessary
x0 = [x(1);0;300*cos(x(2));300*sin(x(2))];
sol = ode45(@cannonfodder,[0,15],x0);
xLast = x;
end
% Now compute objective function
% First find when the projectile hits the ground
zerofnd = fzero(@(r)deval(sol,r,2),[sol.x(2),sol.x(end)]);
% Then compute the x-position at that time
y = deval(sol,zerofnd,1);
y = -y; % take negative of distance
end
function [c,ceq] = constr(x)
ceq = [];
if ~isequal(x,xLast) % Check if computation is necessary
x0 = [x(1);0;300*cos(x(2));300*sin(x(2))];
sol = ode45(@cannonfodder,[0,15],x0);
xLast = x;
end
% Now compute constraint functions
% First find when the projectile crosses x = 0
4-98
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Optimize an ODE in Parallel
zerofnd = fzero(@(r)deval(sol,r,1),[sol.x(1),sol.x(end)]);
% Then find the height there, and subtract from 20
c = 20 - deval(sol,zerofnd,2);
end
end
Reinitialize the problem and time the call to runcannon.
x0 = [-30;pi/3];
tic
[xsolution,distance,eflag,outpt] = runcannon(x0,opts);
toc
Elapsed time is 2.610590 seconds.
The solver ran faster than before. If you examine the solution, you see that
the output is identical.
Step 5. Compute in parallel.
Try to save more time by computing in parallel. Begin by opening a parallel
pool of workers.
parpool
Starting parpool using the 'local' profile ... connected to 4 workers.
ans =
Pool with properties:
Connected:
NumWorkers:
Cluster:
AttachedFiles:
IdleTimeout:
SpmdEnabled:
true
4
local
{}
30 minute(s) (30 minutes remaining)
true
Set the options to use parallel computing, and rerun the solver.
opts = psoptimset(opts,'UseParallel',true);
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4-99
4
Using Direct Search
x0 = [-30;pi/3];
tic
[xsolution,distance,eflag,outpt] = runcannon(x0,opts);
toc
Elapsed time is 3.917971 seconds.
In this case, parallel computing was slower. If you examine the solution,
you see that the output is identical.
Step 6. Compare with the genetic algorithm.
You can also try to solve the problem using the genetic algorithm. However,
the genetic algorithm is usually slower and less reliable.
The runcannonga file calls the genetic algorithm and avoids double
evaluation of the ODE solver. It resembles runcannon, but calls ga instead
of patternsearch, and also checks whether the trajectory reaches the wall.
Copy this file to a folder on your MATLAB path.
function [x,f,eflag,outpt] = runcannonga(opts)
if nargin == 1 % No options supplied
opts = [];
end
xLast = []; % Last place ode solver was called
sol = []; % ODE solution structure
fun = @objfun; % the objective function, nested below
cfun = @constr; % the constraint function, nested below
lb = [-200;0.05];
ub = [-1;pi/2-.05];
% Call ga
[x,f,eflag,outpt] = ga(fun,2,[],[],[],[],lb,ub,cfun,opts);
function y = objfun(x)
if ~isequal(x,xLast) % Check if computation is necessary
x0 = [x(1);0;300*cos(x(2));300*sin(x(2))];
4-100
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Optimize an ODE in Parallel
sol = ode45(@cannonfodder,[0,15],x0);
xLast = x;
end
% Now compute objective function
% First find when the projectile hits the ground
zerofnd = fzero(@(r)deval(sol,r,2),[sol.x(2),sol.x(end)]);
% Then compute the x-position at that time
y = deval(sol,zerofnd,1);
y = -y; % take negative of distance
end
function [c,ceq] = constr(x)
ceq = [];
if ~isequal(x,xLast) % Check if computation is necessary
x0 = [x(1);0;300*cos(x(2));300*sin(x(2))];
sol = ode45(@cannonfodder,[0,15],x0);
xLast = x;
end
% Now compute constraint functions
if sol.y(1,end) <= 0 % projectile never reaches wall
c = 20 - sol.y(2,end);
else
% Find when the projectile crosses x = 0
zerofnd = fzero(@(r)deval(sol,r,1),[sol.x(2),sol.x(end)]);
% Then find the height there, and subtract from 20
c = 20 - deval(sol,zerofnd,2);
end
end
end
Call runcannonga in parallel.
opts = gaoptimset('UseParallel',true);
rng default % for reproducibility
tic
[xsolution,distance,eflag,outpt] = runcannonga(opts)
toc
Optimization terminated: average change in the fitness value less than
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4-101
4
Using Direct Search
options.TolFun and constraint violation is less than options.TolCon.
xsolution =
-17.9172
0.8417
distance =
-116.6263
eflag =
1
outpt =
problemtype:
rngstate:
generations:
funccount:
message:
maxconstraint:
'nonlinearconstr'
[1x1 struct]
5
20212
[1x140 char]
0
Elapsed time is 119.630284 seconds.
The ga solution is not as good as the patternsearch solution: 117 m versus
126 m. ga took much more time: about 120 s versus under 5 s.
Related
Examples
• “Objective and Nonlinear Constraints in the Same Function”
Concepts
• “Parallel Computing”
4-102
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5
Using the Genetic
Algorithm
• “What Is the Genetic Algorithm?” on page 5-3
• “Optimize Using ga” on page 5-4
• “Minimize Rastrigin’s Function” on page 5-8
• “Genetic Algorithm Terminology” on page 5-18
• “How the Genetic Algorithm Works” on page 5-21
• “Mixed Integer Optimization” on page 5-32
• “Solving a Mixed Integer Engineering Design Problem Using the Genetic
Algorithm” on page 5-42
• “Nonlinear Constraint Solver Algorithm” on page 5-52
• “Create Custom Plot Function” on page 5-54
• “Reproduce Results in Optimization App” on page 5-58
• “Resume ga” on page 5-59
• “Options and Outputs” on page 5-66
• “Use Exported Options and Problems” on page 5-70
• “Reproduce Results” on page 5-71
• “Run ga from a File” on page 5-73
• “Population Diversity” on page 5-76
• “Fitness Scaling” on page 5-87
• “Vary Mutation and Crossover” on page 5-91
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5
Using the Genetic Algorithm
• “Global vs. Local Minima Using ga” on page 5-100
• “Include a Hybrid Function” on page 5-107
• “Set Maximum Number of Generations” on page 5-111
• “Vectorize the Fitness Function” on page 5-114
• “Constrained Minimization Using ga” on page 5-116
5-2
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What Is the Genetic Algorithm?
What Is the Genetic Algorithm?
The genetic algorithm is a method for solving both constrained and
unconstrained optimization problems that is based on natural selection, the
process that drives biological evolution. The genetic algorithm repeatedly
modifies a population of individual solutions. At each step, the genetic
algorithm selects individuals at random from the current population to be
parents and uses them to produce the children for the next generation. Over
successive generations, the population "evolves" toward an optimal solution.
You can apply the genetic algorithm to solve a variety of optimization
problems that are not well suited for standard optimization algorithms,
including problems in which the objective function is discontinuous,
nondifferentiable, stochastic, or highly nonlinear. The genetic algorithm can
address problems of mixed integer programming, where some components
are restricted to be integer-valued.
The genetic algorithm uses three main types of rules at each step to create
the next generation from the current population:
• Selection rules select the individuals, called parents, that contribute to the
population at the next generation.
• Crossover rules combine two parents to form children for the next
generation.
• Mutation rules apply random changes to individual parents to form
children.
The genetic algorithm differs from a classical, derivative-based, optimization
algorithm in two main ways, as summarized in the following table.
Classical Algorithm
Genetic Algorithm
Generates a single point at each
iteration. The sequence of points
approaches an optimal solution.
Generates a population of points at
each iteration. The best point in the
population approaches an optimal
solution.
Selects the next point in the sequence
by a deterministic computation.
Selects the next population by
computation which uses random
number generators.
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5-3
5
Using the Genetic Algorithm
Optimize Using ga
In this section...
“Calling the Function ga at the Command Line” on page 5-4
“Use the Optimization App” on page 5-4
Calling the Function ga at the Command Line
To use the genetic algorithm at the command line, call the genetic algorithm
function ga with the syntax
[x fval] = ga(@fitnessfun, nvars, options)
where
• @fitnessfun is a handle to the fitness function.
• nvars is the number of independent variables for the fitness function.
• options is a structure containing options for the genetic algorithm. If you
do not pass in this argument, ga uses its default options.
The results are given by
• x — Point at which the final value is attained
• fval — Final value of the fitness function
For an example, see “Finding the Minimum from the Command Line” on
page 5-12.
Using the function ga is convenient if you want to
• Return results directly to the MATLAB workspace
• Run the genetic algorithm multiple times with different options, by calling
ga from a file
Use the Optimization App
To open the Optimization app, enter
5-4
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Optimize Using ga
optimtool('ga')
at the command line, or enter optimtool and then choose ga from the Solver
menu.
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5-5
5
Using the Genetic Algorithm
You can also start the tool from the MATLAB Apps tab.
5-6
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Optimize Using ga
To use the Optimization app, you must first enter the following information:
• Fitness function — The objective function you want to minimize. Enter
the fitness function in the form @fitnessfun, where fitnessfun.m is a file
that computes the fitness function. “Compute Objective Functions” on
page 2-2 explains how write this file. The @ sign creates a function handle
to fitnessfun.
• Number of variables — The length of the input vector to the fitness
function. For the function my_fun described in “Compute Objective
Functions” on page 2-2, you would enter 2.
You can enter constraints or a nonlinear constraint function for the problem
in the Constraints pane. If the problem is unconstrained, leave these fields
blank.
To run the genetic algorithm, click the Start button. The tool displays the
results of the optimization in the Run solver and view results pane.
You can change the options for the genetic algorithm in the Options pane.
To view the options in one of the categories listed in the pane, click the +
sign next to it.
For more information,
• See “Optimization App” in the Optimization Toolbox documentation.
• See “Minimize Rastrigin’s Function” on page 5-8 for an example of using
the tool.
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5-7
5
Using the Genetic Algorithm
Minimize Rastrigin’s Function
In this section...
“Rastrigin’s Function” on page 5-8
“Finding the Minimum of Rastrigin’s Function” on page 5-10
“Finding the Minimum from the Command Line” on page 5-12
“Displaying Plots” on page 5-13
Rastrigin’s Function
This section presents an example that shows how to find the minimum of
Rastrigin’s function, a function that is often used to test the genetic algorithm.
For two independent variables, Rastrigin’s function is defined as
Ras( x) = 20 + x12 + x22 − 10 ( cos 2πx1 + cos 2πx2 ) .
Global Optimization Toolbox software contains the rastriginsfcn.m file,
which computes the values of Rastrigin’s function. The following figure shows
a plot of Rastrigin’s function.
5-8
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Minimize Rastrigin’s Function
Global minimum at [0 0]
As the plot shows, Rastrigin’s function has many local minima—the “valleys”
in the plot. However, the function has just one global minimum, which occurs
at the point [0 0] in the x-y plane, as indicated by the vertical line in the
plot, where the value of the function is 0. At any local minimum other than
[0 0], the value of Rastrigin’s function is greater than 0. The farther the
local minimum is from the origin, the larger the value of the function is at
that point.
Rastrigin’s function is often used to test the genetic algorithm, because its
many local minima make it difficult for standard, gradient-based methods
to find the global minimum.
The following contour plot of Rastrigin’s function shows the alternating
maxima and minima.
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5-9
5
Using the Genetic Algorithm
1
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Global minimum at [0 0]
Finding the Minimum of Rastrigin’s Function
This section explains how to find the minimum of Rastrigin’s function using
the genetic algorithm.
Note Because the genetic algorithm uses random number generators, the
algorithm returns slightly different results each time you run it.
To find the minimum, do the following steps:
1 Enter optimtool('ga') at the command line to open the Optimization app.
2 Enter the following in the Optimization app:
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Minimize Rastrigin’s Function
• In the Fitness function field, enter @rastriginsfcn.
• In the Number of variables field, enter 2, the number of independent
variables for Rastrigin’s function.
The Fitness function and Number of variables fields should appear
as shown in the following figure.
3 Click the Start button in the Run solver and view results pane, as
shown in the following figure.
While the algorithm is running, the Current iteration field displays
the number of the current generation. You can temporarily pause the
algorithm by clicking the Pause button. When you do so, the button name
changes to Resume. To resume the algorithm from the point at which
you paused it, click Resume.
When the algorithm is finished, the Run solver and view results pane
appears as shown in the following figure. Your numerical results might
differ from those in the figure, since ga is stochastic.
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Using the Genetic Algorithm
The display shows:
• The final value of the fitness function when the algorithm terminated:
Objective function value: 5.550533778020394E-4
Note that the value shown is very close to the actual minimum value of
Rastrigin’s function, which is 0. “Setting the Initial Range” on page 5-76,
“Setting the Amount of Mutation” on page 5-91, and “Set Maximum
Number of Generations” on page 5-111 describe some ways to get a
result that is closer to the actual minimum.
• The reason the algorithm terminated.
Optimization terminated: maximum number of generations exceeded.
• The final point, which in this example is [-0.002 -0.001].
Finding the Minimum from the Command Line
To find the minimum of Rastrigin’s function from the command line, enter
rng(1,'twister') % for reproducibility
[x fval exitflag] = ga(@rastriginsfcn, 2)
This returns
Optimization terminated:
average change in the fitness value less than options.TolFun.
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Minimize Rastrigin’s Function
x =
-0.0017
-0.0185
fval =
0.0682
exitflag =
1
• x is the final point returned by the algorithm.
• fval is the fitness function value at the final point.
• exitflag is integer value corresponding to the reason that the algorithm
terminated.
Note Because the genetic algorithm uses random number generators, the
algorithm returns slightly different results each time you run it.
Displaying Plots
The Optimization app Plot functions pane enables you to display various
plots that provide information about the genetic algorithm while it is running.
This information can help you change options to improve the performance of
the algorithm. For example, to plot the best and mean values of the fitness
function at each generation, select the box next to Best fitness, as shown in
the following figure.
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Using the Genetic Algorithm
When you click Start, the Optimization app displays a plot of the best and
mean values of the fitness function at each generation.
Try this on “Minimize Rastrigin’s Function” on page 5-8:
When the algorithm stops, the plot appears as shown in the following figure.
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Minimize Rastrigin’s Function
The points at the bottom of the plot denote the best fitness values, while
the points above them denote the averages of the fitness values in each
generation. The plot also displays the best and mean values in the current
generation numerically at the top.
To get a better picture of how much the best fitness values are decreasing, you
can change the scaling of the y-axis in the plot to logarithmic scaling. To do so,
1 Select Axes Properties from the Edit menu in the plot window to open
the Property Editor attached to your figure window as shown below.
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Using the Genetic Algorithm
2 Click the Y Axis tab.
3 In the Y Scale pane, select Log.
The plot now appears as shown in the following figure.
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Minimize Rastrigin’s Function
Typically, the best fitness value improves rapidly in the early generations,
when the individuals are farther from the optimum. The best fitness value
improves more slowly in later generations, whose populations are closer
to the optimal point.
Note When you display more than one plot, you can open a larger version
of a plot in a separate window. Right-click (Ctrl-click for Mac) on a blank
area in a plot while ga is running, or after it has stopped, and choose the
sole menu item.
“Plot Options” on page 9-32 describes the types of plots you can create.
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Using the Genetic Algorithm
Genetic Algorithm Terminology
In this section...
“Fitness Functions” on page 5-18
“Individuals” on page 5-18
“Populations and Generations” on page 5-19
“Diversity” on page 5-19
“Fitness Values and Best Fitness Values” on page 5-20
“Parents and Children” on page 5-20
Fitness Functions
The fitness function is the function you want to optimize. For standard
optimization algorithms, this is known as the objective function. The toolbox
software tries to find the minimum of the fitness function.
Write the fitness function as a file or anonymous function, and pass it as a
function handle input argument to the main genetic algorithm function.
Individuals
An individual is any point to which you can apply the fitness function. The
value of the fitness function for an individual is its score. For example, if
the fitness function is
f ( x1 , x2 , x3 ) = ( 2 x1 + 1) + ( 3 x2 + 4 ) + ( x3 − 2 ) ,
2
2
2
the vector (2, -3, 1), whose length is the number of variables in the problem, is
an individual. The score of the individual (2, –3, 1) is f(2, –3, 1) = 51.
An individual is sometimes referred to as a genome and the vector entries of
an individual as genes.
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Genetic Algorithm Terminology
Populations and Generations
A population is an array of individuals. For example, if the size of the
population is 100 and the number of variables in the fitness function is 3,
you represent the population by a 100-by-3 matrix. The same individual can
appear more than once in the population. For example, the individual (2, -3,
1) can appear in more than one row of the array.
At each iteration, the genetic algorithm performs a series of computations
on the current population to produce a new population. Each successive
population is called a new generation.
Diversity
Diversity refers to the average distance between individuals in a population.
A population has high diversity if the average distance is large; otherwise it
has low diversity. In the following figure, the population on the left has high
diversity, while the population on the right has low diversity.
Diversity is essential to the genetic algorithm because it enables the algorithm
to search a larger region of the space.
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Using the Genetic Algorithm
Fitness Values and Best Fitness Values
The fitness value of an individual is the value of the fitness function for that
individual. Because the toolbox software finds the minimum of the fitness
function, the best fitness value for a population is the smallest fitness value
for any individual in the population.
Parents and Children
To create the next generation, the genetic algorithm selects certain individuals
in the current population, called parents, and uses them to create individuals
in the next generation, called children. Typically, the algorithm is more likely
to select parents that have better fitness values.
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How the Genetic Algorithm Works
How the Genetic Algorithm Works
In this section...
“Outline of the Algorithm” on page 5-21
“Initial Population” on page 5-22
“Creating the Next Generation” on page 5-23
“Plots of Later Generations” on page 5-25
“Stopping Conditions for the Algorithm” on page 5-26
“Selection” on page 5-29
“Reproduction Options” on page 5-29
“Mutation and Crossover” on page 5-30
Outline of the Algorithm
The following outline summarizes how the genetic algorithm works:
1 The algorithm begins by creating a random initial population.
2 The algorithm then creates a sequence of new populations. At each step,
the algorithm uses the individuals in the current generation to create the
next population. To create the new population, the algorithm performs
the following steps:
a Scores each member of the current population by computing its fitness
value.
b Scales the raw fitness scores to convert them into a more usable range of
values.
c Selects members, called parents, based on their fitness.
d Some of the individuals in the current population that have lower fitness
are chosen as elite. These elite individuals are passed to the next
population.
e Produces children from the parents. Children are produced either by
making random changes to a single parent—mutation—or by combining
the vector entries of a pair of parents—crossover.
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Using the Genetic Algorithm
f
Replaces the current population with the children to form the next
generation.
3 The algorithm stops when one of the stopping criteria is met. See “Stopping
Conditions for the Algorithm” on page 5-26.
Initial Population
The algorithm begins by creating a random initial population, as shown in
the following figure.
1
Initial population
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In this example, the initial population contains 20 individuals, which is the
default value of Population size in the Population options. Note that all
the individuals in the initial population lie in the upper-right quadrant of the
picture, that is, their coordinates lie between 0 and 1, because the default
value of Initial range in the Population options is [0;1].
If you know approximately where the minimal point for a function lies, you
should set Initial range so that the point lies near the middle of that range.
For example, if you believe that the minimal point for Rastrigin’s function is
near the point [0 0], you could set Initial range to be [-1;1]. However, as
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How the Genetic Algorithm Works
this example shows, the genetic algorithm can find the minimum even with a
less than optimal choice for Initial range.
Creating the Next Generation
At each step, the genetic algorithm uses the current population to create the
children that make up the next generation. The algorithm selects a group of
individuals in the current population, called parents, who contribute their
genes—the entries of their vectors—to their children. The algorithm usually
selects individuals that have better fitness values as parents. You can specify
the function that the algorithm uses to select the parents in the Selection
function field in the Selection options.
The genetic algorithm creates three types of children for the next generation:
• Elite children are the individuals in the current generation with the
best fitness values. These individuals automatically survive to the next
generation.
• Crossover children are created by combining the vectors of a pair of parents.
• Mutation children are created by introducing random changes, or
mutations, to a single parent.
The following schematic diagram illustrates the three types of children.
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Using the Genetic Algorithm
“Mutation and Crossover” on page 5-30 explains how to specify the number of
children of each type that the algorithm generates and the functions it uses
to perform crossover and mutation.
The following sections explain how the algorithm creates crossover and
mutation children.
Crossover Children
The algorithm creates crossover children by combining pairs of parents in
the current population. At each coordinate of the child vector, the default
crossover function randomly selects an entry, or gene, at the same coordinate
from one of the two parents and assigns it to the child. For problems with
linear constraints, the default crossover function creates the child as a
random weighted average of the parents.
Mutation Children
The algorithm creates mutation children by randomly changing the genes of
individual parents. By default, for unconstrained problems the algorithm
adds a random vector from a Gaussian distribution to the parent. For bounded
or linearly constrained problems, the child remains feasible.
The following figure shows the children of the initial population, that is, the
population at the second generation, and indicates whether they are elite,
crossover, or mutation children.
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How the Genetic Algorithm Works
1
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Crossover children
Mutation children
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Plots of Later Generations
The following figure shows the populations at iterations 60, 80, 95, and 100.
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5
Using the Genetic Algorithm
Iteration 60
Iteration 80
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Iteration 95
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Iteration 100
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As the number of generations increases, the individuals in the population get
closer together and approach the minimum point [0 0].
Stopping Conditions for the Algorithm
The genetic algorithm uses the following conditions to determine when to stop:
• Generations — The algorithm stops when the number of generations
reaches the value of Generations.
• Time limit — The algorithm stops after running for an amount of time in
seconds equal to Time limit.
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How the Genetic Algorithm Works
• Fitness limit — The algorithm stops when the value of the fitness function
for the best point in the current population is less than or equal to Fitness
limit.
• Stall generations — The algorithm stops when the average relative
change in the fitness function value over Stall generations is less than
Function tolerance.
• Stall time limit — The algorithm stops if there is no improvement in
the objective function during an interval of time in seconds equal to Stall
time limit.
• Stall test — The stall condition is either average change or geometric
weighted. For geometric weighted, the weighting function is 1/2n, where
n is the number of generations prior to the current. Both stall conditions
apply to the relative change in the fitness function over Stall generations.
• Function Tolerance — The algorithm runs until the average relative
change in the fitness function value over Stall generations is less than
Function tolerance.
• Nonlinear constraint tolerance — The Nonlinear constraint
tolerance is not used as stopping criterion. It is used to determine the
feasibility with respect to nonlinear constraints. Also, a point is feasible
with respect to linear constraints when the constraint violation is below the
square root of Nonlinear constraint tolerance.
The algorithm stops as soon as any one of these conditions is met. You can
specify the values of these criteria in the Stopping criteria pane in the
Optimization app. The default values are shown in the pane.
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5
Using the Genetic Algorithm
When you run the genetic algorithm, the Run solver and view results
panel displays the criterion that caused the algorithm to stop.
The options Stall time limit and Time limit prevent the algorithm from
running too long. If the algorithm stops due to one of these conditions, you
might improve your results by increasing the values of Stall time limit and
Time limit.
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How the Genetic Algorithm Works
Selection
The selection function chooses parents for the next generation based on their
scaled values from the fitness scaling function. An individual can be selected
more than once as a parent, in which case it contributes its genes to more than
one child. The default selection option, Stochastic uniform, lays out a line
in which each parent corresponds to a section of the line of length proportional
to its scaled value. The algorithm moves along the line in steps of equal size.
At each step, the algorithm allocates a parent from the section it lands on.
A more deterministic selection option is Remainder, which performs two steps:
• In the first step, the function selects parents deterministically according
to the integer part of the scaled value for each individual. For example,
if an individual’s scaled value is 2.3, the function selects that individual
twice as a parent.
• In the second step, the selection function selects additional parents using
the fractional parts of the scaled values, as in stochastic uniform selection.
The function lays out a line in sections, whose lengths are proportional to
the fractional part of the scaled value of the individuals, and moves along
the line in equal steps to select the parents.
Note that if the fractional parts of the scaled values all equal 0, as can
occur using Top scaling, the selection is entirely deterministic.
Reproduction Options
Reproduction options control how the genetic algorithm creates the next
generation. The options are
• Elite count — The number of individuals with the best fitness values
in the current generation that are guaranteed to survive to the next
generation. These individuals are called elite children. The default value
of Elite count is 2.
When Elite count is at least 1, the best fitness value can only decrease
from one generation to the next. This is what you want to happen, since the
genetic algorithm minimizes the fitness function. Setting Elite count to a
high value causes the fittest individuals to dominate the population, which
can make the search less effective.
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Using the Genetic Algorithm
• Crossover fraction — The fraction of individuals in the next generation,
other than elite children, that are created by crossover. “Setting the
Crossover Fraction” on page 5-93 describes how the value of Crossover
fraction affects the performance of the genetic algorithm.
Mutation and Crossover
The genetic algorithm uses the individuals in the current generation to create
the children that make up the next generation. Besides elite children, which
correspond to the individuals in the current generation with the best fitness
values, the algorithm creates
• Crossover children by selecting vector entries, or genes, from a pair of
individuals in the current generation and combines them to form a child
• Mutation children by applying random changes to a single individual in the
current generation to create a child
Both processes are essential to the genetic algorithm. Crossover enables the
algorithm to extract the best genes from different individuals and recombine
them into potentially superior children. Mutation adds to the diversity of
a population and thereby increases the likelihood that the algorithm will
generate individuals with better fitness values.
See “Creating the Next Generation” on page 5-23 for an example of how the
genetic algorithm applies mutation and crossover.
You can specify how many of each type of children the algorithm creates as
follows:
• Elite count, in Reproduction options, specifies the number of elite
children.
• Crossover fraction, in Reproduction options, specifies the fraction of
the population, other than elite children, that are crossover children.
For example, if the Population size is 20, the Elite count is 2, and the
Crossover fraction is 0.8, the numbers of each type of children in the next
generation are as follows:
• There are two elite children.
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How the Genetic Algorithm Works
• There are 18 individuals other than elite children, so the algorithm rounds
0.8*18 = 14.4 to 14 to get the number of crossover children.
• The remaining four individuals, other than elite children, are mutation
children.
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5
Using the Genetic Algorithm
Mixed Integer Optimization
In this section...
“Solving Mixed Integer Optimization Problems” on page 5-32
“Characteristics of the Integer ga Solver” on page 5-34
“Integer ga Algorithm” on page 5-40
Solving Mixed Integer Optimization Problems
ga can solve problems when certain variables are integer-valued. Give
IntCon, a vector of the x components that are integers:
[x,fval,exitflag] = ga(fitnessfcn,nvars,A,b,[],[],...
lb,ub,nonlcon,IntCon,options)
IntCon is a vector of positive integers that contains the x components that
are integer-valued. For example, if you want to restrict x(2) and x(10) to
be integers, set IntCon to [2,10].
Note Restrictions exist on the types of problems that ga can solve with
integer variables. In particular, ga does not accept any equality constraints
when there are integer variables. For details, see “Characteristics of the
Integer ga Solver” on page 5-34.
Tip ga solves integer problems best when you provide lower and upper
bounds for every x component.
Mixed Integer Optimization of Rastrigin’s Function
This example shows how to find the minimum of Rastrigin’s function
restricted so the first component of x is an integer. The components of x are
further restricted to be in the region
.
Set up the bounds for your problem
5-32
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Mixed Integer Optimization
lb = [5*pi,-20*pi];
ub = [20*pi,-4*pi];
Set a plot function so you can view the progress of ga
opts = gaoptimset('PlotFcns',@gaplotbestf);
Call the ga solver where x(1) has integer values
rng(1,'twister') % for reproducibility
IntCon = 1;
[x,fval,exitflag] = ga(@rastriginsfcn,2,[],[],[],[],...
lb,ub,[],IntCon,opts)
Optimization terminated: average change in the penalty fitness value less t
and constraint violation is less than options.TolCon.
x =
16.0000
-12.9325
fval =
424.1355
exitflag =
1
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5-33
5
Using the Genetic Algorithm
ga converges quickly to the solution.
Characteristics of the Integer ga Solver
There are some restrictions on the types of problems that ga can solve when
you include integer constraints:
• No linear equality constraints. You must have Aeq = [] and beq = [].
For a possible workaround, see “No Equality Constraints” on page 5-35.
5-34
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Mixed Integer Optimization
• No nonlinear equality constraints. Any nonlinear constraint function must
return [] for the nonlinear equality constraint. For a possible workaround,
see “Example: Integer Programming with a Nonlinear Equality Constraint”
on page 5-36.
• Only doubleVector population type.
• No custom creation function (CreationFcn option), crossover function
(CrossoverFcn option), mutation function (MutationFcn option), or initial
scores (InitialScores option). If you supply any of these, ga overrides
their settings.
• ga uses only the binary tournament selection function (SelectionFcn
option), and overrides any other setting.
• No hybrid function. ga overrides any setting of the HybridFcn option.
• ga ignores the ParetoFraction, DistanceMeasureFcn, InitialPenalty,
and PenaltyFactor options.
The listed restrictions are mainly natural, not arbitrary. For example:
• There are no hybrid functions that support integer constraints. So ga does
not use hybrid functions when there are integer constraints.
• To obtain integer variables, ga uses special creation, crossover, and
mutation functions.
No Equality Constraints
You cannot use equality constraints and integer constraints in the same
problem. You can try to work around this restriction by including two
inequality constraints for each linear equality constraint. For example, to
try to include the constraint
3x1 – 2x2 = 5,
create two inequality constraints:
3x1 – 2x2 ≤ 5
3x1 – 2x2 ≥ 5.
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5-35
5
Using the Genetic Algorithm
To write these constraints in the form A x ≤ b, multiply the second inequality
by -1:
–3x1 + 2x2 ≤ –5.
You can try to include the equality constraint using A = [3,-2;-3,2] and
b = [5;-5].
Be aware that this procedure can fail; ga has difficulty with simultaneous
integer and equality constraints.
Example: Integer Programming with a Nonlinear Equality Constraint.
This example attempts to locate the minimum of the Ackley function in five
dimensions with these constraints:
• x(1), x(3), and x(5) are integers.
• norm(x) = 4.
The Ackley function, described briefly in “Resuming ga From the Final
Population” on page 5-59, is difficult to minimize. Adding integer and equality
constraints increases the difficulty.
To include the nonlinear equality constraint, give a small tolerance tol that
allows the norm of x to be within tol of 4. Without a tolerance, the nonlinear
equality constraint is never satisfied, and the solver does not realize when it
has a feasible solution.
1 Write the expression norm(x) = 4 as two “less than zero” inequalities:
norm(x) - 4 ≤ 0
-(norm(x) - 4) ≤ 0.
2 Allow a small tolerance in the inequalities:
norm(x) - 4 - tol ≤ 0
-(norm(x) - 4) - tol ≤ 0.
3 Write a nonlinear inequality constraint function that implements these
inequalities:
5-36
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Mixed Integer Optimization
function [c, ceq] = eqCon(x)
ceq = [];
rad = 4;
tol = 1e-3;
confcnval = norm(x) - rad;
c = [confcnval - tol;-confcnval - tol];
4 Set options:
• StallGenLimit = 50 — Allow the solver to try for a while.
• TolFun = 1e-10 — Specify a stricter stopping criterion than usual.
• Generations = 300 — Allow more generations than default.
• PlotFcns = @gaplotbestfun — Observe the optimization.
opts = gaoptimset('StallGenLimit',50,'TolFun',1e-10,...
'Generations',300,'PlotFcns',@gaplotbestfun);
5 Set lower and upper bounds to help the solver:
nVar = 5;
lb = -5*ones(1,nVar);
ub = 5*ones(1,nVar);
6 Solve the problem:
rng(2,'twister') % for reproducibility
[x,fval,exitflag] = ga(@ackleyfcn,nVar,[],[],[],[], ...
lb,ub,@eqCon,[1 3 5],opts);
Optimization terminated: average change in the penalty fitness
value less than options.TolFun and constraint violation is
less than options.TolCon.
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5-37
5
Using the Genetic Algorithm
7 Examine the solution:
x,fval,exitflag,norm(x)
x =
0
0.0000
-2.0000
-1.7344
fval =
5.2441
exitflag =
1
5-38
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3.0000
Mixed Integer Optimization
ans =
4.0010
The odd x components are integers, as specified. The norm of x is 4, to within
the given tolerance of 1e-3.
8 Despite the exit flag of 1, the solution is not the global optimum. Run the
problem again and examine the solution:
opts = gaoptimset(opts,'Display','off');
[x2,fval2,exitflag2] = ga(@ackleyfcn,nVar,[],[],[],[], ...
lb,ub,@eqCon,[1 3 5],opts);
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5-39
5
Using the Genetic Algorithm
Examine the second solution:
x2,fval2,exitflag2,norm(x2)
x2 =
1.0000
3.1545
2.0000
1.0207
0
fval2 =
4.6111
exitflag2 =
0
ans =
3.9991
The second run gives a better solution (lower fitness function value). Again,
the odd x components are integers, and the norm of x2 is 4, to within the
given tolerance of 1e-3.
Be aware that this procedure can fail; ga has difficulty with simultaneous
integer and equality constraints.
Integer ga Algorithm
Integer programming with ga involves several modifications of the basic
algorithm (see “How the Genetic Algorithm Works” on page 5-21). For integer
programming:
• Special creation, crossover, and mutation functions enforce variables to be
integers. For details, see Deep et al. [2].
• The genetic algorithm attempts to minimize a penalty function, not the
fitness function. The penalty function includes a term for infeasibility. This
penalty function is combined with binary tournament selection to select
individuals for subsequent generations. The penalty function value of a
member of a population is:
-
5-40
If the member is feasible, the penalty function is the fitness function.
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Mixed Integer Optimization
-
If the member is infeasible, the penalty function is the maximum fitness
function among feasible members of the population, plus a sum of the
constraint violations of the (infeasible) point.
For details of the penalty function, see Deb [1].
References
[1] Deb, Kalyanmoy. An efficient constraint handling method for genetic
algorithms. Computer Methods in Applied Mechanics and Engineering,
186(2–4), pp. 311–338, 2000.
[2] Deep, Kusum, Krishna Pratap Singh, M.L. Kansal, and C. Mohan. A real
coded genetic algorithm for solving integer and mixed integer optimization
problems. Applied Mathematics and Computation, 212(2), pp. 505–518, 2009.
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5-41
5
Using the Genetic Algorithm
Solving a Mixed Integer Engineering Design Problem
Using the Genetic Algorithm
This example shows how to solve a mixed integer engineering design problem
using the Genetic Algorithm (ga) solver in Global Optimization Toolbox.
The problem illustrated in this example involves the design of a stepped
cantilever beam. In particular, the beam must be able to carry a prescribed
end load. We will solve a problem to minimize the beam volume subject to
various engineering design constraints.
In this example we will solve two bounded versions of the problem published
in [1].
Stepped Cantilever Beam Design Problem
A stepped cantilever beam is supported at one end and a load is applied at
the free end, as shown in the figure below. The beam must be able to support
the given load, , at a fixed distance from the support. Designers of the beam
can vary the width ( ) and height ( ) of each section. We will assume that each
section of the cantilever has the same length, .
Volume of the beam
The volume of the beam, , is the sum of the volume of the individual sections
Constraints on the Design : 1 - Bending Stress
Consider a single cantilever beam, with the centre of coordinates at the centre
of its cross section at the free end of the beam. The bending stress at a point
in the beam is given by the following equation
5-42
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Solving a Mixed Integer Engineering Design Problem Using the Genetic Algorithm
where is the bending moment at , is the distance from the end load and is
the area moment of inertia of the beam.
Now, in the stepped cantilever beam shown in the figure, the maximum
moment of each section of the beam is , where is the maximum distance from
the end load, , for each section of the beam. Therefore, the maximum stress
for the -th section of the beam, , is given by
where the maximum stress occurs at the edge of the beam, . The area
moment of inertia of the -th section of the beam is given by
Substituting this into the equation for gives
The bending stress in each part of the cantilever should not exceed the
maximum allowable stress, . Consequently, we can finally state the five
bending stress constraints (one for each step of the cantilever)
Constraints on the Design : 2 - End deflection
The end deflection of the cantilever can be calculated using Castigliano’s
second theorem, which states that
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5-43
5
Using the Genetic Algorithm
where is the deflection of the beam, is the energy stored in the beam due
to the applied force, .
The energy stored in a cantilever beam is given by
where is the moment of the applied force at .
Given that for a cantilever beam, we can write the above equation as
where is the area moment of inertia of the -th part of the cantilever.
Evaluating the integral gives the following expression for .
Applying Castigliano’s theorem, the end deflection of the beam is given by
Now, the end deflection of the cantilever, , should be less than the maximum
allowable deflection, , which gives us the following constraint.
Constraints on the Design : 3 - Aspect ratio
For each step of the cantilever, the aspect ratio must not exceed a maximum
allowable aspect ratio, . That is,
for
State the Optimization Problem
We are now able to state the problem to find the optimal parameters for the
stepped cantilever beam given the stated constraints.
Let , , , , , , , , and
5-44
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Solving a Mixed Integer Engineering Design Problem Using the Genetic Algorithm
Minimize:
Subject to:
The first step of the beam can only be machined to the nearest centimetre.
That is, and must be integer. The remaining variables are continuous. The
bounds on the variables are given below:-
Design Parameters for this Problem
For the problem we will solve in this example, the end load that the beam
must support is .
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5-45
5
Using the Genetic Algorithm
The beam lengths and maximum end deflection are:
• Total beam length,
• Individual section of beam,
• Maximum beam end deflection,
The maximum allowed stress in each step of the beam,
Young’s modulus of each step of the beam,
Solve the Mixed Integer Optimization Problem
We now solve the problem described in State the Optimization Problem.
Define the Fitness and Constraint Functions
Examine the MATLAB files cantileverVolume.m and
cantileverConstraints.m to see how the fitness and constraint
functions are implemented.
A note on the linear constraints: When linear constraints are specified to ga,
you normally specify them via the A, b, Aeq and beq inputs. In this case we
have specified them via the nonlinear constraint function. This is because
later in this example, some of the variables will become discrete. When
there are discrete variables in the problem it is far easier to specify linear
constraints in the nonlinear constraint function. The alternative is to modify
the linear constraint matrices to work in the transformed variable space,
which is not trivial and maybe not possible. Also, in the mixed integer ga
solver, the linear constraints are not treated any differently to the nonlinear
constraints regardless of how they are specified.
Set the Bounds
Create vectors containing the lower bound (lb) and upper bound constraints
(ub).
lb = [1 30 2.4 45 2.4 45 1 30 1 30];
ub = [5 65 3.1 60 3.1 60 5 65 5 65];
5-46
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Solving a Mixed Integer Engineering Design Problem Using the Genetic Algorithm
Set the Options
To obtain a more accurate solution, we increase the PopulationSize, and
Generations options from their default values, and decrease the EliteCount
and TolFun options. These settings cause ga to use a larger population
(increased PopulationSize), to increase the search of the design space (reduced
EliteCount), and to keep going until its best member changes by very little
(small TolFun). We also specify a plot function to monitor the penalty function
value as ga progresses.
Note that there are a restricted set of ga options available when solving
mixed integer problems - see Global Optimization Toolbox User’s Guide for
more details.
opts = gaoptimset(...
'PopulationSize', 150, ...
'Generations', 200, ...
'EliteCount', 10, ...
'TolFun', 1e-8, ...
'PlotFcns', @gaplotbestf);
Call ga to Solve the Problem
We can now call ga to solve the problem. In the problem statement and
are integer variables. We specify this by passing the index vector [1 2] to
ga after the nonlinear constraint input and before the options input. We also
seed and set the random number generator here for reproducibility.
rng(0, 'twister');
[xbest, fbest, exitflag] = ga(@cantileverVolume, 10, [], [], [], [], ...
lb, ub, @cantileverConstraints, [1 2], opts);
Optimization terminated: maximum number of generations exceeded.
Analyze the Results
If a problem has integer constraints, ga reformulates it internally. In
particular, the fitness function in the problem is replaced by a penalty
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5-47
5
Using the Genetic Algorithm
function which handles the constraints. For feasible population members, the
penalty function is the same as the fitness function.
The solution returned from ga is displayed below. Note that the section
nearest the support is constrained to have a width ( ) and height ( ) which is
an integer value and this constraint has been honored by GA.
display(xbest);
xbest =
Columns 1 through 7
3.0000
60.0000
2.8315
56.6125
2.5621
51.2047
2.2244
Columns 8 through 10
44.4723
1.7675
35.3416
We can also ask ga to return the optimal volume of the beam.
fprintf('\nCost function returned by ga = %g\n', fbest);
Cost function returned by ga = 63287.6
Add Discrete Non-Integer Variable Constraints
The engineers are now informed that the second and third steps of the
cantilever can only have widths and heights that are chosen from a standard
set. In this section, we show how to add this constraint to the optimization
problem. Note that with the addition of this constraint, this problem is
identical to that solved in [1].
First, we state the extra constraints that will be added to the above
optimization
5-48
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Solving a Mixed Integer Engineering Design Problem Using the Genetic Algorithm
• The width of the second and third steps of the beam must be chosen from
the following set:- [2.4, 2.6, 2.8, 3.1] cm
• The height of the second and third steps of the beam must be chosen from
the following set:- [45, 50, 55, 60] cm
To solve this problem, we need to be able to specify the variables , , and as
discrete variables. To specify a component as taking discrete values from the
set , optimize with an integer variable taking values from 1 to , and use as
the discrete value. To specify the range (1 to ), set 1 as the lower bound and
as the upper bound.
So, first we transform the bounds on the discrete variables. Each set has 4
members and we will map the discrete variables to an integer in the range
[1, 4]. So, to map these variables to be integer, we set the lower bound to 1
and the upper bound to 4 for each of the variables.
lb = [1 30 1 1 1 1 1 30 1 30];
ub = [5 65 4 4 4 4 5 65 5 65];
Transformed (integer) versions of , , and will now be passed to the fitness
and constraint functions when the ga solver is called. To evaluate these
functions correctly, , , and need to be transformed to a member of the given
discrete set in these functions. To see how this is done, examine the MATLAB
files cantileverVolumeWithDisc.m, cantileverConstraintsWithDisc.m
and cantileverMapVariables.m.
Now we can call ga to solve the problem with discrete variables. In this case
are integers. This means that we pass the index vector 1:6 to ga to define
the integer variables.
rng(0, 'twister');
[xbestDisc, fbestDisc, exitflagDisc] = ga(@cantileverVolumeWithDisc, ...
10, [], [], [], [], lb, ub, @cantileverConstraintsWithDisc, 1:6, opts);
Optimization terminated: maximum number of generations exceeded.
Analyze the Results
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5-49
5
Using the Genetic Algorithm
xbestDisc(3:6) are returned from ga as integers (i.e. in their transformed
state). We need to reverse the transform to retrieve the value in their
engineering units.
xbestDisc = cantileverMapVariables(xbestDisc);
display(xbestDisc);
xbestDisc =
Columns 1 through 7
3.0000
60.0000
3.1000
55.0000
2.6000
50.0000
2.2808
Columns 8 through 10
45.6157
1.7499
34.9990
As before, the solution returned from ga honors the constraint that and are
integers. We can also see that , are chosen from the set [2.4, 2.6, 2.8, 3.1] cm
and , are chosen from the set [45, 50, 55, 60] cm.
Recall that we have added additional constraints on the variables x(3), x(4),
x(5) and x(6). As expected, when there are additional discrete constraints
on these variables, the optimal solution has a higher minimum volume. Note
further that the solution reported in [1] has a minimum volume of and that
we find a solution which is approximately the same as that reported in [1].
fprintf('\nCost function returned by ga = %g\n', fbestDisc);
Cost function returned by ga = 64578.6
Summary
This example illustrates how to use the genetic algorithm solver, ga, to solve a
constrained nonlinear optimization problem which has integer constraints.
The example also shows how to handle problems that have discrete variables
in the problem formulation.
5-50
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Solving a Mixed Integer Engineering Design Problem Using the Genetic Algorithm
References
[1] Survey of discrete variable optimization for structural design, P.B.
Thanedar, G.N. Vanderplaats, J. Struct. Eng., 121 (3), 301-306 (1995)
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5-51
5
Using the Genetic Algorithm
Nonlinear Constraint Solver Algorithm
The genetic algorithm uses the Augmented Lagrangian Genetic Algorithm
(ALGA) to solve nonlinear constraint problems without integer constraints.
The optimization problem solved by the ALGA algorithm is
min f ( x)
x
such that
ci ( x) ≤ 0, i = 1 m
ceqi ( x) = 0, i = m + 1 mt
A⋅x ≤ b
Aeq ⋅ x = beq
lb ≤ x ≤ ub,
where c(x) represents the nonlinear inequality constraints, ceq(x) represents
the equality constraints, m is the number of nonlinear inequality constraints,
and mt is the total number of nonlinear constraints.
The Augmented Lagrangian Genetic Algorithm (ALGA) attempts to solve a
nonlinear optimization problem with nonlinear constraints, linear constraints,
and bounds. In this approach, bounds and linear constraints are handled
separately from nonlinear constraints. A subproblem is formulated by
combining the fitness function and nonlinear constraint function using the
Lagrangian and the penalty parameters. A sequence of such optimization
problems are approximately minimized using the genetic algorithm such that
the linear constraints and bounds are satisfied.
A subproblem formulation is defined as
m
( x,  , s,  )  f ( x)   i si log(si  ci ( x)) 
i1
mt

i m 1
where
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i ceqi ( x) 
 mt
 ceqi ( x)2 ,
2 i m1
Nonlinear Constraint Solver Algorithm
• The components λi of the vector λ are nonnegative and are known as
Lagrange multiplier estimates
• The elements si of the vector s are nonnegative shifts
• ρ is the positive penalty parameter.
The algorithm begins by using an initial value for the penalty parameter
(InitialPenalty).
The genetic algorithm minimizes a sequence of subproblems, each of which
is an approximation of the original problem. Each subproblem has a fixed
value of λ, s, and ρ. When the subproblem is minimized to a required
accuracy and satisfies feasibility conditions, the Lagrangian estimates
are updated. Otherwise, the penalty parameter is increased by a penalty
factor (PenaltyFactor). This results in a new subproblem formulation and
minimization problem. These steps are repeated until the stopping criteria
are met.
Each subproblem solution represents one generation. The number of function
evaluations per generation is therefore much higher when using nonlinear
constraints than otherwise.
For a complete description of the algorithm, see the following references:
References
[1] Conn, A. R., N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent
Augmented Lagrangian Algorithm for Optimization with General Constraints
and Simple Bounds,” SIAM Journal on Numerical Analysis, Volume 28,
Number 2, pages 545–572, 1991.
[2] Conn, A. R., N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent
Augmented Lagrangian Barrier Algorithm for Optimization with General
Inequality Constraints and Simple Bounds,” Mathematics of Computation,
Volume 66, Number 217, pages 261–288, 1997.
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5-53
5
Using the Genetic Algorithm
Create Custom Plot Function
In this section...
“About Custom Plot Functions” on page 5-54
“Creating the Custom Plot Function” on page 5-54
“Using the Plot Function” on page 5-55
“How the Plot Function Works” on page 5-56
About Custom Plot Functions
If none of the plot functions that come with the software is suitable for the
output you want to plot, you can write your own custom plot function, which
the genetic algorithm calls at each generation to create the plot. This example
shows how to create a plot function that displays the change in the best fitness
value from the previous generation to the current generation.
Creating the Custom Plot Function
To create the plot function for this example, copy and paste the following code
into a new file in the MATLAB Editor.
function state = gaplotchange(options, state, flag)
% GAPLOTCHANGE Plots the logarithmic change in the best score from the
% previous generation.
%
persistent last_best % Best score in the previous generation
if(strcmp(flag,'init')) % Set up the plot
set(gca,'xlim',[1,options.Generations],'Yscale','log');
hold on;
xlabel Generation
title('Change in Best Fitness Value')
end
best = min(state.Score); % Best score in the current generation
if state.Generation == 0 % Set last_best to best.
last_best = best;
else
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Create Custom Plot Function
change = last_best - best; % Change in best score
last_best=best;
plot(state.Generation, change, '.r');
title(['Change in Best Fitness Value'])
end
Then save the file as gaplotchange.m in a folder on the MATLAB path.
Using the Plot Function
To use the custom plot function, select Custom in the Plot functions pane
and enter @gaplotchange in the field to the right. To compare the custom plot
with the best fitness value plot, also select Best fitness. Now, if you run the
example described in “Minimize Rastrigin’s Function” on page 5-8, the tool
displays plots similar to those shown in the following figure.
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5
Using the Genetic Algorithm
Note that because the scale of the y-axis in the lower custom plot is
logarithmic, the plot only shows changes that are greater than 0. The
logarithmic scale enables you to see small changes in the fitness function
that the upper plot does not reveal.
How the Plot Function Works
The plot function uses information contained in the following structures,
which the genetic algorithm passes to the function as input arguments:
• options — The current options settings
• state — Information about the current generation
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Create Custom Plot Function
• flag — String indicating the current status of the algorithm
The most important lines of the plot function are the following:
• persistent last_best
Creates the persistent variable last_best—the best score in the previous
generation. Persistent variables are preserved over multiple calls to the
plot function.
• set(gca,'xlim',[1,options.Generations],'Yscale','log');
Sets up the plot before the algorithm starts. options.Generations is the
maximum number of generations.
• best = min(state.Score)
The field state.Score contains the scores of all individuals in the current
population. The variable best is the minimum score. For a complete
description of the fields of the structure state, see “Structure of the Plot
Functions” on page 9-34.
• change = last_best - best
The variable change is the best score at the previous generation minus the
best score in the current generation.
• plot(state.Generation, change, '.r')
Plots the change at the current generation, whose number is contained in
state.Generation.
The code for gaplotchange contains many of the same elements as the code
for gaplotbestf, the function that creates the best fitness plot.
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5-57
5
Using the Genetic Algorithm
Reproduce Results in Optimization App
To reproduce the results of the last run of the genetic algorithm, select the
Use random states from previous run check box. This resets the states of
the random number generators used by the algorithm to their previous values.
If you do not change any other settings in the Optimization app, the next time
you run the genetic algorithm, it returns the same results as the previous run.
Normally, you should leave Use random states from previous run
unselected to get the benefit of randomness in the genetic algorithm. Select
the Use random states from previous run check box if you want to analyze
the results of that particular run or show the exact results to others. After
the algorithm has run, you can clear your results using the Clear Status
button in the Run solver settings.
Note If you select Include information needed to resume this run,
then selecting Use random states from previous run has no effect on the
initial population created when you import the problem and run the genetic
algorithm on it. The latter option is only intended to reproduce results from
the beginning of a new run, not from a resumed run.
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Resume ga
Resume ga
In this section...
“Resuming ga From the Final Population” on page 5-59
“Resuming ga From a Previous Run” on page 5-64
Resuming ga From the Final Population
The following example shows how export a problem so that when you import
it and click Start, the genetic algorithm resumes from the final population
saved with the exported problem. To run the example, enter the following in
the Optimization app:
1 Set Fitness function to @ackleyfcn, which computes Ackley’s function, a
test function provided with the software.
2 Set Number of variables to 10.
3 Select Best fitness in the Plot functions pane.
4 Click Start.
This displays the following plot, or similar.
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5-59
5
Using the Genetic Algorithm
Suppose you want to experiment by running the genetic algorithm with other
options settings, and then later restart this run from its final population with
its current options settings. You can do this using the following steps:
1 Click Export to Workspace.
2 In the dialog box that appears,
• Select Export problem and options to a MATLAB structure
named.
• Enter a name for the problem and options, such as ackley_uniform,
in the text field.
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Resume ga
• Select Include information needed to resume this run.
The dialog box should now appear as in the following figure.
3 Click OK.
This exports the problem and options to a structure in the MATLAB
workspace. You can view the structure in the MATLAB Command Window
by entering
ackley_uniform
ackley_uniform =
fitnessfcn:
nvars:
Aineq:
bineq:
Aeq:
beq:
lb:
ub:
nonlcon:
intcon:
rngstate:
solver:
@ackleyfcn
10
[]
[]
[]
[]
[]
[]
[]
[]
[]
'ga'
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5-61
5
Using the Genetic Algorithm
options: [1x1 struct]
After running the genetic algorithm with different options settings or even a
different fitness function, you can restore the problem as follows:
1 Select Import Problem from the File menu. This opens the dialog box
shown in the following figure.
2 Select ackley_uniform.
3 Click Import.
This sets the Initial population and Initial scores fields in the Population
panel to the final population of the run before you exported the problem.
All other options are restored to their setting during that run. When you
click Start, the genetic algorithm resumes from the saved final population.
The following figure shows the best fitness plots from the original run and
the restarted run.
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Resume ga
Note If, after running the genetic algorithm with the imported problem,
you want to restore the genetic algorithm’s default behavior of generating a
random initial population, delete the population in the Initial population
field.
The version of Ackley’s function in the toolbox differs from the published
version of Ackley’s function in Ackley [1]. The toolbox version has another
exponential applied, leading to flatter regions, so a more difficult optimization
problem.
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5
Using the Genetic Algorithm
References
[1] Ackley, D. H. A connectionist machine for genetic hillclimbing. Kluwer
Academic Publishers, Boston, 1987.
Resuming ga From a Previous Run
By default, ga creates a new initial population each time you run it. However,
you might get better results by using the final population from a previous run
as the initial population for a new run. To do so, you must have saved the
final population from the previous run by calling ga with the syntax
[x,fval,exitflag,output,final_pop] = ga(@fitnessfcn, nvars);
The last output argument is the final population. To run ga using final_pop
as the initial population, enter
options = gaoptimset('InitialPop', final_pop);
[x,fval,exitflag,output,final_pop2] = ...
ga(@fitnessfcn,nvars,[],[],[],[],[],[],[],options);
You can then use final_pop2, the final population from the second run, as
the initial population for a third run.
In Optimization app, you can choose to export a problem in a way that lets
you resume the run. Simply check the box Include information needed
to resume this run when exporting the problem.
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Resume ga
This saves the final population, which becomes the initial population when
imported.
If you want to run a problem that was saved with the final population, but
would rather not use the initial population, simply delete or otherwise change
the initial population in the Options > Population pane.
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5-65
5
Using the Genetic Algorithm
Options and Outputs
In this section...
“Running ga with the Default Options” on page 5-66
“Setting Options at the Command Line” on page 5-67
“Additional Output Arguments” on page 5-69
Running ga with the Default Options
To run the genetic algorithm with the default options, call ga with the syntax
[x fval] = ga(@fitnessfun, nvars)
The input arguments to ga are
• @fitnessfun — A function handle to the file that computes the fitness
function. “Compute Objective Functions” on page 2-2 explains how to write
this file.
• nvars — The number of independent variables for the fitness function.
The output arguments are
• x — The final point
• fval — The value of the fitness function at x
For a description of additional input and output arguments, see the reference
page for ga.
You can run the example described in “Minimize Rastrigin’s Function” on
page 5-8 from the command line by entering
rng(1,'twister') % for reproducibility
[x fval] = ga(@rastriginsfcn, 2)
This returns
Optimization terminated:
average change in the fitness value less than options.TolFun.
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Options and Outputs
x =
-0.0017
-0.0185
fval =
0.0682
Setting Options at the Command Line
You can specify any of the options that are available for ga by passing an
options structure as an input argument to ga using the syntax
[x fval] = ga(@fitnessfun,nvars,[],[],[],[],[],[],[],options)
This syntax does not specify any linear equality, linear inequality, or
nonlinear constraints.
You create the options structure using the function gaoptimset.
options = gaoptimset(@ga)
This returns the structure options with the default values for its fields.
options =
PopulationType:
PopInitRange:
PopulationSize:
EliteCount:
CrossoverFraction:
ParetoFraction:
MigrationDirection:
MigrationInterval:
MigrationFraction:
Generations:
TimeLimit:
FitnessLimit:
StallGenLimit:
StallTest:
StallTimeLimit:
TolFun:
TolCon:
InitialPopulation:
'doubleVector'
[]
'50 when numberOfVariables <= 5, else 200'
'0.05*PopulationSize'
0.8000
[]
'forward'
20
0.2000
'100*numberOfVariables'
Inf
-Inf
50
'averageChange'
Inf
1.0000e-06
1.0000e-06
[]
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Using the Genetic Algorithm
InitialScores:
InitialPenalty:
PenaltyFactor:
PlotInterval:
CreationFcn:
FitnessScalingFcn:
SelectionFcn:
CrossoverFcn:
MutationFcn:
DistanceMeasureFcn:
HybridFcn:
Display:
PlotFcns:
OutputFcns:
Vectorized:
UseParallel:
[]
10
100
1
@gacreationuniform
@fitscalingrank
@selectionstochunif
@crossoverscattered
{[@mutationgaussian]
[]
[]
'final'
[]
[]
'off'
0
[1]
[1]}
The function ga uses these default values if you do not pass in options as an
input argument.
The value of each option is stored in a field of the options structure, such as
options.PopulationSize. You can display any of these values by entering
options. followed by the name of the field. For example, to display the size
of the population for the genetic algorithm, enter
options.PopulationSize
ans =
50 when numberOfVariables <= 5, else 200
To create an options structure with a field value that is different from the
default — for example to set PopulationSize to 100 instead of its default
value 50 — enter
options = gaoptimset('PopulationSize', 100)
This creates the options structure with all values set to their defaults except
for PopulationSize, which is set to 100.
If you now enter,
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Options and Outputs
ga(@fitnessfun,nvars,[],[],[],[],[],[],[],options)
ga runs the genetic algorithm with a population size of 100.
If you subsequently decide to change another field in the options structure,
such as setting PlotFcns to @gaplotbestf, which plots the best fitness
function value at each generation, call gaoptimset with the syntax
options = gaoptimset(options,'PlotFcns',@plotbestf)
This preserves the current values of all fields of options except for PlotFcns,
which is changed to @plotbestf. Note that if you omit the input argument
options, gaoptimset resets PopulationSize to its default value 20.
You can also set both PopulationSize and PlotFcns with the single command
options = gaoptimset('PopulationSize',100,'PlotFcns',@plotbestf)
Additional Output Arguments
To get more information about the performance of the genetic algorithm, you
can call ga with the syntax
[x fval exitflag output population scores] = ga(@fitnessfcn, nvars)
Besides x and fval, this function returns the following additional output
arguments:
• exitflag — Integer value corresponding to the reason the algorithm
terminated
• output — Structure containing information about the performance of the
algorithm at each generation
• population — Final population
• scores — Final scores
See the ga reference page for more information about these arguments.
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Using the Genetic Algorithm
Use Exported Options and Problems
As an alternative to creating an options structure using gaoptimset, you can
set the values of options in the Optimization app and then export the options
to a structure in the MATLAB workspace, as described in the “Importing and
Exporting Your Work” section of the Optimization Toolbox documentation.
If you export the default options in the Optimization app, the resulting
structure options has the same settings as the default structure returned
by the command
options = gaoptimset(@ga)
except that the option 'Display' defaults to 'off' in an exported structure,
and is 'final' in the default at the command line.
If you export a problem structure, ga_problem, from the Optimization app,
you can apply ga to it using the syntax
[x fval] = ga(ga_problem)
The problem structure contains the following fields:
• fitnessfcn — Fitness function
• nvars — Number of variables for the problem
• Aineq — Matrix for inequality constraints
• Bineq — Vector for inequality constraints
• Aeq — Matrix for equality constraints
• Beq — Vector for equality constraints
• LB — Lower bound on x
• UB — Upper bound on x
• nonlcon — Nonlinear constraint function
• options — Options structure
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Reproduce Results
Reproduce Results
Because the genetic algorithm is stochastic—that is, it makes random
choices—you get slightly different results each time you run the genetic
algorithm. The algorithm uses the default MATLAB pseudorandom
number stream. For more information about random number streams, see
RandStream. Each time ga calls the stream, its state changes. So that the
next time ga calls the stream, it returns a different random number. This is
why the output of ga differs each time you run it.
If you need to reproduce your results exactly, you can call ga with an output
argument that contains the current state of the default stream, and then reset
the state to this value before running ga again. For example, to reproduce the
output of ga applied to Rastrigin’s function, call ga with the syntax
rng(1,'twister') % for reproducibility
[x fval exitflag output] = ga(@rastriginsfcn, 2);
Suppose the results are
x,fval
x =
-0.0017
-0.0185
fval =
0.0682
The state of the stream is stored in output.rngstate:
output
output =
problemtype: 'unconstrained'
rngstate: [1x1 struct]
generations: 124
funccount: 6250
message: 'Optimization terminated: average
change in the fitness ...'
To reset the state, enter
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Using the Genetic Algorithm
stream = RandStream.getGlobalStream;
stream.State = output.rngstate.state;
If you now run ga a second time, you get the same results as before:
[x fval exitflag output] = ga(@rastriginsfcn, 2)
Optimization terminated: average change in the fitness value less than opti
x =
-0.0017
-0.0185
fval =
0.0682
exitflag =
1
output =
problemtype: 'unconstrained'
rngstate: [1x1 struct]
generations: 124
funccount: 6250
message: 'Optimization terminated: average
change in the fitness ...'
You can reproduce your run in the Optimization app by checking the box Use
random states from previous run in the Run solver and view results
section.
Note If you do not need to reproduce your results, it is better not to set the
state of the stream, so that you get the benefit of the randomness in the
genetic algorithm.
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Run ga from a File
Run ga from a File
The command-line interface enables you to run the genetic algorithm many
times, with different options settings, using a file. For example, you can run
the genetic algorithm with different settings for Crossover fraction to see
which one gives the best results. The following code runs the function ga 21
times, varying options.CrossoverFraction from 0 to 1 in increments of
0.05, and records the results.
options = gaoptimset('Generations',300,'Display','none');
rng('default') % for reproducibility
record=[];
for n=0:.05:1
options = gaoptimset(options,'CrossoverFraction',n);
[x fval]=ga(@rastriginsfcn,2,[],[],[],[],[],[],[],options);
record = [record; fval];
end
You can plot the values of fval against the crossover fraction with the
following commands:
plot(0:.05:1, record);
xlabel('Crossover Fraction');
ylabel('fval')
The following plot appears.
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Using the Genetic Algorithm
The plot suggests that you get the best results by setting
options.CrossoverFraction to a value somewhere between 0.4 and 0.8.
You can get a smoother plot of fval as a function of the crossover fraction by
running ga 20 times and averaging the values of fval for each crossover
fraction. The following figure shows the resulting plot.
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Run ga from a File
This plot also suggests the range of best choices for
options.CrossoverFraction is 0.4 to 0.8.
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Using the Genetic Algorithm
Population Diversity
In this section...
“Importance of Population Diversity” on page 5-76
“Setting the Initial Range” on page 5-76
“Linearly Constrained Population and Custom Plot Function” on page 5-80
“Setting the Population Size” on page 5-85
Importance of Population Diversity
One of the most important factors that determines the performance of the
genetic algorithm performs is the diversity of the population. If the average
distance between individuals is large, the diversity is high; if the average
distance is small, the diversity is low. Getting the right amount of diversity is
a matter of start and error. If the diversity is too high or too low, the genetic
algorithm might not perform well.
This section explains how to control diversity by setting the Initial range of
the population. “Setting the Amount of Mutation” on page 5-91 describes how
the amount of mutation affects diversity.
This section also explains how to set the population size.
Setting the Initial Range
By default, ga creates a random initial population using a creation function.
You can specify the range of the vectors in the initial population in the Initial
range field in Population options.
Note The initial range restricts the range of the points in the initial
population by specifying the lower and upper bounds. Subsequent generations
can contain points whose entries do not lie in the initial range. Set upper and
lower bounds for all generations in the Bounds fields in the Constraints
panel.
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Population Diversity
If you know approximately where the solution to a problem lies, specify the
initial range so that it contains your guess for the solution. However, the
genetic algorithm can find the solution even if it does not lie in the initial
range, if the population has enough diversity.
The following example shows how the initial range affects the performance
of the genetic algorithm. The example uses Rastrigin’s function, described
in “Minimize Rastrigin’s Function” on page 5-8. The minimum value of the
function is 0, which occurs at the origin.
To run the example, open the ga solver in the Optimization app by entering
optimtool('ga') at the command line. Set the following:
• Set Fitness function to @rastriginsfcn.
• Set Number of variables to 2.
• Select Best fitness in the Plot functions pane of the Options pane.
• Select Distance in the Plot functions pane.
• Set Initial range in the Population pane of the Options pane to [1;1.1].
Click Start in Run solver and view results. Although the results of genetic
algorithm computations are random, your results are similar to the following
figure, with a best fitness function value of approximately 2.
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Using the Genetic Algorithm
The upper plot, which displays the best fitness at each generation, shows
little progress in lowering the fitness value. The lower plot shows the average
distance between individuals at each generation, which is a good measure of
the diversity of a population. For this setting of initial range, there is too little
diversity for the algorithm to make progress.
Next, try setting Initial range to [1;100] and running the algorithm. This
time the results are more variable. You might obtain a plot with a best fitness
value of about 7, as in the following plot. You might obtain different results.
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Population Diversity
This time, the genetic algorithm makes progress, but because the average
distance between individuals is so large, the best individuals are far from
the optimal solution.
Finally, set Initial range to [1;2] and run the genetic algorithm. Again,
there is variability in the result, but you might obtain a result similar to the
following figure. Run the optimization several times, and you eventually
obtain a final point near [0;0], with a fitness function value near 0.
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Using the Genetic Algorithm
The diversity in this case is better suited to the problem, so ga usually returns
a better result than in the previous two cases.
Linearly Constrained Population and Custom Plot
Function
This example shows how the default creation function for linearly constrained
problems, gacreationlinearfeasible, creates a well-dispersed population
that satisfies linear constraints and bounds. It also contains an example of a
custom plot function.
The problem uses the objective function in lincontest6.m, a quadratic:
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Population Diversity
x12
+ x22 − x1 x2 − 2 x1 − 6 x2 .
2
To see code for the function, enter type lincontest6 at the command line.
The constraints are three linear inequalities:
f ( x) =
x1 + x2 ≤ 2,
–x1 + 2x2 ≤ 2,
2x1 + x2 ≤ 3.
Also, the variables xi are restricted to be positive.
1 Create a custom plot function file by cutting and pasting the following code
into a new function file in the MATLAB Editor:
function state = gaplotshowpopulation2(unused,state,flag,fcn)
% This plot function works in 2-d only
if size(state.Population,2) > 2
return;
end
if nargin < 4 % check to see if fitness function exists
fcn = [];
end
% Dimensions to plot
dimensionsToPlot = [1 2];
switch flag
% Plot initialization
case 'init'
pop = state.Population(:,dimensionsToPlot);
plotHandle = plot(pop(:,1),pop(:,2),'*');
set(plotHandle,'Tag','gaplotshowpopulation2')
title('Population plot in two dimension',...
'interp','none')
xlabelStr = sprintf('%s %s','Variable ',...
num2str(dimensionsToPlot(1)));
ylabelStr = sprintf('%s %s','Variable ',...
num2str(dimensionsToPlot(2)));
xlabel(xlabelStr,'interp','none');
ylabel(ylabelStr,'interp','none');
hold on;
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Using the Genetic Algorithm
% plot the inequalities
plot([0 1.5],[2 0.5],'m-.') % x1 + x2 <= 2
plot([0 1.5],[1 3.5/2],'m-.'); % -x1 + 2*x2 <= 2
plot([0 1.5],[3 0],'m-.'); % 2*x1 + x2 <= 3
% plot lower bounds
plot([0 0], [0 2],'m-.'); % lb = [ 0 0];
plot([0 1.5], [0 0],'m-.'); % lb = [ 0 0];
set(gca,'xlim',[-0.7,2.2])
set(gca,'ylim',[-0.7,2.7])
% Contour plot the objective function
if ~isempty(fcn) % if there is a fitness function
range = [-0.5,2;-0.5,2];
pts = 100;
span = diff(range')/(pts - 1);
x = range(1,1): span(1) : range(1,2);
y = range(2,1): span(2) : range(2,2);
pop = zeros(pts * pts,2);
values = zeros(pts,1);
k = 1;
for i = 1:pts
for j = 1:pts
pop(k,:) = [x(i),y(j)];
values(k) = fcn(pop(k,:));
k = k + 1;
end
end
values = reshape(values,pts,pts);
contour(x,y,values);
colorbar
end
% Pause for three seconds to view the initial plot
pause(3);
case 'iter'
pop = state.Population(:,dimensionsToPlot);
plotHandle = findobj(get(gca,'Children'),'Tag',...
'gaplotshowpopulation2');
set(plotHandle,'Xdata',pop(:,1),'Ydata',pop(:,2));
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Population Diversity
end
The custom plot function plots the lines representing the linear inequalities
and bound constraints, plots level curves of the fitness function, and plots
the population as it evolves. This plot function expects to have not only the
usual inputs (options,state,flag), but also a function handle to the
fitness function, @lincontest6 in this example. To generate level curves,
the custom plot function needs the fitness function.
2 At the command line, enter the constraints as a matrix and vectors:
A = [1,1;-1,2;2,1]; b = [2;2;3]; lb = zeros(2,1);
3 Set options to use gaplotshowpopulation2, and pass in @lincontest6 as
the fitness function handle:
options = gaoptimset('PlotFcns',...
{{@gaplotshowpopulation2,@lincontest6}});
4 Run the optimization using options:
[x,fval] = ga(@lincontest6,2,A,b,[],[],lb,[],[],options);
A plot window appears showing the linear constraints, bounds, level curves of
the objective function, and initial distribution of the population:
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Using the Genetic Algorithm
You can see that the initial population is biased to lie on the constraints. This
bias exists when there are linear constraints.
The population eventually concentrates around the minimum point:
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Population Diversity
Setting the Population Size
The Population size field in Population options determines the size of the
population at each generation. Increasing the population size enables the
genetic algorithm to search more points and thereby obtain a better result.
However, the larger the population size, the longer the genetic algorithm
takes to compute each generation.
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Using the Genetic Algorithm
Note You should set Population size to be at least the value of Number
of variables, so that the individuals in each population span the space
being searched.
You can experiment with different settings for Population size that return
good results without taking a prohibitive amount of time to run.
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Fitness Scaling
Fitness Scaling
In this section...
“Scaling the Fitness Scores” on page 5-87
“Comparing Rank and Top Scaling” on page 5-89
Scaling the Fitness Scores
Fitness scaling converts the raw fitness scores that are returned by the
fitness function to values in a range that is suitable for the selection function.
The selection function uses the scaled fitness values to select the parents of
the next generation. The selection function assigns a higher probability of
selection to individuals with higher scaled values.
The range of the scaled values affects the performance of the genetic
algorithm. If the scaled values vary too widely, the individuals with the
highest scaled values reproduce too rapidly, taking over the population gene
pool too quickly, and preventing the genetic algorithm from searching other
areas of the solution space. On the other hand, if the scaled values vary only a
little, all individuals have approximately the same chance of reproduction and
the search will progress very slowly.
The default fitness scaling option, Rank, scales the raw scores based on the
rank of each individual instead of its score. The rank of an individual is its
position in the sorted scores: the rank of the most fit individual is 1, the next
most fit is 2, and so on. The rank scaling function assigns scaled values so that
• The scaled value of an individual with rank n is proportional to 1 / n .
• The sum of the scaled values over the entire population equals the number
of parents needed to create the next generation.
Rank fitness scaling removes the effect of the spread of the raw scores.
The following plot shows the raw scores of a typical population of 20
individuals, sorted in increasing order.
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Using the Genetic Algorithm
Raw Scores of Population
140
130
120
Score
110
100
90
80
70
60
50
0
5
10
Sorted individuals
15
20
The following plot shows the scaled values of the raw scores using rank
scaling.
Scaled Values Using Rank Scaling
4.5
4
Scaled value
3.5
3
2.5
2
1.5
1
0.5
0
5
10
Sorted individuals
15
20
Because the algorithm minimizes the fitness function, lower raw scores have
higher scaled values. Also, because rank scaling assigns values that depend
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Fitness Scaling
only on an individual’s rank, the scaled values shown would be the same for
any population of size 20 and number of parents equal to 32.
Comparing Rank and Top Scaling
To see the effect of scaling, you can compare the results of the genetic
algorithm using rank scaling with one of the other scaling options, such as
Top. By default, top scaling assigns 40 percent of the fittest individuals to the
same scaled value and assigns the rest of the individuals to value 0. Using
the default selection function, only 40 percent of the fittest individuals can
be selected as parents.
The following figure compares the scaled values of a population of size 20 with
number of parents equal to 32 using rank and top scaling.
Comparison of Rank and Top Scaling
Rank scaling
Top scaling
8
7
Scaled value
6
5
4
3
2
1
0
0
5
10
Sorted individuals
15
20
Because top scaling restricts parents to the fittest individuals, it creates
less diverse populations than rank scaling. The following plot compares the
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Using the Genetic Algorithm
variances of distances between individuals at each generation using rank
and top scaling.
Variance of Distance Between Individuals Using Rank and Top Scaling
0.8
Variance using rank scaling
Variance using top scaling
0.7
0.6
Variance
0.5
0.4
0.3
0.2
0.1
0
5-90
0
10
20
30
40
50
60
Generation
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70
80
90
100
Vary Mutation and Crossover
Vary Mutation and Crossover
In this section...
“Setting the Amount of Mutation” on page 5-91
“Setting the Crossover Fraction” on page 5-93
“Comparing Results for Varying Crossover Fractions” on page 5-98
Setting the Amount of Mutation
The genetic algorithm applies mutations using the option that you specify
on the Mutation function pane. The default mutation option, Gaussian,
adds a random number, or mutation, chosen from a Gaussian distribution,
to each entry of the parent vector. Typically, the amount of mutation, which
is proportional to the standard deviation of the distribution, decreases at
each new generation. You can control the average amount of mutation that
the algorithm applies to a parent in each generation through the Scale and
Shrink options:
• Scale controls the standard deviation of the mutation at the first
generation, which is Scale multiplied by the range of the initial population,
which you specify by the Initial range option.
• Shrink controls the rate at which the average amount of mutation
decreases. The standard deviation decreases linearly so that its final value
equals 1 – Shrink times its initial value at the first generation. For
example, if Shrink has the default value of 1, then the amount of mutation
decreases to 0 at the final step.
You can see the effect of mutation by selecting the plot options Distance and
Range, and then running the genetic algorithm on a problem such as the one
described in “Minimize Rastrigin’s Function” on page 5-8. The following figure
shows the plot after setting the random number generator.
rng default % for reproducibility
options = gaoptimset('PlotFcns',{@gaplotdistance,@gaplotrange},...
'StallGenLimit',200); % to get a long run
[x,fval] = ga(@rastriginsfcn,2,[],[],[],[],[],[],[],options);
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Using the Genetic Algorithm
The upper plot displays the average distance between points in each
generation. As the amount of mutation decreases, so does the average distance
between individuals, which is approximately 0 at the final generation. The
lower plot displays a vertical line at each generation, showing the range
from the smallest to the largest fitness value, as well as mean fitness value.
As the amount of mutation decreases, so does the range. These plots show
that reducing the amount of mutation decreases the diversity of subsequent
generations.
For comparison, the following figure shows the plots for Distance and Range
when you set Shrink to 0.5.
options = gaoptimset(options,'MutationFcn',{@mutationgaussian,1,.5});
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Vary Mutation and Crossover
[x,fval] = ga(@rastriginsfcn,2,[],[],[],[],[],[],[],options);
With Shrink set to 0.5, the average amount of mutation decreases by a
factor of 1/2 by the final generation. As a result, the average distance between
individuals decreases less than before.
Setting the Crossover Fraction
The Crossover fraction field, in the Reproduction options, specifies the
fraction of each population, other than elite children, that are made up of
crossover children. A crossover fraction of 1 means that all children other than
elite individuals are crossover children, while a crossover fraction of 0 means
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Using the Genetic Algorithm
that all children are mutation children. The following example show that
neither of these extremes is an effective strategy for optimizing a function.
The example uses the fitness function whose value at a point is the sum of the
absolute values of the coordinates at the points. That is,
f ( x1 , x2 ,..., xn ) = x1 + x2 +  + xn .
You can define this function as an anonymous function by setting Fitness
function to
@(x) sum(abs(x))
To run the example,
• Set Fitness function to @(x) sum(abs(x)).
• Set Number of variables to 10.
• Set Initial range to [-1; 1].
• Select Best fitness and Distance in the Plot functions pane.
Run the example with the default value of 0.8 for Crossover fraction, in the
Options > Reproduction pane. For reproducibility, switch to the command
line and enter
rng(14,'twister')
Switch back to Optimization app, and click Run solver and view results
> Start. This returns the best fitness value of approximately 0.0799 and
displays the following plots.
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Vary Mutation and Crossover
Crossover Without Mutation
To see how the genetic algorithm performs when there is no mutation, set
Crossover fraction to 1.0 and click Start. This returns the best fitness
value of approximately .66 and displays the following plots.
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Using the Genetic Algorithm
In this case, the algorithm selects genes from the individuals in the initial
population and recombines them. The algorithm cannot create any new genes
because there is no mutation. The algorithm generates the best individual
that it can using these genes at generation number 8, where the best fitness
plot becomes level. After this, it creates new copies of the best individual,
which are then are selected for the next generation. By generation number 17,
all individuals in the population are the same, namely, the best individual.
When this occurs, the average distance between individuals is 0. Since the
algorithm cannot improve the best fitness value after generation 8, it stalls
after 50 more generations, because Stall generations is set to 50.
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Vary Mutation and Crossover
Mutation Without Crossover
To see how the genetic algorithm performs when there is no crossover, set
Crossover fraction to 0 and click Start. This returns the best fitness value
of approximately 3 and displays the following plots.
In this case, the random changes that the algorithm applies never improve the
fitness value of the best individual at the first generation. While it improves
the individual genes of other individuals, as you can see in the upper plot by
the decrease in the mean value of the fitness function, these improved genes
are never combined with the genes of the best individual because there is no
crossover. As a result, the best fitness plot is level and the algorithm stalls at
generation number 50.
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5-97
5
Using the Genetic Algorithm
Comparing Results for Varying Crossover Fractions
The example deterministicstudy.m, which is included in the software,
compares the results of applying the genetic algorithm to Rastrigin’s function
with Crossover fraction set to 0, .2, .4, .6, .8, and 1. The example runs
for 10 generations. At each generation, the example plots the means and
standard deviations of the best fitness values in all the preceding generations,
for each value of the Crossover fraction.
To run the example, enter
deterministicstudy
at the MATLAB prompt. When the example is finished, the plots appear as in
the following figure.
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Vary Mutation and Crossover
The lower plot shows the means and standard deviations of the best fitness
values over 10 generations, for each of the values of the crossover fraction.
The upper plot shows a color-coded display of the best fitness values in each
generation.
For this fitness function, setting Crossover fraction to 0.8 yields the
best result. However, for another fitness function, a different setting for
Crossover fraction might yield the best result.
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5-99
5
Using the Genetic Algorithm
Global vs. Local Minima Using ga
In this section...
“Searching for a Global Minimum” on page 5-100
“Running the Genetic Algorithm on the Example” on page 5-102
Searching for a Global Minimum
Sometimes the goal of an optimization is to find the global minimum or
maximum of a function—a point where the function value is smaller or larger
at any other point in the search space. However, optimization algorithms
sometimes return a local minimum—a point where the function value is
smaller than at nearby points, but possibly greater than at a distant point
in the search space. The genetic algorithm can sometimes overcome this
deficiency with the right settings.
As an example, consider the following function
2


x  
 exp   

for x  100,

  100  
f ( x)  



 exp(1)  ( x  100)( x  102) for x  100.
The following figure shows a plot of the function.
5-100
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Global vs. Local Minima Using ga
Code for generating the figure
t = -10:.1:103;
for ii = 1:length(t)
y(ii) = two_min(t(ii));
end
plot(t,y)
The function has two local minima, one at x = 0, where the function value is
–1, and the other at x = 21, where the function value is –1 – 1/e. Since the
latter value is smaller, the global minimum occurs at x = 21.
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5-101
5
Using the Genetic Algorithm
Running the Genetic Algorithm on the Example
To run the genetic algorithm on this example,
1 Copy and paste the following code into a new file in the MATLAB Editor.
function y = two_min(x)
if x<=20
y = -exp(-(x/100).^2);
else
y = -exp(-1)+(x-100)*(x-102);
end
2 Save the file as two_min.m in a folder on the MATLAB path.
3 In the Optimization app,
• Set Fitness function to @two_min.
• Set Number of variables to 1.
• Click Start.
The genetic algorithm returns a point very close to the local minimum at x = 0.
The following custom plot shows why the algorithm finds the local minimum
rather than the global minimum. The plot shows the range of individuals in
each generation and the population mean.
5-102
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Global vs. Local Minima Using ga
Code for Creating the Figure
function state = gaplot1drange(options,state,flag)
%gaplot1drange Plots the mean and the range of the population.
%
STATE = gaplot1drange(OPTIONS,STATE,FLAG) plots the mean and the range
%
(highest and the lowest) of individuals (1-D only).
%
%
Example:
%
Create an options structure that uses gaplot1drange
%
as the plot function
%
options = gaoptimset('PlotFcns',@gaplot1drange);
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5-103
5
Using the Genetic Algorithm
%
Copyright 2012-2014 The MathWorks, Inc.
if isinf(options.Generations) || size(state.Population,2) > 1
title('Plot Not Available','interp','none');
return;
end
generation = state.Generation;
score = state.Population;
smean = mean(score);
Y = smean;
L = smean - min(score);
U = max(score) - smean;
switch flag
case 'init'
set(gca,'xlim',[1,options.Generations+1]);
plotRange = errorbar(generation,Y,L,U);
set(plotRange,'Tag','gaplot1drange');
title('Range of Population, Mean','interp','none')
xlabel('Generation','interp','none')
case 'iter'
plotRange = findobj(get(gca,'Children'),'Tag','gaplot1drange');
newX = [get(plotRange,'Xdata') generation];
newY = [get(plotRange,'Ydata') Y];
newL = [get(plotRange,'Ldata'); L];
newU = [get(plotRange,'Udata'); U];
set(plotRange, 'Xdata',newX,'Ydata',newY,'Ldata',newL,'Udata',newU)
end
Note that all individuals lie between –70 and 70. The population never
explores points near the global minimum at x = 101.
One way to make the genetic algorithm explore a wider range of points—that
is, to increase the diversity of the populations—is to increase the Initial
range. The Initial range does not have to include the point x = 101, but
it must be large enough so that the algorithm generates individuals near
x = 101. Set Initial range to [-10;90] as shown in the following figure.
5-104
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Global vs. Local Minima Using ga
Then click Start. The genetic algorithm returns a point very close to 101.
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5-105
5
Using the Genetic Algorithm
This time, the custom plot shows a much wider range of individuals. There
are individuals near 101 from early on, and the population mean begins to
converge to 101.
5-106
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Include a Hybrid Function
Include a Hybrid Function
A hybrid function is an optimization function that runs after the genetic
algorithm terminates in order to improve the value of the fitness function.
The hybrid function uses the final point from the genetic algorithm as its
initial point. You can specify a hybrid function in Hybrid function options.
This example uses Optimization Toolbox function fminunc, an unconstrained
minimization function. The example first runs the genetic algorithm to find a
point close to the optimal point and then uses that point as the initial point
for fminunc.
The example finds the minimum of Rosenbrock’s function, which is defined by
(
f ( x1 , x2 ) = 100 x2 − x12
)
2
+ (1 − x1 )2 .
The following figure shows a plot of Rosenbrock’s function.
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5-107
5
Using the Genetic Algorithm
3000
2500
2000
1500
1000
500
0
3
2
2
1
1
0
0
−1
−1
−2
Minimum at (1,1)
Global Optimization Toolbox software contains the dejong2fcn.m file, which
computes Rosenbrock’s function. To see a worked example of a hybrid
function, enter
hybriddemo
at the MATLAB prompt.
To explore the example, first enter optimtool('ga') to open the Optimization
app to the ga solver. Enter the following settings:
• Set Fitness function to @dejong2fcn.
• Set Number of variables to 2.
• Optionally, to get the same pseudorandom numbers as this example, switch
to the command line and enter:
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Include a Hybrid Function
rng(1,'twister')
Before adding a hybrid function, click Start to run the genetic algorithm by
itself. The genetic algorithm displays the following results in the Run solver
and view results pane:
The final point is somewhat close to the true minimum at (1, 1). You can
improve this result by setting Hybrid function to fminunc (in the Hybrid
function options).
fminunc uses the final point of the genetic algorithm as its initial point.
It returns a more accurate result, as shown in the Run solver and view
results pane.
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5-109
5
Using the Genetic Algorithm
Specify nondefault options for the hybrid function by creating options
at the command line. Use optimset for fminsearch, psoptimset for
patternsearch, or optimoptions for fmincon or fminunc. For example:
hybridopts = optimoptions('fminunc','Display','iter','Algorithm','quasi-new
In the Optimization app enter the name of your options structure in the
Options box under Hybrid function:
At the command line, the syntax is as follows:
options = gaoptimset('HybridFcn',{@fminunc,hybridopts});
hybridopts must exist before you set options.
5-110
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Set Maximum Number of Generations
Set Maximum Number of Generations
The Generations option in Stopping criteria determines the maximum
number of generations the genetic algorithm runs for—see “Stopping
Conditions for the Algorithm” on page 5-26. Increasing the Generations
option often improves the final result.
As an example, change the settings in the Optimization app as follows:
• Set Fitness function to @rastriginsfcn.
• Set Number of variables to 10.
• Select Best fitness in the Plot functions pane.
• Set Generations to Inf.
• Set Stall generations to Inf.
• Set Stall time limit to Inf.
Run the genetic algorithm for approximately 300 generations and click
Stop. The following figure shows the resulting best fitness plot after 300
generations.
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5-111
5
Using the Genetic Algorithm
Best: 5.0444 Mean: 48.7926
100
90
80
Fitness value
70
60
50
40
30
20
10
0
50
100
150
Generation
200
250
300
Genetic algorithm stalls
Note that the algorithm stalls at approximately generation number 170—that
is, there is no immediate improvement in the fitness function after generation
170. If you restore Stall generations to its default value of 50, the algorithm
could terminate at approximately generation number 220. If the genetic
algorithm stalls repeatedly with the current setting for Generations, you
can try increasing both the Generations and Stall generations options
to improve your results. However, changing other options might be more
effective.
5-112
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Set Maximum Number of Generations
Note When Mutation function is set to Gaussian, increasing the value
of Generations might actually worsen the final result. This can occur
because the Gaussian mutation function decreases the average amount of
mutation in each generation by a factor that depends on the value specified
in Generations. Consequently, the setting for Generations affects the
behavior of the algorithm.
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5-113
5
Using the Genetic Algorithm
Vectorize the Fitness Function
In this section...
“Vectorize for Speed” on page 5-114
“Vectorized Constraints” on page 5-115
Vectorize for Speed
The genetic algorithm usually runs faster if you vectorize the fitness function.
This means that the genetic algorithm only calls the fitness function once, but
expects the fitness function to compute the fitness for all individuals in the
current population at once. To vectorize the fitness function,
• Write the file that computes the function so that it accepts a matrix with
arbitrarily many rows, corresponding to the individuals in the population.
For example, to vectorize the function
f ( x1 , x2 ) = x12 − 2 x1 x2 + 6 x1 + x22 − 6 x2
write the file using the following code:
z =x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + x(:,2).^2 - 6*x(:,2);
The colon in the first entry of x indicates all the rows of x, so that x(:, 1)
is a vector. The .^ and .* operators perform elementwise operations on
the vectors.
• In the User function evaluation pane, set the Evaluate fitness and
constraint functions option to vectorized.
Note The fitness function, and any nonlinear constraint function, must accept
an arbitrary number of rows to use the Vectorize option. ga sometimes
evaluates a single row even during a vectorized calculation.
The following comparison, run at the command line, shows the improvement
in speed with Vectorize set to On.
5-114
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Vectorize the Fitness Function
options = gaoptimset('PopulationSize',2000);
tic;ga(@rastriginsfcn,20,[],[],[],[],[],[],[],options);toc
Optimization terminated: maximum number of generations exceeded.
Elapsed time is 12.054973 seconds.
options=gaoptimset(options,'Vectorize','on');
tic;ga(@rastriginsfcn,20,[],[],[],[],[],[],[],options);toc
Optimization terminated: maximum number of generations exceeded.
Elapsed time is 1.860655 seconds.
Vectorized Constraints
If there are nonlinear constraints, the objective function and the nonlinear
constraints all need to be vectorized in order for the algorithm to compute in
a vectorized manner.
“Vectorize the Objective and Constraint Functions” on page 4-85 contains an
example of how to vectorize both for the solver patternsearch. The syntax is
nearly identical for ga. The only difference is that patternsearch can have
its patterns appear as either row or column vectors; the corresponding vectors
for ga are the population vectors, which are always rows.
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5-115
5
Using the Genetic Algorithm
Constrained Minimization Using ga
Suppose you want to minimize the simple fitness function of two variables
x1 and x2,
(
min f ( x) = 100 x12 − x2
x
)
2
+ (1 − x1 )2
subject to the following nonlinear inequality constraints and bounds
x1 ⋅ x2 + x1 − x2 + 1.5 ≤ 0 (nonlinear constraiint)
10 − x1 ⋅ x2 ≤ 0
(nonlinear constraint)
0 ≤ x1 ≤ 1
0 ≤ x2 ≤ 13
(bound)
(bound)
Begin by creating the fitness and constraint functions. First, create a file
named simple_fitness.m as follows:
function y = simple_fitness(x)
y = 100*(x(1)^2 - x(2))^2 + (1 - x(1))^2;
(simple_fitness.m ships with Global Optimization Toolbox software.)
The genetic algorithm function, ga, assumes the fitness function will take one
input x, where x has as many elements as the number of variables in the
problem. The fitness function computes the value of the function and returns
that scalar value in its one return argument, y.
Then create a file, simple_constraint.m, containing the constraints
function [c, ceq] = simple_constraint(x)
c = [1.5 + x(1)*x(2) + x(1) - x(2);...
-x(1)*x(2) + 10];
ceq = [];
(simple_constraint.m ships with Global Optimization Toolbox software.)
The ga function assumes the constraint function will take one input x, where
x has as many elements as the number of variables in the problem. The
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Constrained Minimization Using ga
constraint function computes the values of all the inequality and equality
constraints and returns two vectors, c and ceq, respectively.
To minimize the fitness function, you need to pass a function handle to the
fitness function as the first argument to the ga function, as well as specifying
the number of variables as the second argument. Lower and upper bounds
are provided as LB and UB respectively. In addition, you also need to pass a
function handle to the nonlinear constraint function.
ObjectiveFunction = @simple_fitness;
nvars = 2;
% Number of variables
LB = [0 0];
% Lower bound
UB = [1 13]; % Upper bound
ConstraintFunction = @simple_constraint;
rng(1,'twister') % for reproducibility
[x,fval] = ga(ObjectiveFunction,nvars,...
[],[],[],[],LB,UB,ConstraintFunction)
Optimization terminated: average change in the fitness value
less than options.TolFun and constraint violation is
less than options.TolCon.
x =
0.8122
12.3122
fval =
1.3578e+004
The genetic algorithm solver handles linear constraints and bounds differently
from nonlinear constraints. All the linear constraints and bounds are satisfied
throughout the optimization. However, ga may not satisfy all the nonlinear
constraints at every generation. If ga converges to a solution, the nonlinear
constraints will be satisfied at that solution.
ga uses the mutation and crossover functions to produce new individuals at
every generation. ga satisfies linear and bound constraints by using mutation
and crossover functions that only generate feasible points. For example, in
the previous call to ga, the mutation function mutationguassian does not
necessarily obey the bound constraints. So when there are bound or linear
constraints, the default ga mutation function is mutationadaptfeasible.
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5-117
5
Using the Genetic Algorithm
If you provide a custom mutation function, this custom function must only
generate points that are feasible with respect to the linear and bound
constraints. All the included crossover functions generate points that satisfy
the linear constraints and bounds except the crossoverheuristic function.
To see the progress of the optimization, use the gaoptimset function to create
an options structure that selects two plot functions. The first plot function is
gaplotbestf, which plots the best and mean score of the population at every
generation. The second plot function is gaplotmaxconstr, which plots the
maximum constraint violation of nonlinear constraints at every generation.
You can also visualize the progress of the algorithm by displaying information
to the command window using the 'Display' option.
options = gaoptimset('PlotFcns',{@gaplotbestf,@gaplotmaxconstr},'Display','iter');
Rerun the ga solver.
[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],...
LB,UB,ConstraintFunction,options)
Best
Generation
f-count
f(x)
max
Stall
constraint
Generations
1
2670
13768.4
0
2
5286
13581.3
2.049e-08
0
0
3
7898
13579.5
0
0
4
12660
13578.3
1.062e-09
0
5
18922
13616.4
0
0
6
28972
13578.5
0
0
Optimization terminated: average change in the fitness value less than options.TolFun
and constraint violation is less than options.TolCon.
x =
0.8122
12.3123
fval =
1.3579e+04
5-118
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Constrained Minimization Using ga
You can provide a start point for the minimization to the ga function by
specifying the InitialPopulation option. The ga function will use the
first individual from InitialPopulation as a start point for a constrained
minimization.
X0 = [0.5 0.5]; % Start point (row vector)
options = gaoptimset(options,'InitialPopulation',X0);
Now, rerun the ga solver.
[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],...
LB,UB,ConstraintFunction,options)
Best
max
Stall
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5
Using the Genetic Algorithm
Generation
f-count
f(x)
constraint
Generations
1
2674
13578.4
0
0
2
5290
13578.2
0
0
3
7902
13578.2
5.702e-11
0
Optimization terminated: average change in the fitness value less than options.TolFun
and constraint violation is less than options.TolCon.
x =
0.8122
12.3122
fval =
1.3578e+04
5-120
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6
Using Simulated Annealing
• “What Is Simulated Annealing?” on page 6-2
• “Optimize Using Simulated Annealing” on page 6-3
• “Minimize Function with Many Local Minima” on page 6-6
• “Simulated Annealing Terminology” on page 6-9
• “How Simulated Annealing Works” on page 6-11
• “Command Line Simulated Annealing Optimization” on page 6-15
• “Minimization Using Simulated Annealing Algorithm” on page 6-20
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6
Using Simulated Annealing
What Is Simulated Annealing?
Simulated annealing is a method for solving unconstrained and
bound-constrained optimization problems. The method models the physical
process of heating a material and then slowly lowering the temperature to
decrease defects, thus minimizing the system energy.
At each iteration of the simulated annealing algorithm, a new point is
randomly generated. The distance of the new point from the current point, or
the extent of the search, is based on a probability distribution with a scale
proportional to the temperature. The algorithm accepts all new points that
lower the objective, but also, with a certain probability, points that raise the
objective. By accepting points that raise the objective, the algorithm avoids
being trapped in local minima, and is able to explore globally for more possible
solutions. An annealing schedule is selected to systematically decrease the
temperature as the algorithm proceeds. As the temperature decreases, the
algorithm reduces the extent of its search to converge to a minimum.
6-2
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Optimize Using Simulated Annealing
Optimize Using Simulated Annealing
In this section...
“Calling simulannealbnd at the Command Line” on page 6-3
“Using the Optimization App” on page 6-4
Calling simulannealbnd at the Command Line
To call the simulated annealing function at the command line, use the syntax
[x fval] = simulannealbnd(@objfun,x0,lb,ub,options)
where
• @objfun is a function handle to the objective function.
• x0 is an initial guess for the optimizer.
• lb and ub are lower and upper bound constraints, respectively, on x.
• options is a structure containing options for the algorithm. If you do not
pass in this argument, simulannealbnd uses its default options.
The results are given by:
• x — Final point returned by the solver
• fval — Value of the objective function at x
The command-line function simulannealbnd is convenient if you want to
• Return results directly to the MATLAB workspace.
• Run the simulated annealing algorithm multiple times with different
options by calling simulannealbnd from a file.
“Command Line Simulated Annealing Optimization” on page 6-15 provides
a detailed description of using the function simulannealbnd and creating
the options structure.
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6-3
6
Using Simulated Annealing
Using the Optimization App
To open the Optimization app, enter
optimtool('simulannealbnd')
at the command line, or enter optimtool and then choose simulannealbnd
from the Solver menu.
Set options
Choose solver
Enter problem
and constraints
Run solver
View results
See final point
6-4
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Expand or contract help
Optimize Using Simulated Annealing
You can also start the tool from the MATLAB Apps tab.
To use the Optimization app, you must first enter the following information:
• Objective function — The objective function you want to minimize. Enter
the fitness function in the form @fitnessfun, where fitnessfun.m is a file
that computes the objective function. “Compute Objective Functions” on
page 2-2 explains how write this file. The @ sign creates a function handle
to fitnessfun.
• Number of variables — The length of the input vector to the fitness
function. For the function my_fun described in “Compute Objective
Functions” on page 2-2, you would enter 2.
You can enter bounds for the problem in the Constraints pane. If the
problem is unconstrained, leave these fields blank.
To run the simulated annealing algorithm, click the Start button. The tool
displays the results of the optimization in the Run solver and view results
pane.
You can change the options for the simulated annealing algorithm in the
Options pane. To view the options in one of the categories listed in the pane,
click the + sign next to it.
For more information,
• See “Optimization App” in the Optimization Toolbox documentation.
• See “Minimize Using the Optimization App” on page 6-7 for an example of
using the tool with the function simulannealbnd.
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6-5
6
Using Simulated Annealing
Minimize Function with Many Local Minima
In this section...
“Description” on page 6-6
“Minimize at the Command Line” on page 6-7
“Minimize Using the Optimization App” on page 6-7
Description
This section presents an example that shows how to find the minimum of the
function using simulated annealing.
De Jong’s fifth function is a two-dimensional function with many (25) local
minima:
dejong5fcn
6-6
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Minimize Function with Many Local Minima
Many standard optimization algorithms get stuck in local minima. Because
the simulated annealing algorithm performs a wide random search, the
chance of being trapped in local minima is decreased.
Note Because simulated annealing uses random number generators, each
time you run this algorithm you can get different results. See “Reproduce
Your Results” on page 6-18 for more information.
Minimize at the Command Line
To run the simulated annealing algorithm without constraints, call
simulannealbnd at the command line using the objective function in
dejong5fcn.m, referenced by anonymous function pointer:
rng(10,'twister') % for reproducibility
fun = @dejong5fcn;
[x fval] = simulannealbnd(fun,[0 0])
This returns
Optimization terminated: change in best function value
less than options.TolFun.
x =
-16.1292
-15.8214
fval =
6.9034
where
• x is the final point returned by the algorithm.
• fval is the objective function value at the final point.
Minimize Using the Optimization App
To run the minimization using the Optimization app,
1 Set up your problem as pictured in the Optimization app
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6-7
6
Using Simulated Annealing
2 Click Start under Run solver and view results:
Your results can differ from the pictured ones, because simulannealbnd
uses a random number stream.
6-8
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Simulated Annealing Terminology
Simulated Annealing Terminology
In this section...
“Objective Function” on page 6-9
“Temperature” on page 6-9
“Annealing Parameter” on page 6-10
“Reannealing” on page 6-10
Objective Function
The objective function is the function you want to optimize. Global
Optimization Toolbox algorithms attempt to find the minimum of the objective
function. Write the objective function as a file or anonymous function, and
pass it to the solver as a function_handle. For more information, see
“Compute Objective Functions” on page 2-2.
Temperature
The temperature is a parameter in simulated annealing that affects two
aspects of the algorithm:
• The distance of a trial point from the current point (See “Outline of the
Algorithm” on page 6-11, Step 1.)
• The probability of accepting a trial point with higher objective function
value (See “Outline of the Algorithm” on page 6-11, Step 2.)
Temperature can be a vector with different values for each component of the
current point. Typically, the initial temperature is a scalar.
Temperature decreases gradually as the algorithm proceeds. You can
specify the initial temperature as a positive scalar or vector in the
InitialTemperature option. You can specify the temperature as a function of
iteration number as a function handle in the TemperatureFcn option. The
temperature is a function of the “Annealing Parameter” on page 6-10, which is
a proxy for the iteration number. The slower the rate of temperature decrease,
the better the chances are of finding an optimal solution, but the longer the
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6-9
6
Using Simulated Annealing
run time. For a list of built-in temperature functions and the syntax of a
custom temperature function, see “Temperature Options” on page 9-60.
Annealing Parameter
The annealing parameter is a proxy for the iteration number. The algorithm
can raise temperature by setting the annealing parameter to a lower value
than the current iteration. (See “Reannealing” on page 6-10.) You can specify
the temperature schedule as a function handle with the TemperatureFcn
option.
Reannealing
Annealing is the technique of closely controlling the temperature when cooling
a material to ensure that it reaches an optimal state. Reannealing raises the
temperature after the algorithm accepts a certain number of new points, and
starts the search again at the higher temperature. Reannealing avoids the
algorithm getting caught at local minima. Specify the reannealing schedule
with the ReannealInterval option.
6-10
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How Simulated Annealing Works
How Simulated Annealing Works
In this section...
“Outline of the Algorithm” on page 6-11
“Stopping Conditions for the Algorithm” on page 6-13
“Bibliography” on page 6-13
Outline of the Algorithm
The simulated annealing algorithm performs the following steps:
1 The algorithm generates a random trial point. The algorithm chooses
the distance of the trial point from the current point by a probability
distribution with a scale depending on the current temperature. You set
the trial point distance distribution as a function with the AnnealingFcn
option. Choices:
• @annealingfast (default) — Step length equals the current temperature,
and direction is uniformly random.
• @annealingboltz — Step length equals the square root of temperature,
and direction is uniformly random.
• @myfun — Custom annealing algorithm, myfun. For custom annealing
function syntax, see “Algorithm Settings” on page 9-61.
2 The algorithm determines whether the new point is better or worse than
the current point. If the new point is better than the current point, it
becomes the next point. If the new point is worse than the current point,
the algorithm can still make it the next point. The algorithm accepts
a worse point based on an acceptance function. Choose the acceptance
function with the AcceptanceFcn option. Choices:
• @acceptancesa (default) — Simulated annealing acceptance function.
The probability of acceptance is
1
Δ
⎛
⎞
1 + exp ⎜
⎟
T
max(
)
⎝
⎠
,
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6-11
6
Using Simulated Annealing
where
Δ = new objective – old objective.
T0 = initial temperature of component i
T = the current temperature.
Since both Δ and T are positive, the probability of acceptance is between
0 and 1/2. Smaller temperature leads to smaller acceptance probability.
Also, larger Δ leads to smaller acceptance probability.
• @myfun — Custom acceptance function, myfun. For custom acceptance
function syntax, see “Algorithm Settings” on page 9-61.
3 The algorithm systematically lowers the temperature, storing the best
point found so far. The TemperatureFcn option specifies the function the
algorithm uses to update the temperature. Let k denote the annealing
parameter. (The annealing parameter is the same as the iteration number
until reannealing.) Options:
• @temperatureexp (default) — T = T0 * 0.95^k.
• @temperaturefast — T = T0 / k.
• @temperatureboltz — T = T0 / log(k).
• @myfun — Custom temperature function, myfun. For custom temperature
function syntax, see “Temperature Options” on page 9-60.
4 simulannealbnd reanneals after it accepts ReannealInterval points.
Reannealing sets the annealing parameters to lower values than the
iteration number, thus raising the temperature in each dimension. The
annealing parameters depend on the values of estimated gradients of the
objective function in each dimension. The basic formula is
⎛
max ( s j ) ⎞
T
j
⎟,
ki = log ⎜⎜ 0
⎟
T
s
i
⎝ i
⎠
where
ki = annealing parameter for component i.
T0 = initial temperature of component i.
6-12
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How Simulated Annealing Works
Ti = current temperature of component i.
si = gradient of objective in direction i times difference of bounds in
direction i.
simulannealbnd safeguards the annealing parameter values against Inf
and other improper values.
5 The algorithm stops when the average change in the objective function
is small relative to the TolFun tolerance, or when it reaches any other
stopping criterion. See “Stopping Conditions for the Algorithm” on page
6-13.
For more information on the algorithm, see Ingber [1].
Stopping Conditions for the Algorithm
The simulated annealing algorithm uses the following conditions to determine
when to stop:
• TolFun — The algorithm runs until the average change in value of the
objective function in StallIterLim iterations is less than the value of
TolFun. The default value is 1e-6.
• MaxIter — The algorithm stops when the number of iterations exceeds this
maximum number of iterations. You can specify the maximum number of
iterations as a positive integer or Inf. The default value is Inf.
• MaxFunEval specifies the maximum number of evaluations of the objective
function. The algorithm stops if the number of function evaluations exceeds
the value of MaxFunEval. The default value is 3000*numberofvariables.
• TimeLimit specifies the maximum time in seconds the algorithm runs
before stopping. The default value is Inf.
• ObjectiveLimit — The algorithm stops when the best objective function
value is less than or equal to the value of ObjectiveLimit. The default
value is -Inf.
Bibliography
[1] Ingber, L. Adaptive simulated annealing (ASA): Lessons learned. Invited
paper to a special issue of the Polish Journal Control and Cybernetics
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6-13
6
Using Simulated Annealing
on “Simulated Annealing Applied to Combinatorial Optimization.” 1995.
Available from http://www.ingber.com/asa96_lessons.ps.gz
6-14
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Command Line Simulated Annealing Optimization
Command Line Simulated Annealing Optimization
In this section...
“Run simulannealbnd With the Default Options” on page 6-15
“Set Options for simulannealbnd at the Command Line” on page 6-16
“Reproduce Your Results” on page 6-18
Run simulannealbnd With the Default Options
To run the simulated annealing algorithm with the default options, call
simulannealbnd with the syntax
[x,fval] = simulannealbnd(@objfun,x0)
The input arguments to simulannealbnd are
• @objfun — A function handle to the file that computes the objective
function. “Compute Objective Functions” on page 2-2 explains how to write
this file.
• x0 — The initial guess of the optimal argument to the objective function.
The output arguments are
• x — The final point.
• fval — The value of the objective function at x.
For a description of additional input and output arguments, see the reference
pages for simulannealbnd.
You can run the example described in “Minimize Function with Many Local
Minima” on page 6-6 from the command line by entering
rng(10,'twister') % for reproducibility
[x,fval] = simulannealbnd(@dejong5fcn,[0 0])
This returns
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6-15
6
Using Simulated Annealing
Optimization terminated: change in best function value
less than options.TolFun.
x =
-16.1292
-15.8214
fval =
6.9034
Additional Output Arguments
To get more information about the performance of the algorithm, you can
call simulannealbnd with the syntax
[x,fval,exitflag,output] = simulannealbnd(@objfun,x0)
Besides x and fval, this function returns the following additional output
arguments:
• exitflag — Flag indicating the reason the algorithm terminated
• output — Structure containing information about the performance of the
algorithm
See the simulannealbnd reference pages for more information about these
arguments.
Set Options for simulannealbnd at the Command Line
You can specify options by passing an options structure as an input argument
to simulannealbnd using the syntax
[x,fval] = simulannealbnd(@objfun,x0,[],[],options)
This syntax does not specify any lower or upper bound constraints.
You create the options structure using the saoptimset function:
options = saoptimset('simulannealbnd')
This returns the structure options with the default values for its fields:
6-16
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Command Line Simulated Annealing Optimization
options =
AnnealingFcn:
TemperatureFcn:
AcceptanceFcn:
TolFun:
StallIterLimit:
MaxFunEvals:
TimeLimit:
MaxIter:
ObjectiveLimit:
Display:
DisplayInterval:
HybridFcn:
HybridInterval:
PlotFcns:
PlotInterval:
OutputFcns:
InitialTemperature:
ReannealInterval:
DataType:
@annealingfast
@temperatureexp
@acceptancesa
1.0000e-006
'500*numberofvariables'
'3000*numberofvariables'
Inf
Inf
-Inf
'final'
10
[]
'end'
[]
1
[]
100
100
'double'
The value of each option is stored in a field of the options structure, such as
options.ReannealInterval. You can display any of these values by entering
options followed by the name of the field. For example, to display the interval
for reannealing used for the simulated annealing algorithm, enter
options.ReannealInterval
ans =
100
To create an options structure with a field value that is different from the
default—for example, to set ReannealInterval to 300 instead of its default
value 100—enter
options = saoptimset('ReannealInterval',300)
This creates the options structure with all values set to their defaults, except
for ReannealInterval, which is set to 300.
If you now enter
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6-17
6
Using Simulated Annealing
simulannealbnd(@objfun,x0,[],[],options)
simulannealbnd runs the simulated annealing algorithm with a reannealing
interval of 300.
If you subsequently decide to change another field in the options structure,
such as setting PlotFcns to @saplotbestf, which plots the best objective
function value at each iteration, call saoptimset with the syntax
options = saoptimset(options,'PlotFcns',@saplotbestf)
This preserves the current values of all fields of options except for PlotFcns,
which is changed to @saplotbestf. Note that if you omit the input argument
options, saoptimset resets ReannealInterval to its default value 100.
You can also set both ReannealInterval and PlotFcns with the single
command
options = saoptimset('ReannealInterval',300, ...
'PlotFcns',@saplotbestf)
Reproduce Your Results
Because the simulated annealing algorithm is stochastic—that is, it makes
random choices—you get slightly different results each time you run it. The
algorithm uses the default MATLAB pseudorandom number stream. For
more information about random number streams, see RandStream. Each
time the algorithm calls the stream, its state changes. So the next time the
algorithm calls the stream, it returns a different random number.
If you need to reproduce your results exactly, call simulannealbnd with
the output argument. The output structure contains the current random
number generator state in the output.rngstate field. Reset the state before
running the function again.
For example, to reproduce the output of simulannealbnd applied to De Jong’s
fifth function, call simulannealbnd with the syntax
rng(10,'twister') % for reproducibility
[x,fval,exitflag,output] = simulannealbnd(@dejong5fcn,[0 0]);
Suppose the results are
6-18
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Command Line Simulated Annealing Optimization
x,fval
x =
-16.1292
-15.8214
fval =
6.9034
The state of the random number generator, rngstate, is stored in
output.rngstate:
output.rngstate
ans =
state: [625x1 uint32]
type: 'mt19937ar'
Reset the stream by entering
stream = RandStream.getGlobalStream;
stream.State = output.rngstate.state;
If you now run simulannealbnd a second time, you get the same results.
You can reproduce your run in the Optimization app by checking the box Use
random states from previous run in the Run solver and view results
section.
Note If you do not need to reproduce your results, it is better not to set the
states of RandStream, so that you get the benefit of the randomness in these
algorithms.
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6-19
6
Using Simulated Annealing
Minimization Using Simulated Annealing Algorithm
This example shows how to create and minimize an objective function using
Simulated Annealing in the Global Optimization Toolbox.
A Simple Objective Function
We want to minimize a simple function of two variables
min f(x) = (4 - 2.1*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-4 + 4*x2^2)*x2^2;
x
The above function is known as ’cam’ as described in L.C.W. Dixon and G.P.
Szego (eds.), Towards Global Optimisation 2, North-Holland, Amsterdam,
1978.
Coding the Objective Function
We create a MATLAB file named simple_objective.m with the following code
in it:
function y = simple_objective(x)
y = (4 - 2.1*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ...
(-4 + 4*x(2)^2)*x(2)^2;
The Simulated Annealing solver assumes the objective function will take one
input x where x has as many elements as the number of variables in the
problem. The objective function computes the scalar value of the objective and
returns it in its single return argument y.
Minimizing Using SIMULANNEALBND
To minimize our objective function using the SIMULANNEALBND function,
we need to pass in a function handle to the objective function as well as
specifying a start point as the second argument.
ObjectiveFunction = @simple_objective;
X0 = [0.5 0.5];
% Starting point
[x,fval,exitFlag,output] = simulannealbnd(ObjectiveFunction,X0)
Optimization terminated: change in best function value less than options.To
6-20
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Minimization Using Simulated Annealing Algorithm
x =
-0.0896
0.7130
fval =
-1.0316
exitFlag =
1
output =
iterations:
funccount:
message:
rngstate:
problemtype:
temperature:
totaltime:
2948
2971
'Optimization terminated: change in best function value l.
[1x1 struct]
'unconstrained'
[2x1 double]
5.0600
The first two output arguments returned by SIMULANNEALBND are x,
the best point found, and fval, the function value at the best point. A third
output argument, exitFlag returns a flag corresponding to the reason
SIMULANNEALBND stopped. SIMULANNEALBND can also return a
fourth argument, output, which contains information about the performance
of the solver.
Bound Constrained Minimization
SIMULANNEALBND can be used to solve problems with bound constraints.
The lower and upper bounds are passed to the solver as vectors. For each
dimension i, the solver ensures that lb(i) <= x(i) <= ub(i), where x is a point
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6-21
6
Using Simulated Annealing
selected by the solver during simulation. We impose the bounds on our
problem by specifying a range -64 <= x(i) <= 64 for x(i).
lb = [-64 -64];
ub = [64 64];
Now, we can rerun the solver with lower and upper bounds as input
arguments.
[x,fval,exitFlag,output] = simulannealbnd(ObjectiveFunction,X0,lb,ub);
fprintf('The number of iterations was : %d\n', output.iterations);
fprintf('The number of function evaluations was : %d\n', output.funccount);
fprintf('The best function value found was : %g\n', fval);
Optimization terminated: change in best function value less than options.To
The number of iterations was : 2428
The number of function evaluations was : 2447
The best function value found was : -1.03163
How Simulated Annealing Works
Simulated annealing mimics the annealing process to solve an optimization
problem. It uses a temperature parameter that controls the search. The
temperature parameter typically starts off high and is slowly "cooled" or
lowered in every iteration. At each iteration a new point is generated and its
distance from the current point is proportional to the temperature. If the new
point has a better function value it replaces the current point and iteration
counter is incremented. It is possible to accept and move forward with a worse
point. The probability of doing so is directly dependent on the temperature.
This unintuitive step sometime helps identify a new search region in hope of
finding a better minimum.
An Objective Function with Additional Arguments
Sometimes we want our objective function to be parameterized by extra
arguments that act as constants during the optimization. For example, in the
previous objective function, say we want to replace the constants 4, 2.1, and 4
with parameters that we can change to create a family of objective functions.
6-22
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Minimization Using Simulated Annealing Algorithm
We can re-write the above function to take three additional parameters to
give the new minimization problem.
min f(x) = (a - b*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-c + c*x2^2)*x2^2;
x
a, b, and c are parameters to the objective function that act as constants
during the optimization (they are not varied as part of the minimization).
One can create a MATLAB file called parameterized_objective.m containing
the following code.
function y = parameterized_objective(x,a,b,c)
y = (a - b*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ...
(-c + c*x(2)^2)*x(2)^2;
Minimizing Using Additional Arguments
Again, we need to pass in a function handle to the objective function as well
as a start point as the second argument.
SIMULANNEALBND will call our objective function with just one argument
x, but our objective function has four arguments: x, a, b, c. We can use an
anonymous function to capture the values of the additional arguments, the
constants a, b, and c. We create a function handle ’ObjectiveFunction’ to an
anonymous function that takes one input x, but calls ’parameterized_objective’
with x, a, b and c. The variables a, b, and c have values when the function
handle ’ObjectiveFunction’ is created, so these values are captured by the
anonymous function.
a = 4; b = 2.1; c = 4;
% define constant values
ObjectiveFunction = @(x) parameterized_objective(x,a,b,c);
X0 = [0.5 0.5];
[x,fval] = simulannealbnd(ObjectiveFunction,X0)
Optimization terminated: change in best function value less than options.To
x =
0.0898
-0.7127
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6-23
6
Using Simulated Annealing
fval =
-1.0316
6-24
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7
Multiobjective Optimization
• “What Is Multiobjective Optimization?” on page 7-2
• “Use gamultiobj” on page 7-5
• “Bibliography” on page 7-14
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7
Multiobjective Optimization
What Is Multiobjective Optimization?
You might need to formulate problems with more than one objective, since a
single objective with several constraints may not adequately represent the
problem being faced. If so, there is a vector of objectives,
F(x) = [F1(x), F2(x),...,Fm(x)],
that must be traded off in some way. The relative importance of these
objectives is not generally known until the system’s best capabilities are
determined and tradeoffs between the objectives fully understood. As the
number of objectives increases, tradeoffs are likely to become complex and
less easily quantified. The designer must rely on his or her intuition and
ability to express preferences throughout the optimization cycle. Thus,
requirements for a multiobjective design strategy must enable a natural
problem formulation to be expressed, and be able to solve the problem and
enter preferences into a numerically tractable and realistic design problem.
Multiobjective optimization is concerned with the minimization of a vector of
objectives F(x) that can be the subject of a number of constraints or bounds:
minn F ( x), subject to
x∈R
Gi ( x) = 0, i = 1,..., ke ; Gi ( x) ≤ 0, i = ke + 1,..., k; l ≤ x ≤ u.
Note that because F(x) is a vector, if any of the components of F(x) are
competing, there is no unique solution to this problem. Instead, the concept
of noninferiority in Zadeh [4] (also called Pareto optimality in Censor [1]
and Da Cunha and Polak [2]) must be used to characterize the objectives.
A noninferior solution is one in which an improvement in one objective
requires a degradation of another. To define this concept more precisely,
consider a feasible region, Ω, in the parameter space. x is an element of the
n-dimensional real numbers x ∈ R n that satisfies all the constraints, i.e.,
{
}
Ω = x∈Rn ,
subject to
7-2
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What Is Multiobjective Optimization?
Gi ( x) = 0, i = 1,..., ke ,
Gi ( x) ≤ 0, i = ke + 1,..., k,
l ≤ x ≤ u.
This allows definition of the corresponding feasible region for the objective
function space Λ:
{
}
Λ = y ∈ R m : y = F ( x), x ∈ Ω .
The performance vector F(x) maps parameter space into objective function
space, as represented in two dimensions in the figure Mapping from
Parameter Space into Objective Function Space on page 7-3.
Figure 7-1:
Mapping from Parameter Space into Objective Function Space
A noninferior solution point can now be defined.
Definition: Point x* ∈ Ω is a noninferior solution if for some neighborhood of
x* there does not exist a Δx such that ( x * + Δx ) ∈ Ω and
Fi ( x * + Δx ) ≤ Fi ( x*), i = 1,..., m, and
F j ( x * + Δx ) < F j ( x*) for at least one j.
In the two-dimensional representation of the figure Set of Noninferior
Solutions on page 7-4, the set of noninferior solutions lies on the curve
between C and D. Points A and B represent specific noninferior points.
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7-3
7
Multiobjective Optimization
Figure 7-2:
Set of Noninferior Solutions
A and B are clearly noninferior solution points because an improvement
in one objective, F1, requires a degradation in the other objective, F2, i.e.,
F1B < F1A, F2B > F2A.
Since any point in Ω that is an inferior point represents a point in which
improvement can be attained in all the objectives, it is clear that such a point
is of no value. Multiobjective optimization is, therefore, concerned with the
generation and selection of noninferior solution points.
Noninferior solutions are also called Pareto optima. A general goal in
multiobjective optimization is constructing the Pareto optima. The algorithm
used in gamultiobj is described in Deb [3].
7-4
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Use gamultiobj
Use gamultiobj
In this section...
“Problem Formulation” on page 7-5
“Use gamultiobj with the Optimization app” on page 7-6
“Multiobjective Optimization with Two Objectives” on page 7-6
“Options and Syntax: Differences from ga” on page 7-13
Problem Formulation
The gamultiobj solver attempts to create a set of Pareto optima for a
multiobjective minimization. You may optionally set bounds and linear
constraints on variables. gamultiobj uses the genetic algorithm for finding
local Pareto optima. As in the ga function, you may specify an initial
population, or have the solver generate one automatically.
The fitness function for use in gamultiobj should return a vector of type
double. The population may be of type double, a bit string vector, or can be
a custom-typed vector. As in ga, if you use a custom population type, you
must write your own creation, mutation, and crossover functions that accept
inputs of that population type, and specify these functions in the following
fields, respectively:
• Creation function (CreationFcn)
• Mutation function (MutationFcn)
• Crossover function (CrossoverFcn)
You can set the initial population in a variety of ways. Suppose that you
choose a population of size m. (The default population size is 15 times the
number of variables n.) You can set the population:
• As an m-by-n matrix, where the rows represent m individuals.
• As a k-by-n matrix, where k < m. The remaining m – k individuals are
generated by a creation function.
• The entire population can be created by a creation function.
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7-5
7
Multiobjective Optimization
Use gamultiobj with the Optimization app
You can access gamultiobj from the Optimization app. Enter
optimtool('gamultiobj')
at the command line, or enter optimtool and then choose gamultiobj from
the Solver menu. You can also start the tool from the MATLAB Apps tab.
If the Quick Reference help pane is closed, you can open it by clicking the
. The help pane briefly explains all
“>>” button on the upper right:
the available options.
You can create an options structure in the Optimization app, export it to the
MATLAB workspace, and use the structure at the command line. For details,
see “Importing and Exporting Your Work” in the Optimization Toolbox
documentation.
Multiobjective Optimization with Two Objectives
This example has a two-objective fitness function f(x), where x is also
two-dimensional:
function f = mymulti1(x)
f(1) = x(1)^4 - 10*x(1)^2+x(1)*x(2) + x(2)^4 -(x(1)^2)*(x(2)^2);
f(2) = x(2)^4 - (x(1)^2)*(x(2)^2) + x(1)^4 + x(1)*x(2);
Create this function file before proceeding.
Performing the Optimization with Optimization App
1 To define the optimization problem, start the Optimization app, and set it
as pictured.
7-6
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Use gamultiobj
2 Set the options for the problem as pictured.
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7-7
7
Multiobjective Optimization
3 Run the optimization by clicking Start under Run solver and view
results.
A plot appears in a figure window.
7-8
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Use gamultiobj
This plot shows the tradeoff between the two components of f. It is plotted
in objective function space; see the figure Set of Noninferior Solutions on
page 7-4.
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7-9
7
Multiobjective Optimization
The results of the optimization appear in the following table containing both
objective function values and the value of the variables.
You can sort the table by clicking a heading. Click the heading again to sort
it in the reverse order. The following figures show the result of clicking the
heading f1.
7-10
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Use gamultiobj
Performing the Optimization at the Command Line
To perform the same optimization at the command line:
1 Set the options:
options = gaoptimset('PopulationSize',60,...
'ParetoFraction',0.7,'PlotFcns',@gaplotpareto);
2 Run the optimization using the options:
[x fval flag output population] = gamultiobj(@mymulti1,2,...
[],[],[],[],[-5,-5],[5,5],options);
Alternate Views
There are other ways of regarding the problem. The following figure contains
a plot of the level curves of the two objective functions, the Pareto frontier
calculated by gamultiobj (boxes), and the x-values of the true Pareto frontier
(diamonds connected by a nearly-straight line). The true Pareto frontier
points are where the level curves of the objective functions are parallel. They
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7-11
7
Multiobjective Optimization
were calculated by finding where the gradients of the objective functions are
parallel. The figure is plotted in parameter space; see the figure Mapping
from Parameter Space into Objective Function Space on page 7-3.
= Frontier points calculated by
gamultiobj
= True Pareto frontier
Contours of objective functions, and Pareto frontier
7-12
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Use gamultiobj
gamultiobj found the ends of the line segment, meaning it found the full
extent of the Pareto frontier.
Options and Syntax: Differences from ga
The syntax and options for gamultiobj are similar to those for ga, with the
following differences:
• gamultiobj does not have nonlinear constraints, so its syntax has fewer
inputs.
• gamultiobj takes an option DistanceMeasureFcn, a function that assigns
a distance measure to each individual with respect to its neighbors.
• gamultiobj takes an option ParetoFraction, a number between 0 and 1
that specifies the fraction of the population on the best Pareto frontier to
be kept during the optimization. If there is only one Pareto frontier, this
option is ignored.
• gamultiobj uses only the Tournament selection function.
• gamultiobj uses elite individuals differently than ga. It sorts noninferior
individuals above inferior ones, so it uses elite individuals automatically.
• gamultiobj has only one hybrid function, fgoalattain.
• gamultiobj does not have a stall time limit.
• gamultiobj has different plot functions available.
• gamultiobj does not have a choice of scaling function.
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7-13
7
Multiobjective Optimization
Bibliography
[1] Censor, Y., “Pareto Optimality in Multiobjective Problems,” Appl. Math.
Optimiz., Vol. 4, pp 41–59, 1977.
[2] Da Cunha, N.O. and E. Polak, “Constrained Minimization Under
Vector-Valued Criteria in Finite Dimensional Spaces,” J. Math. Anal. Appl.,
Vol. 19, pp 103–124, 1967.
[3] Deb, Kalyanmoy, “Multi-Objective Optimization using Evolutionary
Algorithms,” John Wiley & Sons, Ltd, Chichester, England, 2001.
[4] Zadeh, L.A., “Optimality and Nonscalar-Valued Performance Criteria,”
IEEE Trans. Automat. Contr., Vol. AC-8, p. 1, 1963.
7-14
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8
Parallel Processing
• “Background” on page 8-2
• “How to Use Parallel Processing” on page 8-12
• “Minimizing an Expensive Optimization Problem Using Parallel Computing
Toolbox™” on page 8-19
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8
Parallel Processing
Background
In this section...
“Parallel Processing Types in Global Optimization Toolbox” on page 8-2
“How Toolbox Functions Distribute Processes” on page 8-3
Parallel Processing Types in Global Optimization
Toolbox
Parallel processing is an attractive way to speed optimization algorithms. To
use parallel processing, you must have a Parallel Computing Toolbox license,
and have a parallel worker pool (parpool). For more information, see “How to
Use Parallel Processing” on page 8-12.
Global Optimization Toolbox solvers use parallel computing in various ways.
Solver
Parallel?
Parallel Characteristics
GlobalSearch
×
No parallel functionality. However, fmincon can use
parallel gradient estimation when run in GlobalSearch. See
“Using Parallel Computing in Optimization Toolbox” in the
Optimization Toolbox documentation.
MultiStart
Start points distributed to multiple processors. From these
points, local solvers run to completion. For more details, see
“MultiStart” on page 8-5 and “How to Use Parallel Processing”
on page 8-12.
For fmincon, no parallel gradient estimation with parallel
MultiStart.
ga, gamultiobj
Population evaluated in parallel, which occurs once per
iteration. For more details, see “Genetic Algorithm” on page 8-9
and “How to Use Parallel Processing” on page 8-12.
No vectorization of fitness or constraint functions.
8-2
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Background
Solver
Parallel?
Parallel Characteristics
Poll points evaluated in parallel, which occurs once per
iteration. For more details, see “Pattern Search” on page 8-7
and “How to Use Parallel Processing” on page 8-12.
patternsearch
No vectorization of fitness or constraint functions.
simulannealbnd
×
No parallel functionality. However, simulannealbnd can
use a hybrid function that runs in parallel. See “Simulated
Annealing” on page 8-11.
In addition, several solvers have hybrid functions that run after they finish.
Some hybrid functions can run in parallel. Also, most patternsearch search
methods can run in parallel. For more information, see “Parallel Search
Functions or Hybrid Functions” on page 8-15.
How Toolbox Functions Distribute Processes
• “parfor Characteristics and Caveats” on page 8-3
• “MultiStart” on page 8-5
• “GlobalSearch” on page 8-6
• “Pattern Search” on page 8-7
• “Genetic Algorithm” on page 8-9
• “Parallel Computing with gamultiobj” on page 8-10
• “Simulated Annealing” on page 8-11
parfor Characteristics and Caveats
No Nested parfor Loops. Solvers employ the Parallel Computing Toolbox
parfor function to perform parallel computations.
Note parfor does not work in parallel when called from within another
parfor loop.
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8-3
8
Parallel Processing
Suppose, for example, your objective function userfcn calls parfor, and you
want to call fmincon using MultiStart and parallel processing. Suppose also
that the conditions for parallel gradient evaluation of fmincon are satisfied,
as given in “Parallel Optimization Functionality”. The figure When parfor
Runs In Parallel on page 8-4 shows three cases:
1 The outermost loop is parallel MultiStart. Only that loop runs in parallel.
2 The outermost parfor loop is in fmincon. Only fmincon runs in parallel.
3 The outermost parfor loop is in userfcn. In this case, userfcn can use
parfor in parallel.
Bold indicates the function that runs in parallel
1
...
problem = createOptimProblem(fmincon,'objective',@userfcn,...)
ms = MultiStart('UseParallel','always');
x = run(ms,problem,10)
Only the outermost parfor loop
...
runs in parallel
If fmincon UseParallel option = 'always'
fmincon estimates gradients in parallel
2
3
...
x = fmincon(@userfcn,...)
...
If fmincon UseParallel option = 'never'
userfcn can use parfor in parallel
...
x = fmincon(@userfcn,...)
...
When parfor Runs In Parallel
Parallel Random Numbers Are Not Reproducible. Random number
sequences in MATLAB are pseudorandom, determined from a seed, or an
initial setting. Parallel computations use seeds that are not necessarily
controllable or reproducible. For example, each instance of MATLAB has a
default global setting that determines the current seed for random sequences.
For patternsearch, if you select MADS as a poll or search method, parallel
pattern search does not have reproducible runs. If you select the genetic
8-4
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Background
algorithm or Latin hypercube as search methods, parallel pattern search does
not have reproducible runs.
For ga and gamultiobj, parallel population generation gives nonreproducible
results.
MultiStart is different. You can have reproducible runs from parallel
MultiStart. Runs are reproducible because MultiStart generates
pseudorandom start points locally, and then distributes the start points to
parallel processors. Therefore, the parallel processors do not use random
numbers. For more details, see “Parallel Processing and Random Number
Streams” on page 3-82.
Limitations and Performance Considerations. More caveats related to
parfor appear in the “Limitations” section of the Parallel Computing Toolbox
documentation.
For information on factors that affect the speed of parallel computations,
and factors that affect the results of parallel computations, see “Improving
Performance with Parallel Computing” in the Optimization Toolbox
documentation. The same considerations apply to parallel computing with
Global Optimization Toolbox functions.
MultiStart
MultiStart can automatically distribute a problem and start points
to multiple processes or processors. The problems run independently,
and MultiStart combines the distinct local minima into a vector of
GlobalOptimSolution objects. MultiStart uses parallel computing when
you:
• Have a license for Parallel Computing Toolbox software.
• Enable parallel computing with parpool, a Parallel Computing Toolbox
function.
• Set the UseParallel property to true in the MultiStart object:
ms = MultiStart('UseParallel',true);
When these conditions hold, MultiStart distributes a problem and start
points to processes or processors one at a time. The algorithm halts when it
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8-5
8
Parallel Processing
reaches a stopping condition or runs out of start points to distribute. If the
MultiStart Display property is 'iter', then MultiStart displays:
Running the local solvers in parallel.
For an example of parallel MultiStart, see “Parallel MultiStart” on page
3-100.
Implementation Issues in Parallel MultiStart. fmincon cannot estimate
gradients in parallel when used with parallel MultiStart. This lack of
parallel gradient estimation is due to the limitation of parfor described in
“No Nested parfor Loops” on page 8-3.
fmincon can take longer to estimate gradients in parallel rather than in
serial. In this case, using MultiStart with parallel gradient estimation in
fmincon amplifies the slowdown. For example, suppose the ms MultiStart
object has UseParallel set to false. Suppose fmincon takes 1 s longer
to solve problem with problem.options.UseParallel set to true.
Then run(ms,problem,200) takes 200 s longer than the same run with
problem.options.UseParallel set to false
Note When executing serially, parfor loops run slower than for loops.
Therefore, for best performance, set your local solver UseParallel option to
false when the MultiStart UseParallel property is true.
Note Even when running in parallel, a solver occasionally calls the objective
and nonlinear constraint functions serially on the host machine. Therefore,
ensure that your functions have no assumptions about whether they are
evaluated in serial and parallel.
GlobalSearch
GlobalSearch does not distribute a problem and start points to multiple
processes or processors. However, when GlobalSearch runs the fmincon local
solver, fmincon can estimate gradients by parallel finite differences. fmincon
uses parallel computing when you:
8-6
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Background
• Have a license for Parallel Computing Toolbox software.
• Enable parallel computing with parpool, a Parallel Computing Toolbox
function.
• Set the UseParallel option to true with optimoptions. Set this option in
the problem structure:
opts = optimoptions(@fmincon,'UseParallel',true,'Algorithm','sqp');
problem = createOptimProblem('fmincon','objective',@myobj,...
'x0',startpt,'options',opts);
For more details, see “Using Parallel Computing in Optimization Toolbox” in
the Optimization Toolbox documentation.
Pattern Search
patternsearch can automatically distribute the evaluation of objective and
constraint functions associated with the points in a pattern to multiple
processes or processors. patternsearch uses parallel computing when you:
• Have a license for Parallel Computing Toolbox software.
• Enable parallel computing with parpool, a Parallel Computing Toolbox
function.
• Set the following options using psoptimset or the Optimization app:
-
CompletePoll is 'on'.
Vectorized is 'off' (default).
UseParallel is true.
When these conditions hold, the solver computes the objective function and
constraint values of the pattern search in parallel during a poll. Furthermore,
patternsearch overrides the setting of the Cache option, and uses the default
'off' setting.
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Parallel Processing
Note Even when running in parallel, patternsearch occasionally calls the
objective and nonlinear constraint functions serially on the host machine.
Therefore, ensure that your functions have no assumptions about whether
they are evaluated in serial or parallel.
Parallel Search Function. patternsearch can optionally call a search
function at each iteration. The search is parallel when you:
• Set CompleteSearch to 'on'.
• Do not set the search method to @searchneldermead or custom.
• Set the search method to a patternsearch poll method or Latin hypercube
search, and set UseParallel to true.
• Or, if you set the search method to ga, create a search method option
structure with UseParallel set to true.
Implementation Issues in Parallel Pattern Search. The limitations
on patternsearch options, listed in “Pattern Search” on page 8-7, arise
partly from the limitations of parfor, and partly from the nature of parallel
processing:
• Cache is overridden to be 'off' — patternsearch implements Cache as a
persistent variable. parfor does not handle persistent variables, because
the variable could have different settings at different processors.
• CompletePoll is 'on' — CompletePoll determines whether a poll stops as
soon as patternsearch finds a better point. When searching in parallel,
parfor schedules all evaluations simultaneously, and patternsearch
continues after all evaluations complete. patternsearch cannot halt
evaluations after they start.
• Vectorized is 'off' — Vectorized determines whether patternsearch
evaluates all points in a pattern with one function call in a vectorized
fashion. If Vectorized is 'on', patternsearch does not distribute the
evaluation of the function, so does not use parfor.
8-8
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Background
Genetic Algorithm
ga and gamultiobj can automatically distribute the evaluation of objective
and nonlinear constraint functions associated with a population to multiple
processors. ga uses parallel computing when you:
• Have a license for Parallel Computing Toolbox software.
• Enable parallel computing with parpool, a Parallel Computing Toolbox
function.
• Set the following options using gaoptimset or the Optimization app:
-
Vectorized is 'off' (default).
UseParallel is true.
When these conditions hold, ga computes the objective function and nonlinear
constraint values of the individuals in a population in parallel.
Note Even when running in parallel, ga occasionally calls the fitness and
nonlinear constraint functions serially on the host machine. Therefore, ensure
that your functions have no assumptions about whether they are evaluated
in serial or parallel.
Implementation Issues in Parallel Genetic Algorithm. The limitations
on options, listed in “Genetic Algorithm” on page 8-9, arise partly from
limitations of parfor, and partly from the nature of parallel processing:
• Vectorized is 'off' — Vectorized determines whether ga evaluates
an entire population with one function call in a vectorized fashion. If
Vectorized is 'on', ga does not distribute the evaluation of the function,
so does not use parfor.
ga can have a hybrid function that runs after it finishes; see “Include a Hybrid
Function” on page 5-107. If you want the hybrid function to take advantage
of parallel computation, set its options separately so that UseParallel is
true. If the hybrid function is patternsearch, set CompletePoll to 'on' so
that patternsearch runs in parallel.
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8-9
8
Parallel Processing
If the hybrid function is fmincon, set the following options with optimoptions
to have parallel gradient estimation:
• GradObj must not be 'on' — it can be 'off' or [].
• Or, if there is a nonlinear constraint function, GradConstr must not be
'on' — it can be 'off' or [].
To find out how to write options for the hybrid function, see “Parallel Hybrid
Functions” on page 8-17.
Parallel Computing with gamultiobj
Parallel computing with gamultiobj works almost the same as with ga. For
detailed information, see “Genetic Algorithm” on page 8-9.
The difference between parallel computing with gamultiobj and ga has to
do with the hybrid function. gamultiobj allows only one hybrid function,
fgoalattain. This function optionally runs after gamultiobj finishes its run.
Each individual in the calculated Pareto frontier, that is, the final population
found by gamultiobj, becomes the starting point for an optimization using
fgoalattain. These optimizations run in parallel. The number of processors
performing these optimizations is the smaller of the number of individuals
and the size of your parpool.
For fgoalattain to run in parallel, set its options correctly:
fgoalopts = optimoptions(@fgoalattain,'UseParallel',true)
gaoptions = gaoptimset('HybridFcn',{@fgoalattain,fgoalopts});
Run gamultiobj with gaoptions, and fgoalattain runs in parallel. For
more information about setting the hybrid function, see “Hybrid Function
Options” on page 9-52.
gamultiobj calls fgoalattain using a parfor loop, so fgoalattain does
not estimate gradients in parallel when used as a hybrid function with
gamultiobj. For more information, see “No Nested parfor Loops” on page 8-3.
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Background
Simulated Annealing
simulannealbnd does not run in parallel automatically. However, it can
call hybrid functions that take advantage of parallel computing. To find out
how to write options for the hybrid function, see “Parallel Hybrid Functions”
on page 8-17.
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8-11
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Parallel Processing
How to Use Parallel Processing
In this section...
“Multicore Processors” on page 8-12
“Processor Network” on page 8-13
“Parallel Search Functions or Hybrid Functions” on page 8-15
Multicore Processors
If you have a multicore processor, you might see speedup using parallel
processing. You can establish a parallel pool of several workers with a Parallel
Computing Toolbox license. For a description of Parallel Computing Toolbox
software, see “Getting Started with Parallel Computing Toolbox”.
Suppose you have a dual-core processor, and want to use parallel computing:
• Enter
parpool
at the command line. MATLAB starts a pool of workers using the multicore
processor. If you had previously set a nondefault cluster profile, you can
enforce multicore (local) computing:
parpool('local')
Note Depending on your preferences, MATLAB can start a parallel pool
automatically. To enable this feature, check Automatically create a
parallel pool in Home > Parallel > Parallel Preferences.
• Set your solver to use parallel processing.
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How to Use Parallel Processing
MultiStart
Patternsearch
GA
ms =
MultiStart('UseParallel',
true);
options =
psoptimset('UseParallel',
true, 'CompletePoll', 'on',
'Vectorized', 'off');
options =
gaoptimset('UseParallel',
true, 'Vectorized', 'off');
For Optimization app:
For Optimization app:
• Options > User function
evaluation > Evaluate
objective and constraint
functions > in parallel
• Options > User function
evaluation > Evaluate
fitness and constraint
functions > in parallel
or
ms.UseParallel = true
-
Options > Complete poll
> on
When you run an applicable solver with options, applicable solvers
automatically use parallel computing.
To stop computing optimizations in parallel, set UseParallel to false, or
set the Optimization app not to compute in parallel. To halt all parallel
computation, enter
delete(gcp)
Processor Network
If you have multiple processors on a network, use Parallel Computing Toolbox
functions and MATLAB Distributed Computing Server™ software to establish
parallel computation. Here are the steps to take:
1 Make sure your system is configured properly for parallel computing.
Check with your systems administrator, or refer to the Parallel Computing
Toolbox documentation.
To perform a basic check:
a At the command line, enter
parpool(prof)
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8
Parallel Processing
where prof is your cluster profile.
b Workers must be able to access your objective function file and, if
applicable, your nonlinear constraint function file. There are two ways
of ensuring access:
i Distribute the files to the workers using the parpool AttachedFiles
argument. For example, if objfun.m is your objective function file,
and constrfun.m is your nonlinear constraint function file, enter
parpool('AttachedFiles',{'objfun.m','constrfun.m'});
Workers access their own copies of the files.
ii Give a network file path to your files. If network_file_path is the
network path to your objective or constraint function files, enter
pctRunOnAll('addpath network_file_path')
Workers access the function files over the network.
c Check whether a file is on the path of every worker by entering
pctRunOnAll('which filename')
If any worker does not have a path to the file, it reports
filename not found.
2 Set your solver to use parallel processing.
MultiStart
Patternsearch
GA
ms =
MultiStart('UseParallel',
true);
options =
psoptimset('UseParallel',
true, 'CompletePoll', 'on',
'Vectorized', 'off');
options =
gaoptimset('UseParallel',
true, 'Vectorized', 'off');
For Optimization app:
For Optimization app:
• Options > User function
evaluation > Evaluate
• Options > User function
evaluation > Evaluate
or
ms.UseParallel = true
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How to Use Parallel Processing
MultiStart
Patternsearch
GA
objective and constraint
functions > in parallel
fitness and constraint
functions > in parallel
• Options > Complete poll
> on
After you establish your parallel computing environment, applicable solvers
automatically use parallel computing whenever you call them with options.
To stop computing optimizations in parallel, set UseParallel to false, or
set the Optimization app not to compute in parallel. To halt all parallel
computation, enter
delete(gcp)
Parallel Search Functions or Hybrid Functions
To have a patternsearch search function run in parallel, or a hybrid function
for ga or simulannealbnd run in parallel, do the following.
1 Set up parallel processing as described in “Multicore Processors” on page
8-12 or “Processor Network” on page 8-13.
2 Ensure that your search function or hybrid function has the conditions
outlined in these sections:
• “patternsearch Search Function” on page 8-15
• “Parallel Hybrid Functions” on page 8-17
patternsearch Search Function
patternsearch uses a parallel search function under the following conditions:
• CompleteSearch is 'on'.
• The search method is not @searchneldermead or custom.
• If the search method is a patternsearch poll method or Latin hypercube
search, UseParallel is true. Set at the command line with psoptimset:
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8
Parallel Processing
options = psoptimset('UseParallel',true,...
'CompleteSearch','on','SearchMethod',@GPSPositiveBasis2N);
Or you can use the Optimization app.
• If the search method is ga, the search method option structure has
UseParallel set to true. Set at the command line with psoptimset and
gaoptimset:
iterlim = 1; % iteration limit, specifies # ga runs
gaopt = gaoptimset('UseParallel',true);
options = psoptimset('SearchMethod',...
{@searchga,iterlim,gaopt});
In the Optimization app, first create the gaopt structure as above, and then
use these settings:
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How to Use Parallel Processing
For more information about search functions, see “Using a Search Method”
on page 4-33.
Parallel Hybrid Functions
ga and simulannealbnd can have other solvers run after or interspersed
with their iterations. These other solvers are called hybrid functions. For
information on using a hybrid function with gamultiobj, see “Parallel
Computing with gamultiobj” on page 8-10. Both patternsearch and fmincon
can be hybrid functions. You can set options so that patternsearch runs in
parallel, or fmincon estimates gradients in parallel.
Set the options for the hybrid function as described in “Hybrid Function
Options” on page 9-52 for ga, or “Hybrid Function Options” on page 9-63
for simulannealbnd. To summarize:
• If your hybrid function is patternsearch
1 Create a patternsearch options structure:
hybridopts = psoptimset('UseParallel',true,...
'CompletePoll','on');
2 Set the ga or simulannealbnd options to use patternsearch as a hybrid
function:
options = gaoptimset('UseParallel',true); % for ga
options = gaoptimset(options,...
'HybridFcn',{@patternsearch,hybridopts});
% or, for simulannealbnd:
options = saoptimset('HybridFcn',{@patternsearch,hybridopts});
Or use the Optimization app.
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Parallel Processing
For more information on parallel patternsearch, see “Pattern Search”
on page 8-7.
• If your hybrid function is fmincon:
1 Create a fmincon options structure:
hybridopts = optimoptions(@fmincon,'UseParallel',true,...
'Algorithm','interior-point');
% You can use any Algorithm except trust-region-reflective
2 Set the ga or simulannealbnd options to use fmincon as a hybrid function:
options =
options =
% or, for
options =
gaoptimset('UseParallel',true);
gaoptimset(options,'HybridFcn',{@fmincon,hybridopts});
simulannealbnd:
saoptimset('HybridFcn',{@fmincon,hybridopts});
Or use the Optimization app.
For more information on parallel fmincon, see “Parallel Computing”.
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Minimizing an Expensive Optimization Problem Using Parallel Computing Toolbox™
Minimizing an Expensive Optimization Problem Using
Parallel Computing Toolbox™
This example shows how to how to speed up the minimization of an expensive
optimization problem using functions in Optimization Toolbox™ and
Global Optimization Toolbox. In the first part of the example we solve the
optimization problem by evaluating functions in a serial fashion and in the
second part of the example we solve the same problem using the parallel for
loop (parfor) feature by evaluating functions in parallel. We compare the
time taken by the optimization function in both cases.
Expensive Optimization Problem
For the purpose of this example, we solve a problem in four variables, where
the objective and constraint functions are made artificially expensive by
pausing.
type expensive_objfun.m
type expensive_confun.m
function f = expensive_objfun(x)
%EXPENSIVE_OBJFUN An expensive objective function used in optimparfor examp
%
%
Copyright 2007-2013 The MathWorks, Inc.
$Revision: 1.1.8.2 $ $Date: 2013/05/04 00:47:14 $
% Simulate an expensive function by pausing
pause(0.1)
% Evaluate objective function
f = exp(x(1)) * (4*x(3)^2 + 2*x(4)^2 + 4*x(1)*x(2) + 2*x(2) + 1);
function [c,ceq] = expensive_confun(x)
%EXPENSIVE_CONFUN An expensive constraint function used in optimparfor exam
%
%
Copyright 2007-2013 The MathWorks, Inc.
$Revision: 1.1.8.2 $ $Date: 2013/05/04 00:47:13 $
% Simulate an expensive function by pausing
pause(0.1);
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Parallel Processing
% Evaluate constraints
c = [1.5 + x(1)*x(2)*x(3) - x(1) - x(2) - x(4);
-x(1)*x(2) + x(4) - 10];
% No nonlinear equality constraints:
ceq = [];
Minimizing Using fmincon
We are interested in measuring the time taken by fmincon in serial so that
we can compare it to the parallel fmincon evaluation.
startPoint = [-1 1 1 -1];
options = optimoptions('fmincon','Display','iter','Algorithm','sqp');
startTime = tic;
xsol = fmincon(@expensive_objfun,startPoint,[],[],[],[],[],[],@expensive_co
time_fmincon_sequential = toc(startTime);
fprintf('Serial FMINCON optimization takes %g seconds.\n',time_fmincon_sequ
Iter F-count
0
5
1
12
2
17
3
22
4
27
5
32
6
37
7
43
8
48
9
53
10
58
11
63
12
68
13
73
14
78
15
83
f(x) Feasibility
1.839397e+00
1.500e+00
-8.841073e-01
4.019e+00
-1.382832e+00
0.000e+00
-2.241952e+00
0.000e+00
-3.145762e+00
0.000e+00
-5.277523e+00
6.413e+00
-6.310709e+00
0.000e+00
-6.447956e+00
0.000e+00
-7.135133e+00
0.000e+00
-7.162732e+00
0.000e+00
-7.178390e+00
0.000e+00
-7.180399e+00
1.191e-05
-7.180408e+00
0.000e+00
-7.180411e+00
0.000e+00
-7.180412e+00
0.000e+00
-7.180412e+00
0.000e+00
Steplength
4.900e-01
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
7.000e-01
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
Norm of First-ord
step optimali
3.311e+
2.335e+00
7.015e1.142e+00
9.272e2.447e+00
1.481e+
1.756e+00
5.464e+
2.224e+00
1.357e+
1.099e+00
1.309e+
2.191e+00
3.631e+
3.719e-01
1.205e4.083e-01
2.935e1.591e-01
3.110e2.644e-02
1.553e1.140e-02
5.584e1.764e-03
4.677e8.827e-05
1.304e1.528e-06
1.023e-
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
8-20
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Minimizing an Expensive Optimization Problem Using Parallel Computing Toolbox™
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint
Serial FMINCON optimization takes 18.1397 seconds.
Minimizing Using Genetic Algorithm
Since ga usually takes many more function evaluations than fmincon, we
remove the expensive constraint from this problem and perform unconstrained
optimization instead; we pass empty ([]) for constraints. In addition, we limit
the maximum number of generations to 15 for ga so that ga can terminate
in a reasonable amount of time. We are interested in measuring the time
taken by ga so that we can compare it to the parallel ga evaluation. Note that
running ga requires Global Optimization Toolbox.
rng default % for reproducibility
try
gaAvailable = false;
nvar = 4;
gaoptions = gaoptimset('Generations',15,'Display','iter');
startTime = tic;
gasol = ga(@expensive_objfun,nvar,[],[],[],[],[],[],[],gaoptions);
time_ga_sequential = toc(startTime);
fprintf('Serial GA optimization takes %g seconds.\n',time_ga_sequential
gaAvailable = true;
catch ME
warning(message('optimdemos:optimparfor:gaNotFound'));
end
Generation
1
2
3
4
5
6
f-count
100
150
200
250
300
350
Best
f(x)
-6.433e+16
-1.501e+17
-7.878e+26
-8.664e+27
-1.096e+28
-5.422e+33
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Mean
f(x)
-1.287e+15
-7.138e+15
-1.576e+25
-1.466e+26
-2.062e+26
-1.145e+32
Stall
Generations
0
0
0
0
0
0
8-21
8
Parallel Processing
7
400
-1.636e+36
-3.316e+34
0
8
450
-2.933e+36
-1.513e+35
0
9
500
-1.351e+40
-2.705e+38
0
10
550
-1.351e+40
-7.9e+38
1
11
600
-2.07e+40
-2.266e+39
0
12
650
-1.845e+44
-3.696e+42
0
13
700
-2.893e+44
-1.687e+43
0
14
750
-5.076e+44
-6.516e+43
0
15
800
-8.321e+44
-2.225e+44
0
Optimization terminated: maximum number of generations exceeded.
Serial GA optimization takes 87.3686 seconds.
Setting Parallel Computing Toolbox
The finite differencing used by the functions in Optimization Toolbox to
approximate derivatives is done in parallel using the parfor feature if
Parallel Computing Toolbox is available and there is a parallel pool of workers.
Similarly, ga, gamultiobj, and patternsearch solvers in Global Optimization
Toolbox evaluate functions in parallel. To use the parfor feature, we use the
parpool function to set up the parallel environment. The computer on which
this example is published has four cores, so parpool starts four MATLAB®
workers. If there is already a parallel pool when you run this example, we use
that pool; see the documentation for parpool for more information.
if max(size(gcp)) == 0 % parallel pool needed
parpool % create the parallel pool
end
Starting parallel pool (parpool) using the 'local' profile ... connected to
Minimizing Using Parallel fmincon
To minimize our expensive optimization problem using the parallel fmincon
function, we need to explicitly indicate that our objective and constraint
functions can be evaluated in parallel and that we want fmincon to use its
parallel functionality wherever possible. Currently, finite differencing can be
done in parallel. We are interested in measuring the time taken by fmincon
so that we can compare it to the serial fmincon run.
options = optimoptions(options,'UseParallel',true);
8-22
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Minimizing an Expensive Optimization Problem Using Parallel Computing Toolbox™
startTime = tic;
xsol = fmincon(@expensive_objfun,startPoint,[],[],[],[],[],[],@expensive_co
time_fmincon_parallel = toc(startTime);
fprintf('Parallel FMINCON optimization takes %g seconds.\n',time_fmincon_pa
Iter F-count
0
5
1
12
2
17
3
22
4
27
5
32
6
37
7
43
8
48
9
53
10
58
11
63
12
68
13
73
14
78
15
83
f(x) Feasibility
1.839397e+00
1.500e+00
-8.841073e-01
4.019e+00
-1.382832e+00
0.000e+00
-2.241952e+00
0.000e+00
-3.145762e+00
0.000e+00
-5.277523e+00
6.413e+00
-6.310709e+00
0.000e+00
-6.447956e+00
0.000e+00
-7.135133e+00
0.000e+00
-7.162732e+00
0.000e+00
-7.178390e+00
0.000e+00
-7.180399e+00
1.191e-05
-7.180408e+00
0.000e+00
-7.180411e+00
0.000e+00
-7.180412e+00
0.000e+00
-7.180412e+00
0.000e+00
Steplength
4.900e-01
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
7.000e-01
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
1.000e+00
Norm of First-ord
step optimali
3.311e+
2.335e+00
7.015e1.142e+00
9.272e2.447e+00
1.481e+
1.756e+00
5.464e+
2.224e+00
1.357e+
1.099e+00
1.309e+
2.191e+00
3.631e+
3.719e-01
1.205e4.083e-01
2.935e1.591e-01
3.110e2.644e-02
1.553e1.140e-02
5.584e1.764e-03
4.677e8.827e-05
1.304e1.528e-06
1.023e-
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint
Parallel FMINCON optimization takes 8.78988 seconds.
Minimizing Using Parallel Genetic Algorithm
To minimize our expensive optimization problem using the ga function,
we need to explicitly indicate that our objective function can be evaluated
in parallel and that we want ga to use its parallel functionality wherever
possible. To use the parallel ga we also require that the ’Vectorized’ option
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8-23
8
Parallel Processing
be set to the default (i.e., ’off’). We are again interested in measuring the
time taken by ga so that we can compare it to the serial ga run. Though this
run may be different from the serial one because ga uses a random number
generator, the number of expensive function evaluations is the same in both
runs. Note that running ga requires Global Optimization Toolbox.
rng default % to get the same evaluations as the previous run
if gaAvailable
gaoptions = gaoptimset(gaoptions,'UseParallel',true);
startTime = tic;
gasol = ga(@expensive_objfun,nvar,[],[],[],[],[],[],[],gaoptions);
time_ga_parallel = toc(startTime);
fprintf('Parallel GA optimization takes %g seconds.\n',time_ga_parallel
end
Generation
f-count
1
100
2
150
3
200
4
250
5
300
6
350
7
400
8
450
9
500
10
550
11
600
12
650
13
700
14
750
15
800
Optimization terminated:
Parallel GA optimization
Best
Mean
Stall
f(x)
f(x)
Generations
-6.433e+16
-1.287e+15
0
-1.501e+17
-7.138e+15
0
-7.878e+26
-1.576e+25
0
-8.664e+27
-1.466e+26
0
-1.096e+28
-2.062e+26
0
-5.422e+33
-1.145e+32
0
-1.636e+36
-3.316e+34
0
-2.933e+36
-1.513e+35
0
-1.351e+40
-2.705e+38
0
-1.351e+40
-7.9e+38
1
-2.07e+40
-2.266e+39
0
-1.845e+44
-3.696e+42
0
-2.893e+44
-1.687e+43
0
-5.076e+44
-6.516e+43
0
-8.321e+44
-2.225e+44
0
maximum number of generations exceeded.
takes 23.707 seconds.
Compare Serial and Parallel Time
X = [time_fmincon_sequential time_fmincon_parallel];
Y = [time_ga_sequential time_ga_parallel];
8-24
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Minimizing an Expensive Optimization Problem Using Parallel Computing Toolbox™
t = [0 1];
plot(t,X,'r--',t,Y,'k-')
ylabel('Time in seconds')
legend('fmincon','ga')
set(gca,'XTick',[0 1])
set(gca,'XTickLabel',{'Serial' 'Parallel'})
axis([0 1 0 ceil(max([X Y]))])
title('Serial Vs. Parallel Times')
Utilizing parallel function evaluation via parfor improved the efficiency
of both fmincon and ga. The improvement is typically better for expensive
objective and constraint functions.
At last we delete the parallel pool.
if max(size(gcp)) > 0 % parallel pool exists
delete(gcp) % delete the pool
end
Parallel pool using the 'local' profile is shutting down.
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8-25
‫‪Parallel Processing‬‬
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‫‪8‬‬
‫‪8-26‬‬
9
Options Reference
• “GlobalSearch and MultiStart Properties (Options)” on page 9-2
• “Pattern Search Options” on page 9-9
• “Genetic Algorithm Options” on page 9-31
• “Simulated Annealing Options” on page 9-58
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9
Options Reference
GlobalSearch and MultiStart Properties (Options)
In this section...
“How to Set Properties” on page 9-2
“Properties of Both Objects” on page 9-2
“GlobalSearch Properties” on page 9-7
“MultiStart Properties” on page 9-8
How to Set Properties
To create a GlobalSearch or MultiStart object with nondefault properties,
use name-value pairs. For example, to create a GlobalSearch object that has
iterative display and runs only from feasible points with respect to bounds
and inequalities, enter
gs = GlobalSearch('Display','iter', ...
'StartPointsToRun','bounds-ineqs');
To set a property of an existing GlobalSearch or MultiStart object, use dot
notation. For example, if ms is a MultiStart object, and you want to set the
Display property to 'iter', enter
ms.Display = 'iter';
To set multiple properties of an existing object simultaneously, use the
constructor (GlobalSearch or MultiStart) with name-value pairs. For
example, to set the Display property to 'iter' and the MaxTime property
to 100, enter
ms = MultiStart(ms,'Display','iter','MaxTime',100);
For more information on setting properties, see “Changing Global Options”
on page 3-77.
Properties of Both Objects
You can create a MultiStart object from a GlobalSearch object and
vice-versa.
9-2
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GlobalSearch and MultiStart Properties (Options)
The syntax for creating a new object from an existing object is:
ms = MultiStart(gs);
or
gs = GlobalSearch(ms);
The new object contains the properties that apply of the old object. This
section describes those shared properties:
• “Display” on page 9-3
• “MaxTime” on page 9-3
• “OutputFcns” on page 9-3
• “PlotFcns” on page 9-5
• “StartPointsToRun” on page 9-6
• “TolX” on page 9-6
• “TolFun” on page 9-6
Display
Values for the Display property are:
• 'final' (default) — Summary results to command line after last solver
run.
• 'off' — No output to command line.
• 'iter' — Summary results to command line after each local solver run.
MaxTime
The MaxTime property describes a tolerance on the number of seconds since
the solver began its run. Solvers halt when they see MaxTime seconds have
passed since the beginning of the run. Time means wall clock as opposed to
processor cycles. The default is Inf.
OutputFcns
The OutputFcns property directs the global solver to run one or more output
functions after each local solver run completes. The output functions also
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9-3
9
Options Reference
run when the global solver starts and ends. Include a handle to an output
function written in the appropriate syntax, or include a cell array of such
handles. The default is empty ([]).
The syntax of an output function is:
stop = outFcn(optimValues,state)
• stop is a Boolean. When true, the algorithm stops. When false, the
algorithm continues.
Note A local solver can have an output function. The global
solver does not necessarily stop when a local solver output function
causes a local solver run to stop. If you want the global solver to
stop in this case, have the global solver output function stop when
optimValues.localsolution.exitflag=-1.
• optimValues is a structure, described in “optimvalues Structure” on page
9-5.
• state is a string that indicates the current state of the global algorithm:
9-4
-
'init' — The global solver has not called the local solver. The fields
in the optimValues structure are empty, except for localrunindex,
which is 0, and funccount, which contains the number of objective and
constraint function evaluations.
-
'iter' — The global solver calls output functions after each local solver
-
'done' — The global solver finished calling local solvers. The fields
in optimValues generally have the same values as the ones from the
final output function call with state='iter'. However, the value of
optimValues.funccount for GlobalSearch can be larger than the
value in the last function call with 'iter', because the GlobalSearch
algorithm might have performed some function evaluations that were
not part of a local solver. For more information, see “GlobalSearch
Algorithm” on page 3-54.
run.
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GlobalSearch and MultiStart Properties (Options)
For an example using an output function, see “GlobalSearch Output Function”
on page 3-43.
Note Output and plot functions do not run when MultiStart has the
UseParallel option set to true and there is an open parpool.
optimvalues Structure. The optimValues structure contains the following
fields:
• bestx — The current best point
• bestfval — Objective function value at bestx
• funccount — Total number of function evaluations
• localrunindex — Index of the local solver run
• localsolution — A structure containing part of the output of the local
solver call: X, Fval and Exitflag
PlotFcns
The PlotFcns property directs the global solver to run one or more plot
functions after each local solver run completes. Include a handle to a plot
function written in the appropriate syntax, or include a cell array of such
handles. The default is empty ([]).
The syntax of a plot function is the same as that of an output function. For
details, see “OutputFcns” on page 9-3.
There are two predefined plot functions for the global solvers:
• @gsplotbestf plots the best objective function value.
• @gsplotfunccount plots the number of function evaluations.
For an example using a plot function, see “MultiStart Plot Function” on page
3-48.
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9-5
9
Options Reference
If you specify more than one plot function, all plots appear as subplots in
the same window. Right-click any subplot to obtain a larger version in a
separate figure window.
Note Output and plot functions do not run when MultiStart has the
UseParallel option set to true and there is an open parpool.
StartPointsToRun
The StartPointsToRun property directs the solver to exclude certain start
points from being run:
• all — Accept all start points.
• bounds — Reject start points that do not satisfy bounds.
• bounds-ineqs — Reject start points that do not satisfy bounds or inequality
constraints.
TolX
The TolX property describes how close two points must be for solvers to
consider them identical for creating the vector of local solutions. Set TolX to 0
to obtain the results of every local solver run. Set TolX to a larger value to
have fewer results. Solvers compute the distance between a pair of points
with norm, the Euclidean distance.
Solvers consider two solutions identical if they are within TolX distance of
each other and have objective function values within TolFun of each other. If
both conditions are not met, solvers report the solutions as distinct.
TolFun
The TolFun property describes how close two objective function values must
be for solvers to consider them identical for creating the vector of local
solutions. Set TolFun to 0 to obtain the results of every local solver run. Set
TolFun to a larger value to have fewer results.
9-6
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GlobalSearch and MultiStart Properties (Options)
Solvers consider two solutions identical if they are within TolX distance of
each other and have objective function values within TolFun of each other. If
both conditions are not met, solvers report the solutions as distinct.
GlobalSearch Properties
• “NumTrialPoints” on page 9-7
• “NumStageOnePoints” on page 9-7
• “MaxWaitCycle” on page 9-7
• “BasinRadiusFactor” on page 9-8
• “DistanceThresholdFactor” on page 9-8
• “PenaltyThresholdFactor” on page 9-8
NumTrialPoints
Number of potential start points to examine in addition to x0 from the
problem structure. GlobalSearch runs only those potential start points that
pass several tests. For more information, see “GlobalSearch Algorithm” on
page 3-54.
Default: 1000
NumStageOnePoints
Number of start points in Stage 1. For details, see “Obtain Stage 1 Start
Point, Run” on page 3-55.
Default: 200
MaxWaitCycle
A positive integer tolerance appearing in several points in the algorithm.
• If the observed penalty function of MaxWaitCycle consecutive trial points
is at least the penalty threshold, then raise the penalty threshold (see
“PenaltyThresholdFactor” on page 9-8).
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9-7
9
Options Reference
• If MaxWaitCycle consecutive trial points are in a basin, then update that
basin’s radius (see “BasinRadiusFactor” on page 9-8).
Default: 20
BasinRadiusFactor
A basin radius decreases after MaxWaitCycle consecutive start points
are within the basin. The basin radius decreases by a factor of
1–BasinRadiusFactor.
Default: 0.2
DistanceThresholdFactor
A multiplier for determining whether a trial point is in an existing basin of
attraction. For details, see “Examine Stage 2 Trial Point to See if fmincon
Runs” on page 3-56. Default: 0.75
PenaltyThresholdFactor
Determines increase in penalty threshold. For details, see React to Large
Counter Values.
Default: 0.2
MultiStart Properties
UseParallel
The UseParallel property determines whether the solver distributes start
points to multiple processors:
• false (default) — Do not run in parallel.
• true — Run in parallel.
For the solver to run in parallel you must set up a parallel environment with
parpool. For details, see “How to Use Parallel Processing” on page 8-12.
9-8
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Pattern Search Options
Pattern Search Options
In this section...
“Optimization App vs. Command Line” on page 9-9
“Plot Options” on page 9-10
“Poll Options” on page 9-12
“Search Options” on page 9-14
“Mesh Options” on page 9-19
“Constraint Parameters” on page 9-20
“Cache Options” on page 9-21
“Stopping Criteria” on page 9-21
“Output Function Options” on page 9-22
“Display to Command Window Options” on page 9-24
“Vectorize and Parallel Options (User Function Evaluation)” on page 9-25
“Options Table for Pattern Search Algorithms” on page 9-27
Optimization App vs. Command Line
There are two ways to specify options for pattern search, depending
on whether you are using the Optimization app or calling the function
patternsearch at the command line:
• If you are using the Optimization app, you specify the options by selecting
an option from a drop-down list or by entering the value of the option in
the text field.
• If you are calling patternsearch from the command line, you specify the
options by creating an options structure using the function psoptimset,
as follows:
options = psoptimset('Param1',value1,'Param2',value2,...);
See “Set Options” on page 4-48 for examples.
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9-9
9
Options Reference
In this section, each option is listed in two ways:
• By its label, as it appears in the Optimization app
• By its field name in the options structure
For example:
• Poll method refers to the label of the option in the Optimization app.
• PollMethod refers to the corresponding field of the options structure.
Plot Options
Plot options enable you to plot data from the pattern search while it is
running. When you select plot functions and run the pattern search, a plot
window displays the plots on separate axes. You can stop the algorithm at
any time by clicking the Stop button on the plot window.
Plot interval (PlotInterval) specifies the number of iterations between
consecutive calls to the plot function.
You can select any of the following plots in the Plot functions pane.
• Best function value (@psplotbestf) plots the best objective function
value.
• Function count (@psplotfuncount) plots the number of function
evaluations.
• Mesh size (@psplotmeshsize) plots the mesh size.
• Best point (@psplotbestx) plots the current best point.
• Max constraint (@psplotmaxconstr) plots the maximum nonlinear
constraint violation.
• Custom enables you to use your own plot function. To specify the plot
function using the Optimization app,
-
9-10
Select Custom function.
Enter @myfun in the text box, where myfun is the name of your function.
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Pattern Search Options
“Structure of the Plot Functions” on page 9-11 describes the structure of
a plot function.
To display a plot when calling patternsearch from the command line, set the
PlotFcns field of options to be a function handle to the plot function. For
example, to display the best function value, set options as follows
options = psoptimset('PlotFcns', @psplotbestf);
To display multiple plots, use the syntax
options = psoptimset('PlotFcns', {@plotfun1, @plotfun2, ...});
where @plotfun1, @plotfun2, and so on are function handles to the plot
functions (listed in parentheses in the preceding list).
If you specify more than one plot function, all plots appear as subplots in
the same window. Right-click any subplot to obtain a larger version in a
separate figure window.
Structure of the Plot Functions
The first line of a plot function has the form
function stop = plotfun(optimvalues, flag)
The input arguments to the function are
• optimvalues — Structure containing information about the current state
of the solver. The structure contains the following fields:
-
x — Current point
iteration — Iteration number
fval — Objective function value
meshsize — Current mesh size
funccount — Number of function evaluations
method — Method used in last iteration
TolFun — Tolerance on function value in last iteration
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Options Reference
-
TolX — Tolerance on x value in last iteration
nonlinineq — Nonlinear inequality constraints, displayed only when a
nonlinear constraint function is specified
nonlineq — Nonlinear equality constraints, displayed only when a
nonlinear constraint function is specified
• flag — Current state in which the plot function is called. The possible
values for flag are
-
init — Initialization state
iter — Iteration state
interrupt — Intermediate stage
done — Final state
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
The output argument stop provides a way to stop the algorithm at the current
iteration. stop can have the following values:
• false — The algorithm continues to the next iteration.
• true — The algorithm terminates at the current iteration.
Poll Options
Poll options control how the pattern search polls the mesh points at each
iteration.
Poll method (PollMethod) specifies the pattern the algorithm uses to create
the mesh. There are two patterns for each of the classes of direct search
algorithms: the generalized pattern search (GPS) algorithm, the generating
set search (GSS) algorithm, and the mesh adaptive direct search (MADS)
algorithm. These patterns are the Positive basis 2N and the Positive basis
N+1:
• The default pattern, GPS Positive basis 2N (GPSPositiveBasis2N),
consists of the following 2N vectors, where N is the number of independent
variables for the objective function.
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Pattern Search Options
[1 0 0...0][0 1 0...0] ...[0 0 0...1][–1 0 0...0][0 –1 0...0][0 0 0...–1].
For example, if the optimization problem has three independent variables,
the pattern consists of the following six vectors.
[1 0 0][0 1 0][0 0 1][–1 0 0][0 –1 0][0 0 –1].
• The GSS Positive basis 2N pattern (GSSPositiveBasis2N) is similar to
GPS Positive basis 2N, but adjusts the basis vectors to account for linear
constraints. GSS Positive basis 2N is more efficient than GPS Positive
basis 2N when the current point is near a linear constraint boundary.
• The MADS Positive basis 2N pattern (MADSPositiveBasis2N) consists
of 2N randomly generated vectors, where N is the number of independent
variables for the objective function. This is done by randomly generating
N vectors which form a linearly independent set, then using this first set
and the negative of this set gives 2N vectors. As shown above, the GPS
Positive basis 2N pattern is formed using the positive and negative
of the linearly independent identity, however, with the MADS Positive
basis 2N, the pattern is generated using a random permutation of an
N-by-N linearly independent lower triangular matrix that is regenerated
at each iteration.
• The GPS Positive basis NP1 pattern consists of the following N + 1
vectors.
[1 0 0...0][0 1 0...0] ...[0 0 0...1][–1 –1 –1...–1].
For example, if the objective function has three independent variables, the
pattern consists of the following four vectors.
[1 0 0][0 1 0][0 0 1][–1 –1 –1].
• The GSS Positive basis Np1 pattern (GSSPositiveBasisNp1) is similar
to GPS Positive basis Np1, but adjusts the basis vectors to account for
linear constraints. GSS Positive basis Np1 is more efficient than GPS
Positive basis Np1 when the current point is near a linear constraint
boundary.
• The MADS Positive basis Np1 pattern (MADSPositiveBasisNp1) consists
of N randomly generated vectors to form the positive basis, where N is
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Options Reference
the number of independent variables for the objective function. Then, one
more random vector is generated, giving N+1 randomly generated vectors.
Each iteration generates a new pattern when the MADS Positive basis
N+1 is selected.
Complete poll (CompletePoll) specifies whether all the points in the
current mesh must be polled at each iteration. Complete Poll can have
the values On or Off.
• If you set Complete poll to On, the algorithm polls all the points in the
mesh at each iteration and chooses the point with the smallest objective
function value as the current point at the next iteration.
• If you set Complete poll to Off, the default value, the algorithm stops the
poll as soon as it finds a point whose objective function value is less than
that of the current point. The algorithm then sets that point as the current
point at the next iteration.
Polling order (PollingOrder) specifies the order in which the algorithm
searches the points in the current mesh. The options are
• Random — The polling order is random.
• Success — The first search direction at each iteration is the direction in
which the algorithm found the best point at the previous iteration. After
the first point, the algorithm polls the mesh points in the same order as
Consecutive.
• Consecutive — The algorithm polls the mesh points in consecutive order,
that is, the order of the pattern vectors as described in “Poll Method” on
page 4-22.
Search Options
Search options specify an optional search that the algorithm can perform at
each iteration prior to the polling. If the search returns a point that improves
the objective function, the algorithm uses that point at the next iteration and
omits the polling. Please note, if you have selected the same Search method
and Poll method, only the option selected in the Poll method will be used,
although both will be used when the options selected are different.
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Pattern Search Options
Complete search (CompleteSearch) applies when you set Search method
to GPS Positive basis Np1, GPS Positive basis 2N, GSS Positive basis
Np1, GSS Positive basis 2N, MADS Positive basis Np1, MADS Positive
basis 2N, or Latin hypercube. Complete search can have the values On or
Off.
For GPS Positive basis Np1, MADS Positive basis Np1, GPS Positive
basis 2N, or MADS Positive basis 2N, Complete search has the same
meaning as the poll option Complete poll.
Search method (SearchMethod) specifies the optional search step. The
options are
• None ([]) (the default) specifies no search step.
• GPS Positive basis 2N (@GPSPositiveBasis2N)
• GPS Positive basis Np1 (@GPSPositiveBasisNp1)
• GSS Positive basis 2N (@GSSPositiveBasis2N)
• GSS Positive basis Np1 (@GSSPositiveBasisNp1)
• MADS Positive basis 2N (@MADSPositiveBasis2N)
• MADS Positive basis Np1 (@MADSPositiveBasisNp1)
• Genetic Algorithm (@searchga) specifies a search using the genetic
algorithm. If you select Genetic Algorithm, two other options appear:
-
Iteration limit — Positive integer specifying the number of iterations of
the pattern search for which the genetic algorithm search is performed.
The default for Iteration limit is 1.
-
Options — Options structure for the genetic algorithm, which you can
set using gaoptimset.
To change the default values of Iteration limit and Options at the
command line, use the syntax
options = psoptimset('SearchMethod',...
{@searchga,iterlim,optionsGA})
where iterlim is the value of Iteration limit and optionsGA is the
genetic algorithm options structure.
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Options Reference
Note If you set Search method to Genetic algorithm or Nelder-Mead,
we recommend that you leave Iteration limit set to the default value 1,
because performing these searches more than once is not likely to improve
results.
• Latin hypercube (@searchlhs) specifies a Latin hypercube search.
patternsearch generates each point for the search as follows. For each
component, take a random permutation of the vector [1,2,...,k] minus
rand(1,k), divided by k. (k is the number of points.) This yields k points,
with each component close to evenly spaced. The resulting points are then
scaled to fit any bounds. Latin hypercube uses default bounds of -1 and 1.
The way the search is performed depends on the setting for Complete
search:
-
If you set Complete search to On, the algorithm polls all the points that
are randomly generated at each iteration by the Latin hypercube search
and chooses the one with the smallest objective function value.
-
If you set Complete search to Off (the default), the algorithm stops
the poll as soon as it finds one of the randomly generated points whose
objective function value is less than that of the current point, and
chooses that point for the next iteration.
If you select Latin hypercube, two other options appear:
-
Iteration limit — Positive integer specifying the number of iterations
of the pattern search for which the Latin hypercube search is performed.
The default for Iteration limit is 1.
-
Design level — The Design level is the number of points
patternsearch searches, a positive integer. The default for Design
level is 15 times the number of dimensions.
To change the default values of Iteration limit and Design level at the
command line, use the syntax
options=psoptimset('SearchMethod', {@searchlhs,iterlim,level})
where iterlim is the value of Iteration limit and level is the value of
Design level.
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Pattern Search Options
• Nelder-Mead (@searchneldermead) specifies a search using fminsearch,
which uses the Nelder-Mead algorithm. If you select Nelder-Mead, two
other options appear:
-
Iteration limit — Positive integer specifying the number of iterations
of the pattern search for which the Nelder-Mead search is performed.
The default for Iteration limit is 1.
-
Options — Options structure for the function fminsearch, which you
can create using the function optimset.
To change the default values of Iteration limit and Options at the
command line, use the syntax
options=psoptimset('SearchMethod',...
{@searchneldermead,iterlim,optionsNM})
where iterlim is the value of Iteration limit and optionsNM is the
options structure.
• Custom enables you to write your own search function. To specify the
search function using the Optimization app,
-
Set Search function to Custom.
Set Function name to @myfun, where myfun is the name of your
function.
If you are using patternsearch, set
options = psoptimset('SearchMethod', @myfun);
To see a template that you can use to write your own search function, enter
edit searchfcntemplate
The following section describes the structure of the search function.
Structure of the Search Function
Your search function must have the following calling syntax.
function [successSearch,xBest,fBest,funccount] =
searchfcntemplate(fun,x,A,b,Aeq,beq,lb,ub, ...
optimValues,options)
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Options Reference
The search function has the following input arguments:
• fun — Objective function
• x — Current point
• A,b — Linear inequality constraints
• Aeq,beq — Linear equality constraints
• lb,ub — Lower and upper bound constraints
• optimValues — Structure that enables you to set search options. The
structure contains the following fields:
-
x — Current point
fval — Objective function value at x
iteration — Current iteration number
funccount — Counter for user function evaluation
scale — Scale factor used to scale the design points
problemtype — Flag passed to the search routines, indicating
whether the problem is 'unconstrained', 'boundconstraints', or
'linearconstraints'. This field is a subproblem type for nonlinear
constrained problems.
-
meshsize — Current mesh size used in search step
method — Method used in last iteration
• options — Pattern search options structure
The function has the following output arguments:
• successSearch — A Boolean identifier indicating whether the search is
successful or not
• xBest,fBest — Best point and best function value found by search method
• funccount — Number of user function evaluation in search method
See “Using a Search Method” on page 4-33 for an example.
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Pattern Search Options
Mesh Options
Mesh options control the mesh that the pattern search uses. The following
options are available.
Initial size (InitialMeshSize) specifies the size of the initial mesh, which is
the length of the shortest vector from the initial point to a mesh point. Initial
size should be a positive scalar. The default is 1.0.
Max size (MaxMeshSize) specifies a maximum size for the mesh. When the
maximum size is reached, the mesh size does not increase after a successful
iteration. Max size must be a positive scalar, and is only used when a GPS or
GSS algorithm is selected as the Poll or Search method. The default value is
Inf. MADS uses a maximum size of 1.
Accelerator (MeshAccelerator) specifies whether, when the mesh size is
small, the Contraction factor is multiplied by 0.5 after each unsuccessful
iteration. Accelerator can have the values On or Off, the default.
Accelerator applies to the GPS and GSS algorithms.
Rotate (MeshRotate) is only applied when Poll method is set to GPS
Positive basis Np1 or GSS Positive basis Np1. It specifies whether the
mesh vectors are multiplied by –1 when the mesh size is less than 1/100 of the
mesh tolerance (minimum mesh size TolMesh) after an unsuccessful poll. In
other words, after the first unsuccessful poll with small mesh size, instead of
polling in directions ei (unit positive directions) and –Σei, the algorithm polls
in directions –ei and Σei. Rotate can have the values Off or On (the default).
When the problem has equality constraints, Rotate is disabled.
Rotate is especially useful for discontinuous functions.
Note Changing the setting of Rotate has no effect on the poll when Poll
method is set to GPS Positive basis 2N, GSS Positive basis 2N, MADS
Positive basis 2N, or MADS Positive basis Np1.
Scale (ScaleMesh) specifies whether the algorithm scales the mesh points
by carefully multiplying the pattern vectors by constants proportional to the
logarithms of the absolute values of components of the current point (or, for
unconstrained problems, of the initial point). Scale can have the values
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Options Reference
Off or On (the default). When the problem has equality constraints, Scale
is disabled.
Expansion factor (MeshExpansion) specifies the factor by which the mesh
size is increased after a successful poll. The default value is 2.0, which
means that the size of the mesh is multiplied by 2.0 after a successful poll.
Expansion factor must be a positive scalar and is only used when a GPS or
GSS method is selected as the Poll or Search method. MADS uses a factor of
4.0.
Contraction factor (MeshContraction) specifies the factor by which the
mesh size is decreased after an unsuccessful poll. The default value is
0.5, which means that the size of the mesh is multiplied by 0.5 after an
unsuccessful poll. Contraction factor must be a positive scalar and is only
used when a GPS or GSS method is selected as the Poll or Search method.
MADS uses a factor of 0.25.
See “Mesh Expansion and Contraction” on page 4-63 for more information.
Constraint Parameters
For information on the meaning of penalty parameters, see “Nonlinear
Constraint Solver Algorithm” on page 4-39.
• Initial penalty (InitialPenalty) — Specifies an initial value of the
penalty parameter that is used by the nonlinear constraint algorithm.
Initial penalty must be greater than or equal to 1, and has a default of 10.
• Penalty factor (PenaltyFactor) — Increases the penalty parameter when
the problem is not solved to required accuracy and constraints are not
satisfied. Penalty factor must be greater than 1, and has a default of 100.
Bind tolerance (TolBind) specifies the tolerance for the distance from the
current point to the boundary of the feasible region with respect to linear
constraints. This means Bind tolerance specifies when a linear constraint is
active. Bind tolerance is not a stopping criterion. Active linear constraints
change the pattern of points patternsearch uses for polling or searching.
patternsearch always uses points that satisfy linear constraints to within
Bind tolerance. The default value of Bind tolerance is 1e-3.
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Pattern Search Options
Cache Options
The pattern search algorithm can keep a record of the points it has already
polled, so that it does not have to poll the same point more than once. If the
objective function requires a relatively long time to compute, the cache option
can speed up the algorithm. The memory allocated for recording the points is
called the cache. This option should only be used for deterministic objective
functions, but not for stochastic ones.
Cache (Cache) specifies whether a cache is used. The options are On and Off,
the default. When you set Cache to On, the algorithm does not evaluate the
objective function at any mesh points that are within Tolerance of a point
in the cache.
Tolerance (CacheTol) specifies how close a mesh point must be to a point in
the cache for the algorithm to omit polling it. Tolerance must be a positive
scalar. The default value is eps.
Size (CacheSize) specifies the size of the cache. Size must be a positive
scalar. The default value is 1e4.
See “Use Cache” on page 4-81 for more information.
Stopping Criteria
Stopping criteria determine what causes the pattern search algorithm to stop.
Pattern search uses the following criteria:
Mesh tolerance (TolMesh) specifies the minimum tolerance for mesh size.
The GPS and GSS algorithms stop if the mesh size becomes smaller than
Mesh tolerance. MADS 2N stops when the mesh size becomes smaller than
TolMesh^2. MADS Np1 stops when the mesh size becomes smaller than
(TolMesh/nVar)^2, where nVar is the number of elements of x0. The default
value of TolMesh is 1e-6.
Max iteration (MaxIter) specifies the maximum number of iterations the
algorithm performs. The algorithm stops if the number of iterations reaches
Max iteration. You can select either
• 100*numberOfVariables — Maximum number of iterations is 100 times
the number of independent variables (the default).
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Options Reference
• Specify — A positive integer for the maximum number of iterations
Max function evaluations (MaxFunEval) specifies the maximum number
of evaluations of the objective function. The algorithm stops if the number
of function evaluations reaches Max function evaluations. You can select
either
• 2000*numberOfVariables — Maximum number of function evaluations
is 2000 times the number of independent variables.
• Specify — A positive integer for the maximum number of function
evaluations
Time limit (TimeLimit) specifies the maximum time in seconds the pattern
search algorithm runs before stopping. This also includes any specified pause
time for pattern search algorithms.
X tolerance (TolX) specifies the minimum distance between the current
points at two consecutive iterations. The algorithm stops if the distance
between two consecutive points is less than X tolerance and the mesh size is
smaller than X tolerance. The default value is 1e-6.
Function tolerance (TolFun) specifies the minimum tolerance for the
objective function. After a successful poll, the algorithm stops if the difference
between the function value at the previous best point and function value at
the current best point is less than Function tolerance, and if the mesh size
is also smaller than Function tolerance. The default value is 1e-6.
See “Setting Solver Tolerances” on page 4-32 for an example.
Constraint tolerance (TolCon) — The Constraint tolerance is not used
as stopping criterion. It is used to determine the feasibility with respect to
nonlinear constraints.
Output Function Options
Output functions are functions that the pattern search algorithm calls at each
generation. To specify the output function using the Optimization app,
• Select Custom function.
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Pattern Search Options
• Enter @myfun in the text box, where myfun is the name of your function.
• To pass extra parameters in the output function, use “Anonymous
Functions”.
• For multiple output functions, enter a cell array of output function handles:
{@myfun1,@myfun2,...}.
At the command line, set
options = psoptimset('OutputFcns',@myfun);
For multiple output functions, enter a cell array:
options = psoptimset('OutputFcns',{@myfun1,@myfun2,...});
To see a template that you can use to write your own output function, enter
edit psoutputfcntemplate
at the MATLAB command prompt.
The following section describes the structure of the output function.
Structure of the Output Function
Your output function must have the following calling syntax:
[stop,options,optchanged] = myfun(optimvalues,options,flag)
The function has the following input arguments:
• optimvalues — Structure containing information about the current state
of the solver. The structure contains the following fields:
-
x — Current point
iteration — Iteration number
fval — Objective function value
meshsize — Current mesh size
funccount — Number of function evaluations
method — Method used in last iteration
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Options Reference
-
TolFun — Tolerance on function value in last iteration
TolX — Tolerance on x value in last iteration
nonlinineq — Nonlinear inequality constraints, displayed only when a
nonlinear constraint function is specified
nonlineq — Nonlinear equality constraints, displayed only when a
nonlinear constraint function is specified
• options — Options structure
• flag — Current state in which the output function is called. The possible
values for flag are
-
init — Initialization state
iter — Iteration state
interrupt — Intermediate stage
done — Final state
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the output function.
The output function returns the following arguments to ga:
• stop — Provides a way to stop the algorithm at the current iteration. stop
can have the following values.
-
false — The algorithm continues to the next iteration.
true — The algorithm terminates at the current iteration.
• options — Options structure.
• optchanged — Flag indicating changes to options.
Display to Command Window Options
Level of display ('Display') specifies how much information is displayed
at the command line while the pattern search is running. The available
options are
• Off ('off') — No output is displayed.
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Pattern Search Options
• Iterative ('iter') — Information is displayed for each iteration.
• Diagnose ('diagnose') — Information is displayed for each iteration. In
addition, the diagnostic lists some problem information and the options
that are changed from the defaults.
• Final ('final') — The reason for stopping is displayed.
Both Iterative and Diagnose display the following information:
• Iter — Iteration number
• FunEval — Cumulative number of function evaluations
• MeshSize — Current mesh size
• FunVal — Objective function value of the current point
• Method — Outcome of the current poll (with no nonlinear constraint
function specified). With a nonlinear constraint function, Method displays
the update method used after a subproblem is solved.
• Max Constraint — Maximum nonlinear constraint violation (displayed
only when a nonlinear constraint function has been specified)
The default value of Level of display is
• Off in the Optimization app
• 'final' in an options structure created using psoptimset
Vectorize and Parallel Options (User Function
Evaluation)
You can choose to have your objective and constraint functions evaluated in
serial, parallel, or in a vectorized fashion. These options are available in the
User function evaluation section of the Options pane of the Optimization
app, or by setting the 'Vectorized' and 'UseParallel' options with
psoptimset.
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Options Reference
Note You must set CompletePoll to 'on' for patternsearch to use
vectorized or parallel polling. Similarly, set CompleteSearch to 'on' for
vectorized or parallel searching.
• When Evaluate objective and constraint functions ('Vectorized') is
in serial ('Off'), patternsearch calls the objective function on one point
at a time as it loops through the mesh points. (At the command line, this
assumes 'UseParallel' is at its default value of false.)
• When Evaluate objective and constraint functions ('Vectorized') is
vectorized ('On'), patternsearch calls the objective function on all the
points in the mesh at once, i.e., in a single call to the objective function.
If there are nonlinear constraints, the objective function and the nonlinear
constraints all need to be vectorized in order for the algorithm to compute
in a vectorized manner.
For details and an example, see “Vectorize the Objective and Constraint
Functions” on page 4-85.
• When Evaluate objective and constraint functions (UseParallel) is
in parallel (true), patternsearch calls the objective function in parallel,
using the parallel environment you established (see “How to Use Parallel
Processing” on page 8-12). At the command line, set UseParallel to false
to compute serially.
Note You cannot simultaneously use vectorized and parallel computations.
If you set 'UseParallel' to true and 'Vectorized' to 'On', patternsearch
evaluates your objective and constraint functions in a vectorized manner,
not in parallel.
How Objective and Constraint Functions Are Evaluated
9-26
Assume CompletePoll
= 'on'
Vectorized = 'off'
Vectorized = 'on'
UseParallel = false
Serial
Vectorized
UseParallel = true
Parallel
Vectorized
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Pattern Search Options
Options Table for Pattern Search Algorithms
Option Availability Table for All Algorithms
Algorithm
Availability
Option
Description
Cache
With Cache set to 'on',
patternsearch keeps
a history of the mesh
points it polls and does
not poll points close to
them again at subsequent
iterations. Use this option
if patternsearch runs
slowly because it is taking
a long time to compute
the objective function. If
the objective function is
stochastic, it is advised not
to use this option.
All
CacheSize
Size of the cache, in number
of points.
All
CacheTol
Positive scalar specifying
how close the current mesh
point must be to a point
in the cache in order for
patternsearch to avoid
polling it. Available if
'Cache' option is set to
'on'.
All
CompletePoll
Complete poll around
current iterate. Evaluate
all the points in a poll step.
All
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Options Reference
Option Availability Table for All Algorithms (Continued)
9-28
Algorithm
Availability
Option
Description
CompleteSearch
Complete search around
current iterate when the
search method is a poll
method. Evaluate all the
points in a search step.
All
Display
Level of display to
Command Window.
All
InitialMeshSize
Initial mesh size used in
pattern search algorithms.
All
InitialPenalty
Initial value of the penalty
parameter.
All
MaxFunEvals
Maximum number
of objective function
evaluations.
All
MaxIter
Maximum number of
iterations.
All
MaxMeshSize
Maximum mesh size used
in a poll/search step.
GPS and GSS
MeshAccelerator
Accelerate mesh size
contraction.
GPS and GSS
MeshContraction
Mesh contraction factor,
used when iteration is
unsuccessful.
GPS and GSS
MeshExpansion
Mesh expansion factor,
expands mesh when
iteration is successful.
GPS and GSS
MeshRotate
Rotate the pattern before
declaring a point to be
optimum.
GPS Np1 and GSS
Np1
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Pattern Search Options
Option Availability Table for All Algorithms (Continued)
Algorithm
Availability
Option
Description
OutputFcns
User-specified function that
a pattern search calls at
each iteration.
All
PenaltyFactor
Penalty update parameter.
All
PlotFcns
Specifies function to plot at
runtime.
All
PlotInterval
Specifies that plot functions
will be called at every
interval.
All
PollingOrder
Order in which search
directions are polled.
GPS and GSS
PollMethod
Polling strategy used in
pattern search.
All
ScaleMesh
Automatic scaling of
variables.
All
SearchMethod
Specifies search method
used in pattern search.
All
TimeLimit
Total time (in seconds)
allowed for optimization.
Also includes any specified
pause time for pattern
search algorithms.
All
TolBind
Binding tolerance used
to determine if linear
constraint is active.
All
TolCon
Tolerance on nonlinear
constraints.
All
TolFun
Tolerance on function
value.
All
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Options Reference
Option Availability Table for All Algorithms (Continued)
9-30
Option
Description
Algorithm
Availability
TolMesh
Tolerance on mesh size.
All
TolX
Tolerance on independent
variable.
All
UseParallel
When true, compute
objective functions of a
poll or search in parallel.
Disable by setting to false.
All
Vectorized
Specifies whether objective
and constraint functions
are vectorized.
All
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Genetic Algorithm Options
Genetic Algorithm Options
In this section...
“Optimization App vs. Command Line” on page 9-31
“Plot Options” on page 9-32
“Population Options” on page 9-36
“Fitness Scaling Options” on page 9-39
“Selection Options” on page 9-41
“Reproduction Options” on page 9-43
“Mutation Options” on page 9-43
“Crossover Options” on page 9-46
“Migration Options” on page 9-50
“Constraint Parameters” on page 9-51
“Multiobjective Options” on page 9-51
“Hybrid Function Options” on page 9-52
“Stopping Criteria Options” on page 9-53
“Output Function Options” on page 9-54
“Display to Command Window Options” on page 9-55
“Vectorize and Parallel Options (User Function Evaluation)” on page 9-56
Optimization App vs. Command Line
There are two ways to specify options for the genetic algorithm, depending
on whether you are using the Optimization app or calling the functions ga or
at the command line:
• If you are using the Optimization app (optimtool), select an option from a
drop-down list or enter the value of the option in a text field.
• If you are calling ga or gamultiobj from the command line, create an
options structure using the function gaoptimset, as follows:
options = gaoptimset('Param1', value1, 'Param2', value2, ...);
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Options Reference
See “Setting Options at the Command Line” on page 5-67 for examples.
In this section, each option is listed in two ways:
• By its label, as it appears in the Optimization app
• By its field name in the options structure
For example:
• Population type is the label of the option in the Optimization app.
• PopulationType is the corresponding field of the options structure.
Plot Options
Plot options enable you to plot data from the genetic algorithm while it is
running. You can stop the algorithm at any time by clicking the Stop button
on the plot window.
Plot interval (PlotInterval) specifies the number of generations between
consecutive calls to the plot function.
You can select any of the following plot functions in the Plot functions pane:
• Best fitness (@gaplotbestf) plots the best function value versus
generation.
• Expectation (@gaplotexpectation) plots the expected number of children
versus the raw scores at each generation.
• Score diversity (@gaplotscorediversity) plots a histogram of the scores
at each generation.
• Stopping (@gaplotstopping) plots stopping criteria levels.
• Best individual (@gaplotbestindiv) plots the vector entries of the
individual with the best fitness function value in each generation.
• Genealogy (@gaplotgenealogy) plots the genealogy of individuals. Lines
from one generation to the next are color-coded as follows:
-
9-32
Red lines indicate mutation children.
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Genetic Algorithm Options
-
Blue lines indicate crossover children.
Black lines indicate elite individuals.
• Scores (@gaplotscores) plots the scores of the individuals at each
generation.
• Max constraint (@gaplotmaxconstr) plots the maximum nonlinear
constraint violation at each generation.
• Distance (@gaplotdistance) plots the average distance between
individuals at each generation.
• Range (@gaplotrange) plots the minimum, maximum, and mean fitness
function values in each generation.
• Selection (@gaplotselection) plots a histogram of the parents.
• Custom function enables you to use plot functions of your own. To specify
the plot function if you are using the Optimization app,
-
Select Custom function.
Enter @myfun in the text box, where myfun is the name of your function.
See “Structure of the Plot Functions” on page 9-34.
gamultiobj allows Distance, Genealogy , Score diversity, Selection,
Stopping, and Custom function, as well as the following additional choices:
• Pareto front (@gaplotpareto) plots the Pareto front for the first two
objective functions.
• Average Pareto distance (@gaplotparetodistance) plots a bar chart of
the distance of each individual from its neighbors.
• Rank histogram (@gaplotrankhist) plots a histogram of the ranks of the
individuals. Individuals of rank 1 are on the Pareto frontier. Individuals of
rank 2 are lower than at least one rank 1 individual, but are not lower than
any individuals from other ranks, etc.
• Average Pareto spread (@gaplotspread) plots the average spread as a
function of iteration number.
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Options Reference
To display a plot when calling ga from the command line, set the PlotFcns
field of options to be a function handle to the plot function. For example, to
display the best fitness plot, set options as follows
options = gaoptimset('PlotFcns', @gaplotbestf);
To display multiple plots, use the syntax
options =gaoptimset('PlotFcns', {@plotfun1, @plotfun2, ...});
where @plotfun1, @plotfun2, and so on are function handles to the plot
functions.
If you specify more than one plot function, all plots appear as subplots in
the same window. Right-click any subplot to obtain a larger version in a
separate figure window.
Structure of the Plot Functions
The first line of a plot function has the form
function state = plotfun(options,state,flag)
The input arguments to the function are
• options — Structure containing all the current options settings.
• state — Structure containing information about the current generation.
“The State Structure” on page 9-35 describes the fields of state.
• flag — String that tells what stage the algorithm is currently in.
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
The output argument state is a state structure as well. Pass the input
argument, modified if you like. To stop the iterations, set state.StopFlag
to a nonempty string.
9-34
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Genetic Algorithm Options
The State Structure
ga. The state structure for ga, which is an input argument to plot, mutation,
and output functions, contains the following fields:
• Population — Population in the current generation
• Score — Scores of the current population
• Generation — Current generation number
• StartTime — Time when genetic algorithm started
• StopFlag — String containing the reason for stopping
• Selection — Indices of individuals selected for elite, crossover and
mutation
• Expectation — Expectation for selection of individuals
• Best — Vector containing the best score in each generation
• LastImprovement — Generation at which the last improvement in fitness
value occurred
• LastImprovementTime — Time at which last improvement occurred
• NonlinIneq — Nonlinear inequality constraints, displayed only when a
nonlinear constraint function is specified
• NonlinEq — Nonlinear equality constraints, displayed only when a
nonlinear constraint function is specified
gamultiobj. The state structure for gamultiobj, which is an input argument
to plot, mutation, and output functions, contains the following fields:
• Population — Population in the current generation
• Score — Scores of the current population, a Population-by-nObjectives
matrix, where nObjectives is the number of objectives
• Generation — Current generation number
• StartTime — Time when genetic algorithm started
• StopFlag — String containing the reason for stopping
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Options Reference
• Selection — Indices of individuals selected for elite, crossover and
mutation
• Rank — Vector of the ranks of members in the population
• Distance — Vector of distances of each member of the population to the
nearest neighboring member
• AverageDistance — The average of Distance
• Spread — Vector whose entries are the spread in each generation
Population Options
Population options let you specify the parameters of the population that the
genetic algorithm uses.
Population type (PopulationType) specifies the type of input to the fitness
function. Types and their restrictions:
• Double vector ('doubleVector') — Use this option if the individuals
in the population have type double. Use this option for mixed integer
programming. This is the default.
• Bit string ('bitstring') — Use this option if the individuals in the
population have components that are 0 or 1.
Caution The individuals in a Bit string population are vectors of type
double, not strings.
For Creation function (CreationFcn) and Mutation function
(MutationFcn), use Uniform (@gacreationuniform and @mutationuniform)
or Custom. For Crossover function (CrossoverFcn), use Scattered
(@crossoverscattered), Single point (@crossoversinglepoint), Two
point (@crossovertwopoint), or Custom. You cannot use a Hybrid
function, and ga ignores all constraints, including bounds, linear
constraints, and nonlinear constraints.
• Custom — For Crossover function and Mutation function, use
Custom. For Creation function, either use Custom, or provide an
Initial population. You cannot use a Hybrid function, and ga ignores
9-36
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Genetic Algorithm Options
all constraints, including bounds, linear constraints, and nonlinear
constraints.
Population size (PopulationSize) specifies how many individuals there
are in each generation. With a large population size, the genetic algorithm
searches the solution space more thoroughly, thereby reducing the chance
that the algorithm will return a local minimum that is not a global minimum.
However, a large population size also causes the algorithm to run more slowly.
If you set Population size to a vector, the genetic algorithm creates multiple
subpopulations, the number of which is the length of the vector. The size of
each subpopulation is the corresponding entry of the vector. See “Migration
Options” on page 9-50.
Creation function (CreationFcn) specifies the function that creates the
initial population for ga. Do not use with integer problems. You can choose
from the following functions:
• Uniform (@gacreationuniform) creates a random initial population with
a uniform distribution. This is the default when there are no linear
constraints, or when there are integer constraints. The uniform distribution
is in the initial population range (PopInitRange). The default values for
PopInitRange are [-10;10] for every component, or [-9999;10001] when
there are integer constraints. These bounds are shifted and scaled to match
any existing bounds lb and ub.
Caution Do not use @gacreationuniform when you have linear
constraints. Otherwise, your population will not necessarily satisfy the
linear constraints.
• Feasible population (@gacreationlinearfeasible), the default when
there are linear constraints and no integer constraints, creates a random
initial population that satisfies all bounds and linear constraints. If there
are linear constraints, Feasible population creates many individuals
on the boundaries of the constraint region, and creates a well-dispersed
population. Feasible population ignores Initial range (PopInitRange).
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Options Reference
• Custom enables you to write your own creation function, which must
generate data of the type that you specify in Population type. To specify
the creation function if you are using the Optimization app,
-
Set Creation function to Custom.
Set Function name to @myfun, where myfun is the name of your
function.
If you are using ga, set
options = gaoptimset('CreationFcn', @myfun);
Your creation function must have the following calling syntax.
function Population = myfun(GenomeLength, FitnessFcn, options)
The input arguments to the function are
-
Genomelength — Number of independent variables for the fitness
-
FitnessFcn — Fitness function
function
options — Options structure
The function returns Population, the initial population for the genetic
algorithm.
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
Caution When you have bounds or linear constraints, ensure that
your creation function creates individuals that satisfy these constraints.
Otherwise, your population will not necessarily satisfy the constraints.
Initial population (InitialPopulation) specifies an initial population for
the genetic algorithm. The default value is [], in which case ga uses the
default Creation function to create an initial population. If you enter a
nonempty array in the Initial population field, the array must have no more
than Population size rows, and exactly Number of variables columns. In
this case, the genetic algorithm calls a Creation function to generate the
remaining individuals, if required.
9-38
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Genetic Algorithm Options
Initial scores (InitialScores) specifies initial scores for the initial
population. The initial scores can also be partial. Do not use with integer
problems.
Initial range (PopInitRange) specifies the range of the vectors in the initial
population that is generated by the gacreationuniform creation function.
You can set Initial range to be a matrix with two rows and Number of
variables columns, each column of which has the form [lb;ub], where lb is
the lower bound and ub is the upper bound for the entries in that coordinate.
If you specify Initial range to be a 2-by-1 vector, each entry is expanded to a
constant row of length Number of variables. If you do not specify an Initial
range, the default is [-10;10] ([-1e4+1;1e4+1] for integer-constrained
problems), modified to match any existing bounds.
See “Setting the Initial Range” on page 5-76 for an example.
Fitness Scaling Options
Fitness scaling converts the raw fitness scores that are returned by the fitness
function to values in a range that is suitable for the selection function. You
can specify options for fitness scaling in the Fitness scaling pane.
Scaling function (FitnessScalingFcn) specifies the function that performs
the scaling. The options are
• Rank (@fitscalingrank) — The default fitness scaling function, Rank,
scales the raw scores based on the rank of each individual instead of its
score. The rank of an individual is its position in the sorted scores. An
individual with rank r has scaled score proportional to 1 / r . So the scaled
score of the most fit individual is proportional to 1, the scaled score of the
next most fit is proportional to 1 / 2 , and so on. Rank fitness scaling
removes the effect of the spread of the raw scores. The square root makes
poorly ranked individuals more nearly equal in score, compared to rank
scoring. For more information, see “Fitness Scaling” on page 5-87.
• Proportional (@fitscalingprop) — Proportional scaling makes the scaled
value of an individual proportional to its raw fitness score.
• Top (@fitscalingtop) — Top scaling scales the top individuals equally.
Selecting Top displays an additional field, Quantity, which specifies the
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Options Reference
number of individuals that are assigned positive scaled values. Quantity
can be an integer between 1 and the population size or a fraction between 0
and 1 specifying a fraction of the population size. The default value is 0.4.
Each of the individuals that produce offspring is assigned an equal scaled
value, while the rest are assigned the value 0. The scaled values have the
form [01/n 1/n 0 0 1/n 0 0 1/n ...].
To change the default value for Quantity at the command line, use the
following syntax
options = gaoptimset('FitnessScalingFcn', {@fitscalingtop,
quantity})
where quantity is the value of Quantity.
• Shift linear (@fitscalingshiftlinear) — Shift linear scaling scales
the raw scores so that the expectation of the fittest individual is equal to
a constant multiplied by the average score. You specify the constant in
the Max survival rate field, which is displayed when you select Shift
linear. The default value is 2.
To change the default value of Max survival rate at the command line,
use the following syntax
options = gaoptimset('FitnessScalingFcn',
{@fitscalingshiftlinear, rate})
where rate is the value of Max survival rate.
• Custom enables you to write your own scaling function. To specify the
scaling function using the Optimization app,
-
Set Scaling function to Custom.
Set Function name to @myfun, where myfun is the name of your
function.
If you are using ga at the command line, set
options = gaoptimset('FitnessScalingFcn', @myfun);
Your scaling function must have the following calling syntax:
function expectation = myfun(scores, nParents)
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Genetic Algorithm Options
The input arguments to the function are
-
scores — A vector of scalars, one for each member of the population
nParents — The number of parents needed from this population
The function returns expectation, a column vector of scalars of the
same length as scores, giving the scaled values of each member of the
population. The sum of the entries of expectation must equal nParents.
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
See “Fitness Scaling” on page 5-87 for more information.
Selection Options
Selection options specify how the genetic algorithm chooses parents for the
next generation. You can specify the function the algorithm uses in the
Selection function (SelectionFcn) field in the Selection options pane. Do
not use with integer problems. The options are
• Stochastic uniform (@selectionstochunif) — The default selection
function, Stochastic uniform, lays out a line in which each parent
corresponds to a section of the line of length proportional to its scaled value.
The algorithm moves along the line in steps of equal size. At each step, the
algorithm allocates a parent from the section it lands on. The first step is a
uniform random number less than the step size.
• Remainder (@selectionremainder) — Remainder selection assigns parents
deterministically from the integer part of each individual’s scaled value and
then uses roulette selection on the remaining fractional part. For example,
if the scaled value of an individual is 2.3, that individual is listed twice as
a parent because the integer part is 2. After parents have been assigned
according to the integer parts of the scaled values, the rest of the parents
are chosen stochastically. The probability that a parent is chosen in this
step is proportional to the fractional part of its scaled value.
• Uniform (@selectionuniform) — Uniform selection chooses parents using
the expectations and number of parents. Uniform selection is useful for
debugging and testing, but is not a very effective search strategy.
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Options Reference
• Roulette (@selectionroulette) — Roulette selection chooses parents
by simulating a roulette wheel, in which the area of the section of the
wheel corresponding to an individual is proportional to the individual’s
expectation. The algorithm uses a random number to select one of the
sections with a probability equal to its area.
• Tournament (@selectiontournament) — Tournament selection chooses
each parent by choosing Tournament size players at random and then
choosing the best individual out of that set to be a parent. Tournament
size must be at least 2. The default value of Tournament size is 4.
To change the default value of Tournament size at the command line,
use the syntax
options = gaoptimset('SelectionFcn',...
{@selectiontournament,size})
where size is the value of Tournament size.
• Custom enables you to write your own selection function. To specify the
selection function using the Optimization app,
-
Set Selection function to Custom.
Set Function name to @myfun, where myfun is the name of your
function.
If you are using ga at the command line, set
options = gaoptimset('SelectionFcn', @myfun);
Your selection function must have the following calling syntax:
function parents = myfun(expectation, nParents, options)
The input arguments to the function are
-
expectation — Expected number of children for each member of the
population
nParents— Number of parents to select
options — Genetic algorithm options structure
The function returns parents, a row vector of length nParents containing
the indices of the parents that you select.
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Genetic Algorithm Options
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
See “Selection” on page 5-29 for more information.
Reproduction Options
Reproduction options specify how the genetic algorithm creates children for
the next generation.
Elite count (EliteCount) specifies the number of individuals that are
guaranteed to survive to the next generation. Set Elite count to be a positive
integer less than or equal to the population size. The default value is 2 for
noninteger problems, and 0.05*min(max(10*nvars,40),100) for integer
problems.
Crossover fraction (CrossoverFraction) specifies the fraction of the next
generation, other than elite children, that are produced by crossover. Set
Crossover fraction to be a fraction between 0 and 1, either by entering the
fraction in the text box or moving the slider. The default value is 0.8.
See “Setting the Crossover Fraction” on page 5-93 for an example.
Mutation Options
Mutation options specify how the genetic algorithm makes small random
changes in the individuals in the population to create mutation children.
Mutation provides genetic diversity and enable the genetic algorithm to
search a broader space. You can specify the mutation function in the
Mutation function (MutationFcn) field in the Mutation options pane. Do
not use with integer problems. You can choose from the following functions:
• Gaussian (mutationgaussian) — The default mutation function for
unconstrained problems, Gaussian, adds a random number taken from a
Gaussian distribution with mean 0 to each entry of the parent vector. The
standard deviation of this distribution is determined by the parameters
Scale and Shrink, which are displayed when you select Gaussian, and by
the Initial range setting in the Population options.
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Options Reference
-
The Scale parameter determines the standard deviation at the first
generation. If you set Initial range to be a 2-by-1 vector v, the initial
standard deviation is the same at all coordinates of the parent vector,
and is given by Scale*(v(2)-v(1)).
If you set Initial range to be a vector v with two rows and Number of
variables columns, the initial standard deviation at coordinate i of the
parent vector is given by Scale*(v(i,2) - v(i,1)).
-
The Shrink parameter controls how the standard deviation shrinks
as generations go by. If you set Initial range to be a 2-by-1 vector,
the standard deviation at the kth generation, σk, is the same at all
coordinates of the parent vector, and is given by the recursive formula
k
⎛
⎞
 k =  k−1 ⎜ 1 − Shrink
⎟.
Generations ⎠
⎝
If you set Initial range to be a vector with two rows and Number of
variables columns, the standard deviation at coordinate i of the parent
vector at the kth generation, σi,k, is given by the recursive formula
k
⎛
⎞
 i,k =  i,k−1 ⎜ 1 − Shrink
⎟.
Generations
⎝
⎠
If you set Shrink to 1, the algorithm shrinks the standard deviation
in each coordinate linearly until it reaches 0 at the last generation is
reached. A negative value of Shrink causes the standard deviation to
grow.
The default value of both Scale and Shrink is 1. To change the default
values at the command line, use the syntax
options = gaoptimset('MutationFcn', ...
{@mutationgaussian, scale, shrink})
where scale and shrink are the values of Scale and Shrink, respectively.
Caution Do not use mutationgaussian when you have bounds or linear
constraints. Otherwise, your population will not necessarily satisfy the
constraints.
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Genetic Algorithm Options
• Uniform (mutationuniform) — Uniform mutation is a two-step process.
First, the algorithm selects a fraction of the vector entries of an individual
for mutation, where each entry has a probability Rate of being mutated.
The default value of Rate is 0.01. In the second step, the algorithm
replaces each selected entry by a random number selected uniformly from
the range for that entry.
To change the default value of Rate at the command line, use the syntax
options = gaoptimset('MutationFcn', {@mutationuniform, rate})
where rate is the value of Rate.
Caution Do not use mutationuniform when you have bounds or linear
constraints. Otherwise, your population will not necessarily satisfy the
constraints.
• Adaptive Feasible (mutationadaptfeasible), the default mutation
function when there are constraints, randomly generates directions that
are adaptive with respect to the last successful or unsuccessful generation.
The mutation chooses a direction and step length that satisfies bounds
and linear constraints.
• Custom enables you to write your own mutation function. To specify the
mutation function using the Optimization app,
-
Set Mutation function to Custom.
Set Function name to @myfun, where myfun is the name of your
function.
If you are using ga, set
options = gaoptimset('MutationFcn', @myfun);
Your mutation function must have this calling syntax:
function mutationChildren = myfun(parents, options, nvars,
FitnessFcn, state, thisScore, thisPopulation)
The arguments to the function are
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Options Reference
-
parents — Row vector of parents chosen by the selection function
options — Options structure
nvars — Number of variables
FitnessFcn — Fitness function
state — Structure containing information about the current generation.
“The State Structure” on page 9-35 describes the fields of state.
thisScore — Vector of scores of the current population
thisPopulation — Matrix of individuals in the current population
The function returns mutationChildren—the mutated offspring—as a
matrix whose rows correspond to the children. The number of columns of
the matrix is Number of variables.
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
Caution When you have bounds or linear constraints, ensure that your
mutation function creates individuals that satisfy these constraints.
Otherwise, your population will not necessarily satisfy the constraints.
Crossover Options
Crossover options specify how the genetic algorithm combines two individuals,
or parents, to form a crossover child for the next generation.
Crossover function (CrossoverFcn) specifies the function that performs
the crossover. Do not use with integer problems. You can choose from the
following functions:
• Scattered (@crossoverscattered), the default crossover function for
problems without linear constraints, creates a random binary vector and
selects the genes where the vector is a 1 from the first parent, and the
genes where the vector is a 0 from the second parent, and combines the
genes to form the child. For example, if p1 and p2 are the parents
p1 = [a b c d e f g h]
p2 = [1 2 3 4 5 6 7 8]
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Genetic Algorithm Options
and the binary vector is [1 1 0 0 1 0 0 0], the function returns the following
child:
child1 = [a b 3 4 e 6 7 8]
Caution Do not use @crossoverscattered when you have linear
constraints. Otherwise, your population will not necessarily satisfy the
constraints.
• Single point (@crossoversinglepoint) chooses a random integer n
between 1 and Number of variables and then
-
Selects vector entries numbered less than or equal to n from the first
parent.
-
Selects vector entries numbered greater than n from the second parent.
Concatenates these entries to form a child vector.
For example, if p1 and p2 are the parents
p1 = [a b c d e f g h]
p2 = [1 2 3 4 5 6 7 8]
and the crossover point is 3, the function returns the following child.
child = [a b c 4 5 6 7 8]
Caution Do not use @crossoversinglepoint when you have linear
constraints. Otherwise, your population will not necessarily satisfy the
constraints.
• Two point (@crossovertwopoint) selects two random integers m and n
between 1 and Number of variables. The function selects
-
Vector entries numbered less than or equal to m from the first parent
Vector entries numbered from m+1 to n, inclusive, from the second parent
Vector entries numbered greater than n from the first parent.
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Options Reference
The algorithm then concatenates these genes to form a single gene. For
example, if p1 and p2 are the parents
p1 = [a b c d e f g h]
p2 = [1 2 3 4 5 6 7 8]
and the crossover points are 3 and 6, the function returns the following
child.
child = [a b c 4 5 6 g h]
Caution Do not use @crossovertwopoint when you have linear
constraints. Otherwise, your population will not necessarily satisfy the
constraints.
• Intermediate (@crossoverintermediate), the default crossover function
when there are linear constraints, creates children by taking a weighted
average of the parents. You can specify the weights by a single parameter,
Ratio, which can be a scalar or a row vector of length Number of
variables. The default is a vector of all 1’s. The function creates the child
from parent1 and parent2 using the following formula.
child = parent1 + rand * Ratio * ( parent2 - parent1)
If all the entries of Ratio lie in the range [0, 1], the children produced are
within the hypercube defined by placing the parents at opposite vertices. If
Ratio is not in that range, the children might lie outside the hypercube. If
Ratio is a scalar, then all the children lie on the line between the parents.
To change the default value of Ratio at the command line, use the syntax
options = gaoptimset('CrossoverFcn', ...
{@crossoverintermediate, ratio});
where ratio is the value of Ratio.
• Heuristic (@crossoverheuristic) returns a child that lies on the line
containing the two parents, a small distance away from the parent with the
better fitness value in the direction away from the parent with the worse
fitness value. You can specify how far the child is from the better parent
by the parameter Ratio, which appears when you select Heuristic. The
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Genetic Algorithm Options
default value of Ratio is 1.2. If parent1 and parent2 are the parents, and
parent1 has the better fitness value, the function returns the child
child = parent2 + R * (parent1 - parent2);
To change the default value of Ratio at the command line, use the syntax
options=gaoptimset('CrossoverFcn',...
{@crossoverheuristic,ratio});
where ratio is the value of Ratio.
• Arithmetic (@crossoverarithmetic) creates children that are the
weighted arithmetic mean of two parents. Children are always feasible
with respect to linear constraints and bounds.
• Custom enables you to write your own crossover function. To specify the
crossover function using the Optimization app,
-
Set Crossover function to Custom.
Set Function name to @myfun, where myfun is the name of your
function.
If you are using ga, set
options = gaoptimset('CrossoverFcn',@myfun);
Your crossover function must have the following calling syntax.
xoverKids = myfun(parents, options, nvars, FitnessFcn, ...
unused,thisPopulation)
The arguments to the function are
-
parents — Row vector of parents chosen by the selection function
options — options structure
nvars — Number of variables
FitnessFcn — Fitness function
unused — Placeholder not used
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Options Reference
-
thisPopulation — Matrix representing the current population. The
number of rows of the matrix is Population size and the number of
columns is Number of variables.
The function returns xoverKids—the crossover offspring—as a matrix
whose rows correspond to the children. The number of columns of the
matrix is Number of variables.
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
Caution When you have bounds or linear constraints, ensure that your
crossover function creates individuals that satisfy these constraints.
Otherwise, your population will not necessarily satisfy the constraints.
Migration Options
Note Subpopulations refer to a form of parallel processing for the genetic
algorithm. ga currently does not support this form. In subpopulations, each
worker hosts a number of individuals. These individuals are a subpopulation.
The worker evolves the subpopulation independently of other workers, except
when migration causes some individuals to travel between workers.
Because ga does not currently support this form of parallel processing,
there is no benefit to setting PopulationSize to a vector, or to setting the
MigrationDirection, MigrationInterval, or MigrationFraction options.
Migration options specify how individuals move between subpopulations.
Migration occurs if you set Population size to be a vector of length greater
than 1. When migration occurs, the best individuals from one subpopulation
replace the worst individuals in another subpopulation. Individuals that
migrate from one subpopulation to another are copied. They are not removed
from the source subpopulation.
You can control how migration occurs by the following three fields in the
Migration options pane:
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Genetic Algorithm Options
• Direction (MigrationDirection) — Migration can take place in one or
both directions.
-
If you set Direction to Forward ('forward'), migration takes place
toward the last subpopulation. That is, the nth subpopulation migrates
into the (n+1)th subpopulation.
-
If you set Direction to Both ('both'), the nth subpopulation migrates
into both the (n–1)th and the (n+1)th subpopulation.
Migration wraps at the ends of the subpopulations. That is, the last
subpopulation migrates into the first, and the first may migrate into the
last.
• Interval (MigrationInterval) — Specifies how many generation pass
between migrations. For example, if you set Interval to 20, migration
takes place every 20 generations.
• Fraction (MigrationFraction) — Specifies how many individuals move
between subpopulations. Fraction specifies the fraction of the smaller of
the two subpopulations that moves. For example, if individuals migrate
from a subpopulation of 50 individuals into a subpopulation of 100
individuals and you set Fraction to 0.1, the number of individuals that
migrate is 0.1*50=5.
Constraint Parameters
Constraint parameters refer to the nonlinear constraint solver. For more
information on the algorithm, see “Nonlinear Constraint Solver Algorithm”
on page 5-52.
• Initial penalty (InitialPenalty) — Specifies an initial value of the
penalty parameter that is used by the nonlinear constraint algorithm.
Initial penalty must be greater than or equal to 1, and has a default of 10.
• Penalty factor (PenaltyFactor) — Increases the penalty parameter when
the problem is not solved to required accuracy and constraints are not
satisfied. Penalty factor must be greater than 1, and has a default of 100.
Multiobjective Options
Multiobjective options define parameters characteristic of the multiobjective
genetic algorithm. You can specify the following parameters:
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Options Reference
• DistanceMeasureFcn — Defines a handle to the function that computes
distance measure of individuals, computed in decision variable or design
space (genotype) or in function space (phenotype). For example, the default
distance measure function is distancecrowding in function space, or
{@distancecrowding,'phenotype'}.
• ParetoFraction — Sets the fraction of individuals to keep on the first
Pareto front while the solver selects individuals from higher fronts. This
option is a scalar between 0 and 1.
Hybrid Function Options
A hybrid function is another minimization function that runs after the genetic
algorithm terminates. You can specify a hybrid function in Hybrid function
(HybridFcn) options. Do not use with integer problems. The choices are
• [] — No hybrid function.
• fminsearch (@fminsearch) — Uses the MATLAB function fminsearch to
perform unconstrained minimization.
• patternsearch (@patternsearch) — Uses a pattern search to perform
constrained or unconstrained minimization.
• fminunc (@fminunc) — Uses the Optimization Toolbox function fminunc to
perform unconstrained minimization.
• fmincon (@fmincon) — Uses the Optimization Toolbox function fmincon to
perform constrained minimization.
You can set separate options for the hybrid function. Use optimset for
fminsearch, psoptimset for patternsearch, or optimoptions for fmincon or
fminunc. For example:
hybridopts = optimoptions('fminunc','Display','iter','Algorithm','quasi-new
Include the hybrid options in the Genetic Algorithm options structure as
follows:
options = gaoptimset(options,'HybridFcn',{@fminunc,hybridopts});
hybridopts must exist before you set options.
See “Include a Hybrid Function” on page 5-107 for an example.
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Genetic Algorithm Options
Stopping Criteria Options
Stopping criteria determine what causes the algorithm to terminate. You can
specify the following options:
• Generations (Generations) — Specifies the maximum number
of iterations for the genetic algorithm to perform. The default is
100*numberOfVariables.
• Time limit (TimeLimit) — Specifies the maximum time in seconds the
genetic algorithm runs before stopping, as measured by cputime.
• Fitness limit (FitnessLimit) — The algorithm stops if the best fitness
value is less than or equal to the value of Fitness limit.
• Stall generations (StallGenLimit) — The algorithm stops if the average
relative change in the best fitness function value over Stall generations
is less than or equal to Function tolerance. (If the StallTest option
is 'geometricWeighted', then the test is for a geometric weighted
average relative change.) For a problem with nonlinear constraints, Stall
generations applies to the subproblem (see “Nonlinear Constraint Solver
Algorithm” on page 5-52).
For gamultiobj, if the weighted average relative change in the spread
of the Pareto solutions over Stall generations is less than Function
tolerance, and the spread is smaller than the average spread over the last
Stall generations, then the algorithm stops. The spread is a measure of
the movement of the Pareto front.
• Stall time limit (StallTimeLimit) — The algorithm stops if there is no
improvement in the best fitness value for an interval of time in seconds
specified by Stall time limit, as measured by cputime.
• Function tolerance (TolFun) — The algorithm stops if the average
relative change in the best fitness function value over Stall generations
is less than or equal to Function tolerance. (If the StallTest option is
'geometricWeighted', then the test is for a geometric weighted average
relative change.)
For gamultiobj, if the weighted average relative change in the spread
of the Pareto solutions over Stall generations is less than Function
tolerance, and the spread is smaller than the average spread over the last
Stall generations, then the algorithm stops. The spread is a measure of
the movement of the Pareto front.
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Options Reference
• Constraint tolerance (TolCon) — The Constraint tolerance is not used
as stopping criterion. It is used to determine the feasibility with respect to
nonlinear constraints. Also, max(sqrt(eps),sqrt(TolCon)) determines
feasibility with respect to linear constraints.
See “Set Maximum Number of Generations” on page 5-111 for an example.
Output Function Options
Output functions are functions that the genetic algorithm calls at each
generation. To specify the output function using the Optimization app,
• Select Custom function.
• Enter @myfun in the text box, where myfun is the name of your function.
Write myfun with appropriate syntax.
• To pass extra parameters in the output function, use “Anonymous
Functions”.
• For multiple output functions, enter a cell array of output function handles:
{@myfun1,@myfun2,...}.
At the command line, set
options = gaoptimset('OutputFcns',@myfun);
For multiple output functions, enter a cell array:
options = gaoptimset('OutputFcns',{@myfun1,@myfun2,...});
To see a template that you can use to write your own output functions, enter
edit gaoutputfcntemplate
at the MATLAB command line.
Structure of the Output Function
The output function has the following calling syntax.
[state,options,optchanged] = myfun(options,state,flag)
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Genetic Algorithm Options
The function has the following input arguments:
• options — Options structure
• state — Structure containing information about the current generation.
“The State Structure” on page 9-35 describes the fields of state.
• flag — String indicating the current status of the algorithm as follows:
-
'init' — Initial stage
'iter' — Algorithm running
'interrupt' — Intermediate stage
'done' — Algorithm terminated
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the function.
The output function returns the following arguments to ga:
• state — Structure containing information about the current generation.
“The State Structure” on page 9-35 describes the fields of state. To stop
the iterations, set state.StopFlag to a nonempty string.
• options — Options structure modified by the output function. This
argument is optional.
• optchanged — Flag indicating changes to options
Display to Command Window Options
Level of display ('Display') specifies how much information is displayed
at the command line while the genetic algorithm is running. The available
options are
• Off ('off') — No output is displayed.
• Iterative ('iter') — Information is displayed at each iteration.
• Diagnose ('diagnose') — Information is displayed at each iteration. In
addition, the diagnostic lists some problem information and the options
that have been changed from the defaults.
• Final ('final') — The reason for stopping is displayed.
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Options Reference
Both Iterative and Diagnose display the following information:
• Generation — Generation number
• f-count — Cumulative number of fitness function evaluations
• Best f(x) — Best fitness function value
• Mean f(x) — Mean fitness function value
• Stall generations — Number of generations since the last improvement
of the fitness function
When a nonlinear constraint function has been specified, Iterative and
Diagnose will not display the Mean f(x), but will additionally display:
• Max Constraint — Maximum nonlinear constraint violation
The default value of Level of display is
• Off in the Optimization app
• 'final' in an options structure created using gaoptimset
Vectorize and Parallel Options (User Function
Evaluation)
You can choose to have your fitness and constraint functions evaluated in
serial, parallel, or in a vectorized fashion. These options are available in the
User function evaluation section of the Options pane of the Optimization
app, or by setting the 'Vectorized' and 'UseParallel' options with
gaoptimset.
• When Evaluate fitness and constraint functions ('Vectorized') is in
serial ('off'), ga calls the fitness function on one individual at a time
as it loops through the population. (At the command line, this assumes
'UseParallel' is at its default value of false.)
• When Evaluate fitness and constraint functions ('Vectorized') is
vectorized ('on'), ga calls the fitness function on the entire population at
once, i.e., in a single call to the fitness function.
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Genetic Algorithm Options
If there are nonlinear constraints, the fitness function and the nonlinear
constraints all need to be vectorized in order for the algorithm to compute
in a vectorized manner.
See “Vectorize the Fitness Function” on page 5-114 for an example.
• When Evaluate fitness and constraint functions (UseParallel) is in
parallel (true), ga calls the fitness function in parallel, using the parallel
environment you established (see “How to Use Parallel Processing” on page
8-12). At the command line, set UseParallel to false to compute serially.
Note You cannot simultaneously use vectorized and parallel computations.
If you set 'UseParallel' to true and 'Vectorized' to 'on', ga evaluates
your fitness and constraint functions in a vectorized manner, not in parallel.
How Fitness and Constraint Functions Are Evaluated
Vectorized = 'Off'
Vectorized = 'On'
UseParallel = false
Serial
Vectorized
UseParallel = true
Parallel
Vectorized
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Options Reference
Simulated Annealing Options
In this section...
“saoptimset At The Command Line” on page 9-58
“Plot Options” on page 9-58
“Temperature Options” on page 9-60
“Algorithm Settings” on page 9-61
“Hybrid Function Options” on page 9-63
“Stopping Criteria Options” on page 9-64
“Output Function Options” on page 9-64
“Display Options” on page 9-66
saoptimset At The Command Line
Specify options by creating an options structure using the saoptimset
function as follows:
options = saoptimset('Param1',value1,'Param2',value2, ...);
See “Set Options for simulannealbnd at the Command Line” on page 6-16
for examples.
Each option in this section is listed by its field name in the options structure.
For example, InitialTemperature refers to the corresponding field of the
options structure.
Plot Options
Plot options enable you to plot data from the simulated annealing solver while
it is running.
PlotInterval specifies the number of iterations between consecutive calls to
the plot function.
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Simulated Annealing Options
To display a plot when calling simulannealbnd from the command line, set
the PlotFcns field of options to be a function handle to the plot function. You
can specify any of the following plots:
• @saplotbestf plots the best objective function value.
• @saplotbestx plots the current best point.
• @saplotf plots the current function value.
• @saplotx plots the current point.
• @saplotstopping plots stopping criteria levels.
• @saplottemperature plots the temperature at each iteration.
• @myfun plots a custom plot function, where myfun is the name of your
function. See “Structure of the Plot Functions” on page 9-11 for a
description of the syntax.
For example, to display the best objective plot, set options as follows
options = saoptimset('PlotFcns',@saplotbestf);
To display multiple plots, use the cell array syntax
options = saoptimset('PlotFcns',{@plotfun1,@plotfun2, ...});
where @plotfun1, @plotfun2, and so on are function handles to the plot
functions.
If you specify more than one plot function, all plots appear as subplots in
the same window. Right-click any subplot to obtain a larger version in a
separate figure window.
Structure of the Plot Functions
The first line of a plot function has the form
function stop = plotfun(options,optimvalues,flag)
The input arguments to the function are
• options — Options structure created using saoptimset.
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Options Reference
• optimvalues — Structure containing information about the current state
of the solver. The structure contains the following fields:
-
x — Current point
fval — Objective function value at x
bestx — Best point found so far
bestfval — Objective function value at best point
temperature — Current temperature
iteration — Current iteration
funccount — Number of function evaluations
t0 — Start time for algorithm
k — Annealing parameter
• flag — Current state in which the plot function is called. The possible
values for flag are
-
'init' — Initialization state
'iter' — Iteration state
'done' — Final state
The output argument stop provides a way to stop the algorithm at the current
iteration. stop can have the following values:
• false — The algorithm continues to the next iteration.
• true — The algorithm terminates at the current iteration.
Temperature Options
Temperature options specify how the temperature will be lowered at each
iteration over the course of the algorithm.
• InitialTemperature — Initial temperature at the start of the algorithm.
The default is 100.
• TemperatureFcn — Function used to update the temperature schedule. Let
k denote the annealing parameter. (The annealing parameter is the same
as the iteration number until reannealing.) The options are:
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Simulated Annealing Options
-
@temperatureexp — The temperature is equal to InitialTemperature *
0.95^k. This is the default.
-
@temperaturefast — The temperature is equal to InitialTemperature /
-
k.
@temperatureboltz — The temperature is equal to InitialTemperature /
ln(k).
@myfun — Uses a custom function, myfun, to update temperature. The
syntax is:
temperature = myfun(optimValues,options)
where optimValues is a structure described in “Structure of the Plot
Functions” on page 9-59. options is either the structure created with
saoptimset, or the structure of default options, if you did not create an
options structure. Both the annealing parameter optimValues.k and
the temperature optimValues.temperature are vectors with length
equal to the number of elements of the current point x. For example,
the function temperaturefast is:
temperature = options.InitialTemperature./optimValues.k;
Algorithm Settings
Algorithm settings define algorithmic specific parameters used in generating
new points at each iteration.
Parameters that can be specified for simulannealbnd are:
• DataType — Type of data to use in the objective function. Choices:
-
'double' (default) — A vector of type double.
'custom' — Any other data type. You must provide a 'custom'
annealing function. You cannot use a hybrid function.
• AnnealingFcn — Function used to generate new points for the next
iteration. The choices are:
-
@annealingfast — The step has length temperature, with direction
uniformly at random. This is the default.
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Options Reference
-
@annealingboltz — The step has length square root of temperature,
with direction uniformly at random.
-
@myfun — Uses a custom annealing algorithm, myfun. The syntax is:
newx = myfun(optimValues,problem)
where optimValues is a structure described in “Structure of the Output
Function” on page 9-65, and problem is a structure containing the
following information:
• objective:
• x0:
function handle to the objective function
the start point
• nvar:
number of decision variables
• lb:
lower bound on decision variables
• ub:
upper bound on decision variables
For example, the current position is optimValues.x, and the current
objective function value is problem.objective(optimValues.x).
• ReannealInterval — Number of points accepted before reannealing. The
default value is 100.
• AcceptanceFcn — Function used to determine whether a new point is
accepted or not. The choices are:
-
@acceptancesa — Simulated annealing acceptance function, the default.
If the new objective function value is less than the old, the new point is
always accepted. Otherwise, the new point is accepted at random with a
probability depending on the difference in objective function values and
on the current temperature. The acceptance probability is
1
,
Δ
⎛
⎞
1 + exp ⎜
⎟
⎝ max(T ) ⎠
where Δ = new objective – old objective, and T is the current temperature.
Since both Δ and T are positive, the probability of acceptance is between
0 and 1/2. Smaller temperature leads to smaller acceptance probability.
Also, larger Δ leads to smaller acceptance probability.
9-62
@myfun — A custom acceptance function, myfun. The syntax is:
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Simulated Annealing Options
acceptpoint = myfun(optimValues,newx,newfval);
where optimValues is a structure described in “Structure of the Output
Function” on page 9-65, newx is the point being evaluated for acceptance,
and newfval is the objective function at newx. acceptpoint is a Boolean,
with value true to accept newx, and false to reject newx.
Hybrid Function Options
A hybrid function is another minimization function that runs during or at the
end of iterations of the solver. HybridInterval specifies the interval (if not
never or end) at which the hybrid function is called. You can specify a hybrid
function using the HybridFcn option. The choices are:
• [] — No hybrid function.
• @fminsearch — Uses the MATLAB function fminsearch to perform
unconstrained minimization.
• @patternsearch — Uses patternsearch to perform constrained or
unconstrained minimization.
• @fminunc — Uses the Optimization Toolbox function fminunc to perform
unconstrained minimization.
• @fmincon — Uses the Optimization Toolbox function fmincon to perform
constrained minimization.
You can set separate options for the hybrid function. Use optimset for
fminsearch, psoptimset for patternsearch, or optimoptions for fmincon or
fminunc. For example:
hybridopts = optimoptions('fminunc','Display','iter','Algorithm','quasi-new
Include the hybrid options in the simulannealbnd options structure as
follows:
options = saoptimset(options,'HybridFcn',{@fminunc,hybridopts});
hybridopts must exist before you set options.
See “Include a Hybrid Function” on page 5-107 for an example.
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Options Reference
Stopping Criteria Options
Stopping criteria determine what causes the algorithm to terminate. You can
specify the following options:
• TolFun — The algorithm runs until the average change in value of the
objective function in StallIterLim iterations is less than TolFun. The
default value is 1e-6.
• MaxIter — The algorithm stops if the number of iterations exceeds this
maximum number of iterations. You can specify the maximum number of
iterations as a positive integer or Inf. Inf is the default.
• MaxFunEval specifies the maximum number of evaluations of the objective
function. The algorithm stops if the number of function evaluations exceeds
the maximum number of function evaluations. The allowed maximum is
3000*numberofvariables.
• TimeLimit specifies the maximum time in seconds the algorithm runs
before stopping.
• ObjectiveLimit — The algorithm stops if the best objective function value
is less than or equal to the value of ObjectiveLimit.
Output Function Options
Output functions are functions that the algorithm calls at each iteration.
The default value is to have no output function, []. You must first create
an output function using the syntax described in “Structure of the Output
Function” on page 9-65.
Using the Optimization app:
• Specify Output function as @myfun, where myfun is the name of your
function.
• To pass extra parameters in the output function, use “Anonymous
Functions”.
• For multiple output functions, enter a cell array of output function handles:
{@myfun1,@myfun2,...}.
At the command line:
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Simulated Annealing Options
• options = saoptimset('OutputFcns',@myfun);
• For multiple output functions, enter a cell array:
options = saoptimset('OutputFcns',{@myfun1,@myfun2,...});
To see a template that you can use to write your own output functions, enter
edit saoutputfcntemplate
at the MATLAB command line.
Structure of the Output Function
The output function has the following calling syntax.
[stop,options,optchanged] = myfun(options,optimvalues,flag)
The function has the following input arguments:
• options — Options structure created using saoptimset.
• optimvalues — Structure containing information about the current state
of the solver. The structure contains the following fields:
-
x — Current point
fval — Objective function value at x
bestx — Best point found so far
bestfval — Objective function value at best point
temperature — Current temperature, a vector the same length as x
iteration — Current iteration
funccount — Number of function evaluations
t0 — Start time for algorithm
k — Annealing parameter, a vector the same length as x
• flag — Current state in which the output function is called. The possible
values for flag are
-
'init' — Initialization state
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Options Reference
-
'iter' — Iteration state
'done' — Final state
“Passing Extra Parameters” in the Optimization Toolbox documentation
explains how to provide additional parameters to the output function.
The output function returns the following arguments:
• stop — Provides a way to stop the algorithm at the current iteration. stop
can have the following values:
-
false — The algorithm continues to the next iteration.
true — The algorithm terminates at the current iteration.
• options — Options structure modified by the output function.
• optchanged — A boolean flag indicating changes were made to options.
This must be set to true if options are changed.
Display Options
Use the Display option to specify how much information is displayed at the
command line while the algorithm is running. The available options are
• off — No output is displayed. This is the default value for an options
structure created using saoptimset.
• iter — Information is displayed at each iteration.
• diagnose — Information is displayed at each iteration. In addition, the
diagnostic lists some problem information and the options that have been
changed from the defaults.
• final — The reason for stopping is displayed. This is the default.
Both iter and diagnose display the following information:
• Iteration — Iteration number
• f-count — Cumulative number of objective function evaluations
• Best f(x) — Best objective function value
• Current f(x) — Current objective function value
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Simulated Annealing Options
• Mean Temperature — Mean temperature function value
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‫‪Options Reference‬‬
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‫‪9‬‬
‫‪9-68‬‬
10
Functions — Alphabetical
List
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createOptimProblem
Purpose
Create optimization problem structure
Syntax
problem = createOptimProblem('solverName')
problem = createOptimProblem('solverName','ParameterName',
ParameterValue,...)
Description
problem = createOptimProblem('solverName') creates an empty
optimization problem structure for the solverName solver.
problem = createOptimProblem('solverName','ParameterName',
ParameterValue,...) accepts one or more comma-separated
parameter name/value pairs. Specify ParameterName inside single
quotes.
Input
Arguments
solverName
Name of the solver. For a GlobalSearch problem, use 'fmincon'. For
a MultiStart problem, use 'fmincon', 'fminunc', 'lsqcurvefit'
or 'lsqnonlin'.
Parameter Name/Value Pairs
’Aeq’
Matrix for linear equality constraints. The constraints have the form:
Aeq x = beq
’Aineq’
Matrix for linear inequality constraints. The constraints have the form:
Aineq x ≤ bineq
’beq’
Vector for linear equality constraints. The constraints have the form:
Aeq x = beq
’bineq’
10-2
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createOptimProblem
Vector for linear inequality constraints. The constraints have the form:
Aineq x ≤ bineq
’lb’
Vector of lower bounds.
lb can also be an array; see “Matrix Arguments”.
’nonlcon’
Function handle to the nonlinear constraint function. The constraint
function must accept a vector x and return two vectors: c, the nonlinear
inequality constraints, and ceq, the nonlinear equality constraints. If
one of these constraint functions is empty, nonlcon must return []
for that function.
If the GradConstr option is 'on', then in addition nonlcon must return
two additional outputs, gradc and gradceq. The gradc parameter is
a matrix with one column for the gradient of each constraint, as is
gradceq.
For more information, see “Write Constraints” on page 2-7.
’objective’
Function handle to the objective function. For all solvers except
lsqnonlin and lsqcurvefit, the objective function must accept a
vector x and return a scalar. If the GradObj option is 'on', then the
objective function must return a second output, a vector, representing
the gradient of the objective. For lsqnonlin, the objective function
must accept a vector x and return a vector. lsqnonlin sums the squares
of the objective function values. For lsqcurvefit, the objective function
must accept two inputs, x and xdata, and return a vector.
For more information, see “Compute Objective Functions” on page 2-2.
’options’
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10-3
createOptimProblem
Optimization options. Create options with optimoptions, or by
exporting from the Optimization app.
’ub’
Vector of upper bounds.
ub can also be an array; see “Matrix Arguments”.
’x0’
A vector, a potential starting point for the optimization. Gives the
dimensionality of the problem.
x0 can also be an array; see “Matrix Arguments”.
’xdata’
Vector of data points for lsqcurvefit.
’ydata’
Vector of data points for lsqcurvefit.
Output
Arguments
problem
Examples
Create a problem structure using Rosenbrock’s function as objective
(see “Include a Hybrid Function” on page 5-107), the interior-point
algorithm for fmincon, and bounds with absolute value 2:
Optimization problem structure.
anonrosen = @(x)(100*(x(2) - x(1)^2)^2 + (1-x(1))^2);
opts = optimoptions(@fmincon,'Algorithm','interior-point');
problem = createOptimProblem('fmincon','x0',randn(2,1),...
'objective',anonrosen,'lb',[-2;-2],'ub',[2;2],...
'options',opts);
10-4
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createOptimProblem
Alternatives
You can create a problem structure by exporting from the Optimization
app (optimtool), as described in “Exporting from the Optimization
app” on page 3-10.
See Also
optimtool | MultiStart | GlobalSearch
Tutorials
• “Create Problem Structure” on page 3-7
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10-5
CustomStartPointSet
Purpose
User-supplied start points
Description
An object wrapper of a matrix whose rows represent start points for
MultiStart.
Construction
tpoints = CustomStartPointSet(ptmatrix) generates a
CustomStartPointSet object from the ptmatrix matrix. Each row of
ptmatrix represents one start point.
Properties
DimStartPoints
Dimension of each start point, a read-only property.
DimStartPoints is the number of columns in ptmatrix.
DimStartPoints should be the same as the number of elements in
problem.x0, the problem structure you pass to run.
NumStartPoints
Number of start points, a read-only property. This is the number
of rows in ptmatrix.
Methods
list
List custom start points in set
Copy
Semantics
Value. To learn how value classes affect copy operations, see Copying
Objects in the MATLAB Programming Fundamentals documentation.
Examples
Create a CustomStartPointSet object with 40 three-dimensional rows.
Each row represents a normally distributed random variable with mean
[10,10,10] and variance diag([4,4,4]):
fortypts = 10*ones(40,3) + 4*randn(40,3); % a matrix
startpts = CustomStartPointSet(fortypts);
startpts is the fortypts matrix in an object wrapper.
10-6
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CustomStartPointSet
Get the fortypts matrix from the startpts object of the previous
example:
fortypts = list(startpts);
See Also
RandomStartPointSet | MultiStart | list
Tutorials
• “CustomStartPointSet Object for Start Points” on page 3-21
How To
• Class Attributes
• Property Attributes
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10-7
ga
Purpose
Find minimum of function using genetic algorithm
Syntax
x = ga(fitnessfcn,nvars)
x = ga(fitnessfcn,nvars,A,b)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB,nonlcon)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB,nonlcon,options)
x = ga(fitnessfcn,nvars,A,b,[],[],LB,UB,nonlcon,IntCon)
x =
ga(fitnessfcn,nvars,A,b,[],[],LB,UB,nonlcon,IntCon,options)
x = ga(problem)
[x,fval] = ga(fitnessfcn,nvars,...)
[x,fval,exitflag] = ga(fitnessfcn,nvars,...)
[x,fval,exitflag,output] = ga(fitnessfcn,nvars,...)
[x,fval,exitflag,output,population] =
ga(fitnessfcn,nvars,...)
[x,fval,exitflag,output,population,scores] =
ga(fitnessfcn,nvars,...)
Description
x = ga(fitnessfcn,nvars) finds a local unconstrained minimum, x,
to the objective function, fitnessfcn. nvars is the dimension (number
of design variables) of fitnessfcn.
x = ga(fitnessfcn,nvars,A,b) finds a local minimum x to
fitnessfcn, subject to the linear inequalities A*x ≤ b. ga evaluates the
matrix product A*x as if x is transposed (A*x').
x = ga(fitnessfcn,nvars,A,b,Aeq,beq) finds a local minimum x to
fitnessfcn, subject to the linear equalities Aeq*x = beq as well as
A*x ≤ b. (Set A=[] and b=[] if no linear inequalities exist.) ga evaluates
the matrix product Aeq*x as if x is transposed (Aeq*x').
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB) defines a set of lower
and upper bounds on the design variables, x, so that a solution is
found in the range LB ≤ x ≤ UB. (Set Aeq=[] and beq=[] if no linear
equalities exist.)
10-8
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ga
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB,nonlcon) subjects
the minimization to the constraints defined in nonlcon. The function
nonlcon accepts x and returns vectors C and Ceq, representing the
nonlinear inequalities and equalities respectively. ga minimizes the
fitnessfcn such that C(x) ≤ 0 and Ceq(x) = 0. (Set LB=[] and UB=[]
if no bounds exist.)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB,nonlcon,options)
minimizes with the default optimization parameters replaced by values
in the structure options, which can be created using the gaoptimset
function. (Set nonlcon=[] if no nonlinear constraints exist.)
x = ga(fitnessfcn,nvars,A,b,[],[],LB,UB,nonlcon,IntCon)
requires that the variables listed in IntCon take integer values.
Note When there are integer constraints, ga does not accept linear or
nonlinear equality constraints, only inequality constraints.
x =
ga(fitnessfcn,nvars,A,b,[],[],LB,UB,nonlcon,IntCon,options)
minimizes with integer constraints and with the default optimization
parameters replaced by values in the options structure.
x = ga(problem) finds the minimum for problem, where problem is
a structure.
[x,fval] = ga(fitnessfcn,nvars,...) returns fval, the value of
the fitness function at x.
[x,fval,exitflag] = ga(fitnessfcn,nvars,...) returns exitflag,
an integer identifying the reason the algorithm terminated.
[x,fval,exitflag,output] = ga(fitnessfcn,nvars,...) returns
output, a structure that contains output from each generation and
other information about the performance of the algorithm.
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10-9
ga
[x,fval,exitflag,output,population] =
ga(fitnessfcn,nvars,...) returns the matrix, population, whose
rows are the final population.
[x,fval,exitflag,output,population,scores] =
ga(fitnessfcn,nvars,...) returns scores the scores of
the final population.
Tips
• To write a function with additional parameters to the independent
variables that can be called by ga, see “Passing Extra Parameters”
in the Optimization Toolbox documentation.
• For problems that use the population type Double Vector (the
default), ga does not accept functions whose inputs are of type
complex. To solve problems involving complex data, write your
functions so that they accept real vectors, by separating the real and
imaginary parts.
Input
Arguments
fitnessfcn
Handle to the fitness function. The fitness function should accept a row
vector of length nvars and return a scalar value.
When the 'Vectorized' option is 'on', fitnessfcn should accept a
pop-by-nvars matrix, where pop is the current population size. In
this case fitnessfcn should return a vector the same length as pop
containing the fitness function values. fitnessfcn should not assume
any particular size for pop, since ga can pass a single member of a
population even in a vectorized calculation.
nvars
Positive integer representing the number of variables in the problem.
A
Matrix for linear inequality constraints of the form
A*x ≤ b.
10-10
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ga
If the problem has m linear inequality constraints and nvars variables,
then
• A is a matrix of size m-by-nvars.
• b is a vector of length m.
ga evaluates the matrix product A*x as if x is transposed (A*x').
Note ga does not enforce linear constraints to be satisfied when the
PopulationType option is 'bitString' or 'custom'.
b
Vector for linear inequality constraints of the form
A*x ≤ b.
If the problem has m linear inequality constraints and nvars variables,
then
• A is a matrix of size m-by-nvars.
• b is a vector of length m.
Aeq
Matrix for linear equality constraints of the form
Aeq*x = beq.
If the problem has m linear equality constraints and nvars variables,
then
• Aeq is a matrix of size m-by-nvars.
• beq is a vector of length m.
ga evaluates the matrix product Aeq*x as if x is transposed (Aeq*x').
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10-11
ga
Note ga does not enforce linear constraints to be satisfied when the
PopulationType option is 'bitString' or 'custom'.
beq
Vector for linear equality constraints of the form
Aeq*x = beq.
If the problem has m linear equality constraints and nvars variables,
then
• Aeq is a matrix of size m-by-nvars.
• beq is a vector of length m.
LB
Vector of lower bounds. ga enforces that iterations stay above LB. Set
LB(i) = –Inf if x(i)is unbounded below.
Note ga does not enforce bounds to be satisfied when the
PopulationType option is 'bitString' or 'custom'.
UB
Vector of upper bounds. ga enforces that iterations stay below UB. Set
UB(i) = Inf if x(i) is unbounded above.
nonlcon
Function handle that returns two outputs:
[c,ceq] = nonlcon(x)
10-12
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ga
ga attempts to achieve c ≤ 0 and ceq = 0. c and ceq are row vectors
when there are multiple constraints. Set unused outputs to [].
You can write nonlcon as a function handle to a file, such as
nonlcon = @constraintfile
where constraintfile.m is a file on your MATLAB path.
To learn how to use vectorized constraints, see “Vectorized Constraints”
on page 2-8.
Note ga does not enforce nonlinear constraints to be satisfied when
the PopulationType option is set to 'bitString' or 'custom'.
If IntCon is not empty, the second output of nonlcon (ceq) must be
empty ([]).
For information on how ga uses nonlcon, see “Nonlinear Constraint
Solver Algorithm” on page 5-52.
options
Structure containing optimization options. Create options using
gaoptimset, or by exporting options from the Optimization app as
described in “Importing and Exporting Your Work” in the Optimization
Toolbox documentation.
IntCon
Vector of positive integers taking values from 1 to nvars. Each value in
IntCon represents an x component that is integer-valued.
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10-13
ga
Note When IntCon is nonempty, Aeq and beq must be empty ([]), and
nonlcon must return empty for ceq. For more information on integer
programming, see “Mixed Integer Optimization” on page 5-32.
problem
Structure containing the following fields:
fitnessfcn
Fitness function
nvars
Number of design variables
Aineq
A matrix for linear inequality constraints
Bineq
b vector for linear inequality constraints
Aeq
Aeq matrix for linear equality constraints
Beq
beq vector for linear equality constraints
lb
Lower bound on x
ub
Upper bound on x
nonlcon
Nonlinear constraint function
rngstate
Optional field to reset the state of the
random number generator
intcon
Index vector of integer variables
solver
'ga'
options
Options structure created using gaoptimset
or the Optimization app
Create problem by exporting a problem from the Optimization app, as
described in “Importing and Exporting Your Work” in the Optimization
Toolbox documentation.
10-14
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ga
Output
Arguments
x
Best point that ga located during its iterations.
fval
Fitness function evaluated at x.
exitflag
Integer giving the reason ga stopped iterating:
Exit Flag
Meaning
1
Without nonlinear constraints — Average
cumulative change in value of the fitness function over
StallGenLimit generations is less than TolFun, and the
constraint violation is less than TolCon.
With nonlinear constraints — Magnitude of the
complementarity measure (see “Definitions” on page
10-17) is less than sqrt(TolCon), the subproblem is
solved using a tolerance less than TolFun, and the
constraint violation is less than TolCon.
2
Fitness limit reached and the constraint violation is less
than TolCon.
3
Value of the fitness function did not change in
StallGenLimit generations and the constraint violation
is less than TolCon.
4
Magnitude of step smaller than machine precision and
the constraint violation is less than TolCon.
5
Minimum fitness limit FitnessLimit reached and the
constraint violation is less than TolCon.
0
Maximum number of generations Generations exceeded.
-1
Optimization terminated by an output function or plot
function.
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10-15
ga
Exit Flag
Meaning
-2
No feasible point found.
-4
Stall time limit StallTimeLimit exceeded.
-5
Time limit TimeLimit exceeded.
When there are integer constraints, ga uses the penalty fitness value
instead of the fitness value for stopping criteria.
output
Structure containing output from each generation and other information
about algorithm performance. The output structure contains the
following fields:
• problemtype — String describing the type of problem, one of:
-
'unconstrained'
'boundconstraints'
'linearconstraints'
'nonlinearconstr'
'integerconstraints'
• rngstate — State of the MATLAB random number generator, just
before the algorithm started. You can use the values in rngstate to
reproduce the output of ga. See “Reproduce Results” on page 5-71.
• generations — Number of generations computed.
• funccount — Number of evaluations of the fitness function.
• message — Reason the algorithm terminated.
• maxconstraint — Maximum constraint violation, if any.
population
Matrix whose rows contain the members of the final population.
10-16
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ga
scores
Column vector of the fitness values (scores for integerconstraints
problems) of the final population.
Definitions
Complementarity Measure
In the nonlinear constraint solver, the complementarity measure is the
norm of the vector whose elements are ciλi, where ci is the nonlinear
inequality constraint violation, and λi is the corresponding Lagrange
multiplier.
Examples
Given the following inequality constraints and lower bounds
⎡ 1 1⎤
⎡2⎤
⎢ −1 2⎥ ⎡ x1 ⎤ ≤ ⎢2⎥ ,
⎢
⎥ ⎢x ⎥ ⎢ ⎥
⎢⎣ 2 1 ⎥⎦ ⎣ 2 ⎦ ⎢⎣3⎥⎦
x1 ≥ 0, x2 ≥ 0,
use this code to find the minimum of the lincontest6 function, which
is provided in your software:
A = [1 1; -1 2; 2 1];
b = [2; 2; 3];
lb = zeros(2,1);
[x,fval,exitflag] = ga(@lincontest6,...
2,A,b,[],[],lb)
Optimization terminated: average change in
the fitness value less than options.TolFun.
x =
0.6700
1.3310
fval =
-8.2218
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10-17
ga
exitflag =
1
Optimize a function where some variables must be integers:
fun = @(x) (x(1) - 0.2)^2 + ...
(x(2) - 1.7)^2 + (x(3) - 5.1)^2;
x = ga(fun,3,[],[],[],[],[],[],[], ...
[2 3]) % variables 2 and 3 are integers
Optimization terminated: average change
in the penalty fitness value less
than options.TolFun and constraint violation
is less than options.TolCon.
x =
0.2000
Algorithms
2.0000
5.0000
For a description of the genetic algorithm, see “How the Genetic
Algorithm Works” on page 5-21.
For a description of the mixed integer programming algorithm, see
“Integer ga Algorithm” on page 5-40.
For a description of the nonlinear constraint algorithm, see “Nonlinear
Constraint Solver Algorithm” on page 5-52.
References
[1] Goldberg, David E., Genetic Algorithms in Search, Optimization &
Machine Learning, Addison-Wesley, 1989.
[2] A. R. Conn, N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent
Augmented Lagrangian Algorithm for Optimization with General
Constraints and Simple Bounds”, SIAM Journal on Numerical Analysis,
Volume 28, Number 2, pages 545–572, 1991.
[3] A. R. Conn, N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent
Augmented Lagrangian Barrier Algorithm for Optimization with
10-18
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ga
General Inequality Constraints and Simple Bounds”, Mathematics of
Computation, Volume 66, Number 217, pages 261–288, 1997.
Alternatives
For problems without integer constraints, consider using
patternsearch instead of ga.
See Also
gamultiobj | gaoptimset | patternsearch
How To
• “Genetic Algorithm”
• “Getting Started with Global Optimization Toolbox”
• “Optimization Problem Setup”
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10-19
gamultiobj
Purpose
Find minima of multiple functions using genetic algorithm
Syntax
X = gamultiobj(FITNESSFCN,NVARS)
X = gamultiobj(FITNESSFCN,NVARS,A,b)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,options)
X = gamultiobj(problem)
[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = gamultiobj(FITNESSFCN, ...)
[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] = gamultiobj(FITNESSFCN,
...)
Description
gamultiobj implements the genetic algorithm at the command line to
minimize a multicomponent objective function.
X = gamultiobj(FITNESSFCN,NVARS) finds a local Pareto set X of
the objective functions defined in FITNESSFCN. For details on writing
FITNESSFCN, see “Compute Objective Functions” on page 2-2. NVARS
is the dimension of the optimization problem (number of decision
variables). X is a matrix with NVARS columns. The number of rows in X
is the same as the number of Pareto solutions. All solutions in a Pareto
set are equally optimal; it is up to the designer to select a solution in
the Pareto set depending on the application.
X = gamultiobj(FITNESSFCN,NVARS,A,b) finds a local Pareto set X of
the objective functions defined in FITNESSFCN, subject to the linear
inequalities A ∗ x ≤ b , see “Linear Inequality Constraints”. Linear
constraints are supported only for the default PopulationType option
('doubleVector'). Other population types, e.g., 'bitString' and
'custom', are not supported.
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq) finds a local
Pareto set X of the objective functions defined in FITNESSFCN, subject
to the linear equalities Aeq ∗ x = beq as well as the linear inequalities
10-20
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gamultiobj
A ∗ x ≤ b , see “Linear Equality Constraints”. (Set A=[] and b=[] if no
inequalities exist.) Linear constraints are supported only for the default
PopulationType option ('doubleVector'). Other population types,
e.g., 'bitString' and 'custom', are not supported.
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB) defines
a set of lower and upper bounds on the design variables X so that
a local Pareto set is found in the range LB ≤ x ≤ UB , see “Bound
Constraints”. Use empty matrices for LB and UB if no bounds exist.
Bound constraints are supported only for the default PopulationType
option ('doubleVector'). Other population types, e.g., 'bitString'
and 'custom', are not supported.
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,options)
finds a Pareto set X with the default optimization parameters replaced
by values in the structure options. options can be created with the
gaoptimset function.
X = gamultiobj(problem) finds the Pareto set for problem, where
problem is a structure containing the following fields:
fitnessfcn
Fitness functions
nvars
Number of design variables
Aineq
A matrix for linear inequality constraints
bineq
b vector for linear inequality constraints
Aeq
Aeq matrix for linear equality constraints
beq
beq vector for linear equality constraints
lb
Lower bound on x
ub
Upper bound on x
solver
'gamultiobj'
rngstate
Optional field to reset the state of the
random number generator
options
Options structure created using gaoptimset
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10-21
gamultiobj
Create the structure problem by exporting a problem from Optimization
app, as described in “Importing and Exporting Your Work” in the
Optimization Toolbox documentation.
[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...) returns a matrix
FVAL, the value of all the objective functions defined in FITNESSFCN
at all the solutions in X. FVAL has numberOfObjectives columns and
same number of rows as does X.
[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...) returns
EXITFLAG, which describes the exit condition of gamultiobj. Possible
values of EXITFLAG and the corresponding exit conditions are listed
in this table.
EXITFLAG Exit Condition
Value
1
Average change in value of the spread over
options.StallGenLimit generations less than
options.TolFun, and the final spread is less than the
average spread over the past options.StallGenLimit
generations
0
Maximum number of generations exceeded
-1
Optimization terminated by an output function or plot
function
-2
No feasible point found
-5
Time limit exceeded
[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS,
...) returns a structure OUTPUT with the following fields:
10-22
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gamultiobj
OUTPUT Field
Meaning
problemtype
Type of problem:
• 'unconstrained' — No constraints
• 'boundconstraints' — Only bound constraints
• 'linearconstraints' — Linear constraints,
with or without bound constraints
rngstate
State of the MATLAB random number generator,
just before the algorithm started. You can use the
values in rngstate to reproduce the output of ga.
See “Reproduce Results” on page 5-71.
generations
funccount
Total number of generations, excluding HybridFcn
iterations
Total number of function evaluations
message
gamultiobj termination message
averagedistance Average “distance,” which by default is the
standard deviation of the norm of the difference
between Pareto front members and their mean
spread
Combination of the “distance,” and a measure of
the movement of the points on the Pareto front
between the final two iterations
[X,FVAL,EXITFLAG,OUTPUT,POPULATION] =
gamultiobj(FITNESSFCN, ...) returns the final POPULATION at
termination.
[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] =
gamultiobj(FITNESSFCN, ...) returns the SCORE of the
final POPULATION.
Examples
This example optimizes two objectives defined by Schaffer’s second
function, which has two objectives and a scalar input argument. The
Pareto front is disconnected. Define this function in a file:
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10-23
gamultiobj
function y = schaffer2(x) % y has two columns
% Initialize y for two objectives and for all x
y = zeros(length(x),2); % ready for vectorization
% Evaluate first objective.
% This objective is piecewise continuous.
for i = 1:length(x)
if x(i) <= 1
y(i,1) = -x(i);
elseif x(i) <=3
y(i,1) = x(i) -2;
elseif x(i) <=4
y(i,1) = 4 - x(i);
else
y(i,1) = x(i) - 4;
end
end
% Evaluate second objective
y(:,2) = (x -5).^2;
First, plot the two objectives:
x = -1:0.1:8;
y = schaffer2(x);
plot(x,y(:,1),'.r'); hold on
plot(x,y(:,2),'.b');
The two component functions compete in the range [1, 3] and [4, 5]. But
the Pareto-optimal front consists of only two disconnected regions: [1, 2]
and [4, 5]. This is because the region [2, 3] is inferior to [1, 2].
Next, impose a bound constraint on x, −5 ≤ x ≤ 10 setting
lb = -5;
ub = 10;
10-24
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gamultiobj
The best way to view the results of the genetic algorithm is to visualize
the Pareto front directly using the @gaplotpareto option. To optimize
Schaffer’s function, a larger population size than the default (15) is
needed, because of the disconnected front. This example uses 60. Set
the optimization options as:
options = gaoptimset('PopulationSize',60,'PlotFcns',...
@gaplotpareto);
Now call gamultiobj, specifying one independent variable and only the
bound constraints:
[x,f,exitflag] = gamultiobj(@schaffer2,1,[],[],[],[],...
lb,ub,options);
Optimization terminated: average change in the spread of
Pareto solutions less than options.TolFun.
exitflag
exitflag = 1
The vectors x, f(:,1), and f(:,2) respectively contain the Pareto set
and both objectives evaluated on the Pareto set.
Examples Included in the Toolbox
The gamultiobjfitness example solves a simple problem with one
decision variable and two objectives.
The gamultiobjoptionsdemo example shows how to set options for
multiobjective optimization.
Algorithms
gamultiobj uses a controlled elitist genetic algorithm (a variant of
NSGA-II [1]). An elitist GA always favors individuals with better fitness
value (rank). A controlled elitist GA also favors individuals that can
help increase the diversity of the population even if they have a lower
fitness value. It is important to maintain the diversity of population
for convergence to an optimal Pareto front. Diversity is maintained
by controlling the elite members of the population as the algorithm
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10-25
gamultiobj
progresses. Two options, ParetoFraction and DistanceFcn, control the
elitism. ParetoFraction limits the number of individuals on the Pareto
front (elite members). The distance function, selected by DistanceFcn,
helps to maintain diversity on a front by favoring individuals that are
relatively far away on the front. The algorithm stops if the spread, a
measure of the movement of the Pareto front, is small.
References
[1] Deb, Kalyanmoy. Multi-Objective Optimization Using Evolutionary
Algorithms. John Wiley & Sons, 2001.
See Also
ga | gaoptimget | gaoptimset | patternsearch | Special
Characters | rand | randn
10-26
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gaoptimget
Purpose
Obtain values of genetic algorithm options structure
Syntax
val = gaoptimget(options, 'name')
val = gaoptimget(options, 'name', default)
Description
val = gaoptimget(options, 'name') returns the value of the
parameter name from the genetic algorithm options structure options.
gaoptimget(options, 'name') returns an empty matrix [] if the
value of name is not specified in options. It is only necessary to type
enough leading characters of name to uniquely identify it. gaoptimget
ignores case in parameter names.
val = gaoptimget(options, 'name', default) returns the 'name'
parameter, but will return the default value if the name parameter is
not specified (or is []) in options.
See Also
ga | gamultiobj | gaoptimset
How To
• “Genetic Algorithm Options” on page 9-31
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10-27
gaoptimset
Purpose
Create genetic algorithm options structure
Syntax
gaoptimset
options = gaoptimset
options = gaoptimset(@ga)
options = gaoptimset(@gamultiobj)
options = gaoptimset('param1',value1,'param2',value2,...)
options = gaoptimset(oldopts,'param1',value1,...)
options = gaoptimset(oldopts,newopts)
Description
gaoptimset with no input or output arguments displays a complete list
of parameters with their valid values.
options = gaoptimset (with no input arguments) creates a structure
called options that contains the options, or parameters, for the genetic
algorithm and sets parameters to [], indicating default values will
be used.
options = gaoptimset(@ga) creates a structure called options that
contains the default options for the genetic algorithm.
options = gaoptimset(@gamultiobj) creates a structure called
options that contains the default options for gamultiobj.
options = gaoptimset('param1',value1,'param2',value2,...)
creates a structure called options and sets the value of 'param1' to
value1, 'param2' to value2, and so on. Any unspecified parameters are
set to their default values. It is sufficient to type only enough leading
characters to define the parameter name uniquely. Case is ignored
for parameter names.
options = gaoptimset(oldopts,'param1',value1,...) creates a
copy of oldopts, modifying the specified parameters with the specified
values.
options = gaoptimset(oldopts,newopts) combines an existing
options structure, oldopts, with a new options structure, newopts.
Any parameters in newopts with nonempty values overwrite the
corresponding old parameters in oldopts.
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gaoptimset
Options
The following table lists the options you can set with gaoptimset. See
“Genetic Algorithm Options” on page 9-31 for a complete description
of these options and their values. Values in {} denote the default
value. {}* means the default when there are linear constraints, and
for MutationFcn also when there are bounds. You can also view the
optimization parameters and defaults by typing gaoptimset at the
command line. I* indicates that ga ignores or overwrites the option for
mixed integer optimization problems.
Option
Description
Values
CreationFcn
I* Handle to the function
that creates the initial
population. See “Population
Options” on page 9-36.
{@gacreationuniform} |
{@gacreationlinearfeasible}*
CrossoverFcn
I* Handle to the function
that the algorithm uses to
create crossover children.
See “Crossover Options” on
page 9-46.
@crossoverheuristic |
{@crossoverscattered} |
{@crossoverintermediate}*
| @crossoversinglepoint
| @crossovertwopoint |
@crossoverarithmetic
CrossoverFraction
The fraction of the
population at the next
generation, not including
elite children, that is created
by the crossover function
Positive scalar | {0.8}
Display
Level of display
'off' | 'iter' | 'diagnose' |
{'final'}
DistanceMeasureFcn
I* Handle to the function
that computes distance
measure of individuals,
computed in decision
variable or design space
(genotype) or in function
space (phenotype)
{@distancecrowding,'phenotype'}
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gaoptimset
10-30
Option
Description
Values
EliteCount
Positive integer specifying
how many individuals in
the current generation are
guaranteed to survive to the
next generation. Not used
in gamultiobj.
Positive integer |
FitnessLimit
Scalar. If the fitness
function attains the value
of FitnessLimit, the
algorithm halts.
Scalar | {-Inf}
FitnessScalingFcn
Handle to the function that
scales the values of the
fitness function
@fitscalingshiftlinear
| @fitscalingprop
| @fitscalingtop |
{@fitscalingrank}
Generations
Positive integer specifying
the maximum number
of iterations before the
algorithm halts
Positive integer
|{100*numberOfVariables}
HybridFcn
I* Handle to a function that
continues the optimization
after ga terminates
Function handle | @fminsearch
or
or
Cell array specifying the
hybrid function and its
options structure
1-by-2 cell array | {@solver,
hybridoptions}, where solver
= fminsearch, patternsearch,
fminunc, or fmincon {[]}
InitialPenalty
I* Initial value of penalty
parameter
Positive scalar | {10}
InitialPopulation
Initial population used to
seed the genetic algorithm;
can be partial
Matrix | {[]}
{floor(.05*PopulationSize)}
— {0.05*(default population
size)} for mixed integer problems
| @patternsearch | @fminunc |
@fmincon | {[]}
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gaoptimset
Option
Description
Values
InitialScores
I* Initial scores used to
determine fitness; can be
partial
Column vector | {[]}
MigrationDirection
Direction of migration —
see “Migration Options” on
page 9-50
'both' | {'forward'}
MigrationFraction
Scalar between 0 and 1
specifying the fraction
of individuals in each
subpopulation that migrates
to a different subpopulation
— see “Migration Options”
on page 9-50
Scalar | {0.2}
MigrationInterval
Positive integer specifying
the number of generations
that take place between
migrations of individuals
between subpopulations —
see “Migration Options” on
page 9-50
Positive integer | {20}
MutationFcn
I* Handle to the function
that produces mutation
children. See “Mutation
Options” on page 9-43.
@mutationuniform |
{@mutationadaptfeasible}*
| {@mutationgaussian}
OutputFcns
Functions that ga calls at
each iteration
Function handle or cell array of
function handles | {[]}
ParetoFraction
I* Scalar between 0 and 1
specifying the fraction of
individuals to keep on the
first Pareto front while the
solver selects individuals
from higher fronts
Scalar | {0.35}
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10-31
gaoptimset
Option
Description
Values
PenaltyFactor
I* Penalty update
parameter
Positive scalar | {100}
PlotFcns
Array of handles to
functions that plot data
computed by the algorithm
@gaplotbestf |
@gaplotbestindiv |
@gaplotdistance |
@gaplotexpectation
| @gaplotgenealogy
| @gaplotmaxconstr
| @gaplotrange |
@gaplotselection |
@gaplotscorediversity
| @gaplotscores |
@gaplotstopping | {[]}
For gamultiobj there are
additional choices: @gaplotpareto
| @gaplotparetodistance
| @gaplotrankhist |
@gaplotspread
10-32
PlotInterval
Positive integer specifying
the number of generations
between consecutive calls to
the plot functions
Positive integer | {1}
PopInitRange
Matrix or vector
specifying the range of
the individuals in the
initial population. Applies
to gacreationuniform
creation function. ga shifts
and scales the default initial
range to match any finite
bounds.
Matrix or vector | {[-10;10]}
for unbounded components,
{[-1e4+1;1e4+1]} for unbounded
components of integer-constrained
problems, {[lb;ub]} for bounded
components, with the default range
modified to match one-sided bounds.
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gaoptimset
Option
Description
Values
PopulationSize
Size of the population
Positive integer | {50}
when numberOfVariables
<= 5, {200} otherwise |
{min(max(10*nvars,40),100)} for
mixed integer problems
PopulationType
String describing the data
type of the population —
must be 'doubleVector'
for mixed integer problems
'bitstring' | 'custom' |
{'doubleVector'}
ga ignores all constraints when
PopulationType is set to
'bitString' or 'custom'. See
“Population Options” on page 9-36.
SelectionFcn
I* Handle to the function
that selects parents of
crossover and mutation
children
@selectionremainder |
@selectionuniform |
{@selectionstochunif}
| @selectionroulette |
@selectiontournament
StallGenLimit
Positive integer. The
algorithm stops if the
average relative change
in the best fitness
function value over
StallGenLimit generations
is less than or equal to
TolFun. If StallTest is
'geometricWeighted',
then the algorithm stops
if the weighted average
relative change is less than
or equal to TolFun.
Positive integer | {50}
StallTest
String describing the
stopping test.
'geometricWeighted' |
{'totalChange'}
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10-33
gaoptimset
Option
Description
Values
StallTimeLimit
Positive scalar. The
algorithm stops if there
is no improvement in
the objective function for
StallTimeLimit seconds,
as measured by cputime.
Positive scalar | {Inf}
TimeLimit
Positive scalar. The
algorithm stops after
running for TimeLimit
seconds, as measured by
cputime.
Positive scalar | {Inf}
TolCon
Positive scalar. TolCon
is used to determine the
feasibility with respect to
nonlinear constraints. Also,
Positive scalar | {1e-6}
max(sqrt(eps),sqrt(TolCon))
determines feasibility
with respect to linear
constraints.
TolFun
10-34
Positive scalar. The
algorithm stops if the
average relative change
in the best fitness
function value over
StallGenLimit generations
is less than or equal to
TolFun. If StallTest is
'geometricWeighted',
then the algorithm stops
if the weighted average
relative change is less than
or equal to TolFun.
Positive scalar | {1e-6}
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gaoptimset
Option
Description
Values
UseParallel
Compute fitness and
nonlinear constraint
functions in parallel,
see “Vectorize and Parallel
Options (User Function
Evaluation)” on page 9-56
and “How to Use Parallel
Processing” on page 8-12.
true | {false}
Vectorized
Specifies whether functions
are vectorized, see
“Vectorize and Parallel
Options (User Function
Evaluation)” on page 9-56
and “Vectorize the Fitness
Function” on page 5-114.
'on' | {'off'}
See Also
ga | gamultiobj | gaoptimget
How To
• “Genetic Algorithm Options” on page 9-31
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10-35
GlobalOptimSolution
Purpose
Optimization solution
Description
Information on a local minimum, including location, objective function
value, and start point or points that lead to the minimum.
GlobalSearch and MultiStart generate a vector of
GlobalOptimSolution objects. The vector is ordered by
objective function value, from lowest (best) to highest (worst).
Construction
When you run them, GlobalSearch and MultiStart create
GlobalOptimSolution objects as output.
Properties
Exitflag
An integer describing the result of the local solver run.
For the meaning of the exit flag, see the description in the
appropriate local solver function reference page:
• fmincon
• fminunc
• lsqcurvefit
• lsqnonlin
Fval
Objective function value at the solution.
Output
Output structure returned by the local solver.
X
Solution point, with the same dimensions as the initial point.
X0
Cell array of start points that led to the solution.
10-36
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GlobalOptimSolution
Copy
Semantics
Value. To learn how value classes affect copy operations, see Copying
Objects in the MATLAB Programming Fundamentals documentation.
Examples
Use MultiStart to create a vector of GlobalOptimSolution objects:
ms = MultiStart;
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3]);
[xmin,fmin,flag,outpt,allmins] = run(ms,problem,30);
allmins is the vector of GlobalOptimSolution objects:
allmins
allmins =
1x30 GlobalOptimSolution
Properties:
X
Fval
Exitflag
Output
X0
See Also
MultiStart | MultiStart.run | GlobalSearch | GlobalSearch.run
Tutorials
• “Multiple Solutions” on page 3-30
• “Visualize the Basins of Attraction” on page 3-40
How To
• Class Attributes
• Property Attributes
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10-37
GlobalSearch
Purpose
Find global minimum
Description
A GlobalSearch object contains properties (options) that affect how
the run method searches for a global minimum, or generates a
GlobalOptimSolution object.
Construction
gs = GlobalSearch constructs a new global search optimization solver
with its properties set to the defaults.
gs = GlobalSearch('PropertyName',PropertyValue,...)
constructs the object using options, specified as property name and
value pairs.
gs = GlobalSearch(oldgs,'PropertyName',PropertyValue,...)
constructs a copy of the GlobalSearch solver oldgs. The gs object has
the named properties altered with the specified values.
gs = GlobalSearch(ms) constructs gs, a GlobalSearch solver, with
common parameter values from the ms MultiStart solver.
Properties
BasinRadiusFactor
A basin radius decreases after MaxWaitCycle consecutive start
points are within the basin. The basin radius decreases by a
factor of 1 – BasinRadiusFactor.
Set BasinRadiusFactor to 0 to disable updates of the basin
radius.
Default: 0.2
Display
Detail level of iterative display. Possible values:
• 'final' — Report summary results after run finishes.
• 'iter' — Report results after the initial fmincon run, after
Stage 1, after every 200 start points, and after every run of
fmincon, in addition to the final summary.
10-38
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GlobalSearch
• 'off' — No display.
Default: 'final'
DistanceThresholdFactor
A multiplier for determining whether a trial point is in an existing
basin of attraction. For details, see “Examine Stage 2 Trial Point
to See if fmincon Runs” on page 3-56.
Default: 0.75
MaxTime
Time in seconds for a run. GlobalSearch halts when it sees
MaxTime seconds have passed since the beginning of the run.
Default: Inf
MaxWaitCycle
A positive integer tolerance appearing in several points in the
algorithm:
• If the observed penalty function of MaxWaitCycle consecutive
trial points is at least the penalty threshold, then raise the
penalty threshold (see PenaltyThresholdFactor).
• If MaxWaitCycle consecutive trial points are in a basin, then
update that basin’s radius (see BasinRadiusFactor).
Default: 20
NumStageOnePoints
Number of start points in Stage 1. For details, see “Obtain Stage
1 Start Point, Run” on page 3-55.
Default: 200
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10-39
GlobalSearch
NumTrialPoints
Number of potential start points to examine in addition to x0 from
the problem structure. GlobalSearch runs only those potential
start points that pass several tests. For more information, see
“GlobalSearch Algorithm” on page 3-54.
Default: 1000
OutputFcns
A function handle or cell array of function handles to output
functions. Output functions run after each local solver call. They
also run when the global solver starts and ends. Write your
output functions using the syntax described in “OutputFcns” on
page 9-3. See “GlobalSearch Output Function” on page 3-43.
Default: []
PenaltyThresholdFactor
Determines increase in the penalty threshold. For details, see
React to Large Counter Values.
Default: 0.2
PlotFcns
A function handle or cell array of function handles to plot
functions. Plot functions run after each local solver call. They
also run when the global solver starts and ends. Write your plot
functions using the syntax described in “OutputFcns” on page 9-3.
There are two built-in plot functions:
• @gsplotbestf plots the best objective function value.
• @gsplotfunccount plots the number of function evaluations.
See “MultiStart Plot Function” on page 3-48.
10-40
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GlobalSearch
Default: []
StartPointsToRun
Directs the solver to exclude certain start points from being run:
• all — Accept all start points.
• bounds — Reject start points that do not satisfy bounds.
• bounds-ineqs — Reject start points that do not satisfy bounds
or inequality constraints.
GlobalSearch checks the StartPointsToRun property only
during Stage 2 of the GlobalSearch algorithm (the main loop).
For more information, see “GlobalSearch Algorithm” on page 3-54.
Default: 'all'
TolFun
Describes how close two objective function values must be for
solvers to consider them identical for creating the vector of local
solutions. Solvers consider two solutions identical if they are
within TolX distance of each other and have objective function
values within TolFun of each other. If both conditions are not
met, solvers report the solutions as distinct. Set TolFun to 0 to
obtain the results of every local solver run. Set TolFun to a larger
value to have fewer results.
Default: 1e-6
TolX
Describes how close two points must be for solvers to consider
them identical for creating the vector of local solutions. Solvers
compute the distance between a pair of points with norm, the
Euclidean distance. Solvers consider two solutions identical if
they are within TolX distance of each other and have objective
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10-41
GlobalSearch
function values within TolFun of each other. If both conditions
are not met, solvers report the solutions as distinct. Set TolX to
0 to obtain the results of every local solver run. Set TolX to a
larger value to have fewer results.
Default: 1e-6
Methods
run
Find global minimum
Copy
Semantics
Value. To learn how value classes affect copy operations, see Copying
Objects in the MATLAB Programming Fundamentals documentation.
Examples
Solve a problem using a default GlobalSearch object:
opts = optimoptions(@fmincon,'Algorithm','interior-point');
problem = createOptimProblem('fmincon','objective',...
@(x) x.^2 + 4*sin(5*x),'x0',3,'lb',-5,'ub',5,'options',opts);
gs = GlobalSearch;
[x,f] = run(gs,problem)
Algorithms
A detailed description of the algorithm appears in “GlobalSearch
Algorithm” on page 3-54. Ugray et al. [1] describes both the algorithm
and the scatter-search method of generating trial points.
References
[1] Ugray, Zsolt, Leon Lasdon, John Plummer, Fred Glover, James
Kelly, and Rafael Martí. Scatter Search and Local NLP Solvers: A
Multistart Framework for Global Optimization. INFORMS Journal on
Computing, Vol. 19, No. 3, 2007, pp. 328–340.
See Also
MultiStart | createOptimProblem | GlobalOptimSolution
Tutorials
• “Global or Multiple Starting Point Search”
10-42
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CustomStartPointSet.list
Purpose
List custom start points in set
Syntax
tpts = list(CS)
Description
tpts = list(CS) returns the matrix of start points in the CS
CustomStartPoints object.
Input
Arguments
CS
Output
Arguments
tpts
Examples
Create a CustomStartPointSet containing 40 seven-dimensional
normally distributed points, then use list to get the matrix of points
from the object:
A CustomStartPointSet object.
Matrix of start points. The rows of tpts represent the start points.
startpts = randn(40,7) % 40 seven-dimensional start points
cs = CustomStartPointSet(startpts); % cs is an object
startpts2 = list(cs) % startpts2 = startpts
See Also
CustomStartPointSet | createOptimProblem
Tutorials
• “CustomStartPointSet Object for Start Points” on page 3-21
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10-43
RandomStartPointSet.list
Purpose
Generate start points
Syntax
points = list(RandSet,problem)
Description
points = list(RandSet,problem) generates pseudorandom start
points using the parameters in the RandSet RandomStartPointSet
object, and information from the problem problem structure.
Input
Arguments
RandSet
A RandomStartPointSet object. This contains parameters for
generating the points: number of points, and artificial bounds.
problem
An optimization problem structure. list generates points
uniformly within the bounds of the problem structure. If a
component is unbounded, list uses the artificial bounds from
RandSet. list takes the dimension of the points from the x0 field
in problem.
Output
Arguments
points
Examples
Create a matrix representing 40 seven-dimensional start points:
A k-by-n matrix. The number of rows k is the number of start
points that RandSet specifies. The number of columns n is the
dimension of the start points. n is equal to the number of elements
in the x0 field in problem. The MultiStart algorithm uses each
row of points as an initial point in an optimization.
rs = RandomStartPointSet('NumStartPoints',40); % 40 points
problem = createOptimProblem('fminunc','x0',ones(7,1),...
'objective',@rosenbrock);
ptmatrix = list(rs,problem); % matrix values between
% -1000 and 1000 since those are the default bounds
% for unconstrained dimensions
10-44
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RandomStartPointSet.list
Algorithms
The list method generates a pseudorandom random matrix using the
default random number stream. Each row of the matrix represents a
start point to run. list generates points that satisfy the bounds in
problem. If lb is the vector of lower bounds, ub is the vector of upper
bounds, there are n dimensions in a point, and there are k rows in the
matrix, the random matrix is
lb + (ub - lb).*rand(k,n)
• If a component has no bounds, RandomStartPointSet uses
a lower bound of -ArtificialBound, and an upper bound of
ArtificialBound.
• If a component has a lower bound lb, but no upper
bound, RandomStartPointSet uses an upper bound of
lb + 2*ArtificialBound.
• Similarly, if a component has an upper bound ub, but no
lower bound, RandomStartPointSet uses a lower bound of
ub - 2*ArtificialBound.
The default value of ArtificialBound is 1000.
To obtain identical pseudorandom results, reset the default random
number stream. See “Reproduce Results” on page 3-79.
See Also
RandomStartPointSet | createOptimProblem | MultiStart
Tutorials
• “RandomStartPointSet Object for Start Points” on page 3-21
• “Reproduce Results” on page 3-79
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MultiStart
Purpose
Find multiple local minima
Description
A MultiStart object contains properties (options) that affect how
the run method repeatedly runs a local solver, or generates a
GlobalOptimSolution object.
Construction
MS = MultiStart constructs MS, a MultiStart solver with its properties
set to the defaults.
MS = MultiStart('PropertyName',PropertyValue,...) constructs
MS using options, specified as property name and value pairs.
MS = MultiStart(oldMS,'PropertyName',PropertyValue,...)
creates a copy of the oldMS MultiStart solver, with the named
properties changed to the specified values.
MS = MultiStart(GS) constructs MS, a MultiStart solver, with
common parameter values from the GS GlobalSearch solver.
Properties
Display
Detail level of the output to the Command Window:
• 'final' — Report summary results after run finishes.
• 'iter' — Report results after each local solver run, in addition
to the final summary.
• 'off' — No display.
Default: final
MaxTime
Tolerance on the time MultiStart runs. MultiStart and its
local solvers halt when they see MaxTime seconds have passed
since the beginning of the run. Time means wall clock as opposed
to processor cycles.
Default: Inf
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MultiStart
OutputFcns
A function handle or cell array of function handles to output
functions. Output functions run after each local solver call. They
also run when the global solver starts and ends. Write your
output functions using the syntax described in “OutputFcns” on
page 9-3. See “GlobalSearch Output Function” on page 3-43.
Default: []
PlotFcns
A function handle or cell array of function handles to plot
functions. Plot functions run after each local solver call. They
also run when the global solver starts and ends. Write your plot
functions using the syntax described in “OutputFcns” on page 9-3.
There are two built-in plot functions:
• @gsplotbestf plots the best objective function value.
• @gsplotfunccount plots the number of function evaluations.
See “MultiStart Plot Function” on page 3-48.
Default: []
StartPointsToRun
Directs the solver to exclude certain start points from being run:
• all — Accept all start points.
• bounds — Reject start points that do not satisfy bounds.
• bounds-ineqs — Reject start points that do not satisfy bounds
or inequality constraints.
Default: all
TolFun
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MultiStart
Describes how close two objective function values must be for
solvers to consider them identical for creating the vector of local
solutions. Solvers consider two solutions identical if they are
within TolX distance of each other and have objective function
values within TolFun of each other. If both conditions are not
met, solvers report the solutions as distinct. Set TolFun to 0 to
obtain the results of every local solver run. Set TolFun to a larger
value to have fewer results.
Default: 1e-6
TolX
Describes how close two points must be for solvers to consider
them identical for creating the vector of local solutions. Solvers
compute the distance between a pair of points with norm, the
Euclidean distance. Solvers consider two solutions identical if
they are within TolX distance of each other and have objective
function values within TolFun of each other. If both conditions
are not met, solvers report the solutions as distinct. Set TolX to
0 to obtain the results of every local solver run. Set TolX to a
larger value to have fewer results.
Default: 1e-6
UseParallel
Distribute local solver calls to multiple processors:
• true — Distribute the local solver calls to multiple processors.
• false — Cannot not run in parallel.
Default: false
Methods
10-48
run
Run local solver from multiple
points
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MultiStart
Copy
Semantics
Value. To learn how value classes affect copy operations, see Copying
Objects in the MATLAB Programming Fundamentals documentation.
Examples
Run MultiStart on 20 instances of a problem using the fmincon sqp
algorithm:
opts = optimoptions(@fmincon,'Algorithm','sqp');
problem = createOptimProblem('fmincon','objective',...
@(x) x.^2 + 4*sin(5*x),'x0',3,'lb',-5,'ub',5,'options',opts);
ms = MultiStart;
[x,f] = run(ms,problem,20)
Algorithms
A detailed description of the algorithm appears in “MultiStart
Algorithm” on page 3-59.
See Also
GlobalSearch | createOptimProblem | RandomStartPointSet |
CustomStartPointSet | GlobalOptimSolution
Tutorials
• “Global or Multiple Starting Point Search”
• “Parallel Computing”
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10-49
patternsearch
Purpose
Find minimum of function using pattern search
Syntax
x = patternsearch(fun,x0)
x = patternsearch(fun,x0,A,b)
x = patternsearch(fun,x0,A,b,Aeq,beq)
x = patternsearch(fun,x0,A,b,Aeq,beq,LB,UB)
x = patternsearch(fun,x0,A,b,Aeq,beq,LB,UB,nonlcon)
x = patternsearch(fun,x0,A,b,Aeq,beq,LB,UB,nonlcon,options)
x = patternsearch(problem)
[x,fval] = patternsearch(fun,x0, ...)
[x,fval,exitflag] = patternsearch(fun,x0, ...)
[x,fval,exitflag,output] = patternsearch(fun,x0, ...)
Description
patternsearch finds the minimum of a function using a pattern search.
x = patternsearch(fun,x0) finds a local minimum, x, to the function
handle fun that computes the values of the objective function. For
details on writing fun, see “Compute Objective Functions” on page 2-2.
x0 is an initial point for the pattern search algorithm, a real vector.
Note To write a function with additional parameters to the
independent variables that can be called by patternsearch, see the
section on “Passing Extra Parameters” in the Optimization Toolbox
documentation.
x = patternsearch(fun,x0,A,b) finds a local minimum x to the
function fun, subject to the linear inequality constraints represented in
matrix form by Ax ≤ b , see “Linear Inequality Constraints”.
If the problem has m linear inequality constraints and n variables, then
• A is a matrix of size m-by-n.
• b is a vector of length m.
x = patternsearch(fun,x0,A,b,Aeq,beq) finds a local minimum x to
the function fun, starting at x0, and subject to the constraints
10-50
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patternsearch
A*x  b
Aeq * x  beq,
where Aeq ∗ x = beq represents the linear equality constraints in matrix
form, see “Linear Equality Constraints”. If the problem has r linear
equality constraints and n variables, then
• Aeq is a matrix of size r-by-n.
• beq is a vector of length r.
If there are no inequality constraints, pass empty matrices, [], for
A and b.
x = patternsearch(fun,x0,A,b,Aeq,beq,LB,UB) defines a set of
lower and upper bounds on the design variables, x, so that a solution
is found in the range LB ≤ x ≤ UB, see “Bound Constraints”. If the
problem has n variables, LB and UB are vectors of length n. If LB or UB is
empty (or not provided), it is automatically expanded to -Inf or Inf,
respectively. If there are no inequality or equality constraints, pass
empty matrices for A, b, Aeq and beq.
x = patternsearch(fun,x0,A,b,Aeq,beq,LB,UB,nonlcon) subjects
the minimization to the constraints defined in nonlcon, a nonlinear
constraint function. The function nonlcon accepts x and returns
the vectors C and Ceq, representing the nonlinear inequalities and
equalities respectively. patternsearch minimizes fun such that
C(x) ≤ 0 and Ceq(x) = 0. (Set LB=[] and UB=[] if no bounds exist.)
x = patternsearch(fun,x0,A,b,Aeq,beq,LB,UB,nonlcon,options)
minimizes fun with the default optimization parameters replaced
by values in options. The structure options can be created using
psoptimset.
x = patternsearch(problem) finds the minimum for problem, where
problem is a structure containing the following fields:
• objective — Objective function
• X0 — Starting point
• Aineq — Matrix for linear inequality constraints
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10-51
patternsearch
• bineq — Vector for linear inequality constraints
• Aeq — Matrix for linear equality constraints
• beq — Vector for linear equality constraints
• lb — Lower bound for x
• ub — Upper bound for x
• nonlcon — Nonlinear constraint function
• solver — 'patternsearch'
• options — Options structure created with psoptimset
• rngstate — Optional field to reset the state of the random number
generator
Create the structure problem by exporting a problem from the
Optimization app, as described in “Importing and Exporting Your
Work” in the Optimization Toolbox documentation.
Note problem must have all the fields as specified above.
[x,fval] = patternsearch(fun,x0, ...) returns the value of the
objective function fun at the solution x.
[x,fval,exitflag] = patternsearch(fun,x0, ...) returns
exitflag, which describes the exit condition of patternsearch.
Possible values of exitflag and the corresponding conditions are
10-52
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patternsearch
Exit
Flag
Meaning
1
Without nonlinear constraints — Magnitude of the
mesh size is less than the specified tolerance and constraint
violation is less than TolCon.
With nonlinear constraints — Magnitude of the
complementarity measure (defined after this table) is less
than sqrt(TolCon), the subproblem is solved using a mesh
finer than TolMesh, and the constraint violation is less than
TolCon.
2
Change in x and the mesh size are both less than the
specified tolerance, and the constraint violation is less than
TolCon.
3
Change in fval and the mesh size are both less than the
specified tolerance, and the constraint violation is less than
TolCon.
4
Magnitude of step smaller than machine precision and the
constraint violation is less than TolCon.
0
Maximum number of function evaluations or iterations
reached.
-1
Optimization terminated by an output function or plot
function.
-2
No feasible point found.
In the nonlinear constraint solver, the complementarity measure is the
norm of the vector whose elements are ciλi, where ci is the nonlinear
inequality constraint violation, and λi is the corresponding Lagrange
multiplier.
[x,fval,exitflag,output] = patternsearch(fun,x0, ...) returns
a structure output containing information about the search. The output
structure contains the following fields:
• function — Objective function.
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10-53
patternsearch
• problemtype — String describing the type of problem, one of:
-
'unconstrained'
'boundconstraints'
'linearconstraints'
'nonlinearconstr'
• pollmethod — Polling technique.
• searchmethod — Search technique used, if any.
• iterations — Total number of iterations.
• funccount — Total number of function evaluations.
• meshsize — Mesh size at x.
• maxconstraint — Maximum constraint violation, if any.
• rngstate — State of the MATLAB random number generator, just
before the algorithm started. You can use the values in rngstate
to reproduce the output when you use a random search method or
random poll method. See “Reproduce Results” on page 5-71, which
discusses the identical technique for ga.
• message — Reason why the algorithm terminated.
Note patternsearch does not accept functions whose inputs are of
type complex. To solve problems involving complex data, write your
functions so that they accept real vectors, by separating the real and
imaginary parts.
10-54
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patternsearch
Examples
Given the following constraints
⎡ 1 1⎤
⎡2⎤
⎢ −1 2⎥ ⎡ x1 ⎤ ≤ ⎢2⎥ ,
⎢
⎥ ⎢x ⎥ ⎢ ⎥
⎢⎣ 2 1 ⎥⎦ ⎣ 2 ⎦ ⎢⎣3⎥⎦
x1 ≥ 0, x2 ≥ 0,
the following code finds the minimum of the function, lincontest6,
that is provided with your software:
A = [1 1; -1 2; 2 1];
b = [2; 2; 3];
lb = zeros(2,1);
[x,fval,exitflag] = patternsearch(@lincontest6,[0 0],...
A,b,[],[],lb)
Optimization terminated: mesh size less than
options.TolMesh.
x =
0.6667
1.3333
fval =
-8.2222
exitflag =
1
References
[1] Audet, Charles and J. E. Dennis Jr. “Analysis of Generalized
Pattern Searches.” SIAM Journal on Optimization, Volume 13, Number
3, 2003, pp. 889–903.
[2] Conn, A. R., N. I. M. Gould, and Ph. L. Toint. “A Globally
Convergent Augmented Lagrangian Barrier Algorithm for Optimization
with General Inequality Constraints and Simple Bounds.” Mathematics
of Computation, Volume 66, Number 217, 1997, pp. 261–288.
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10-55
patternsearch
[3] Abramson, Mark A. Pattern Search Filter Algorithms for Mixed
Variable General Constrained Optimization Problems. Ph.D. Thesis,
Department of Computational and Applied Mathematics, Rice
University, August 2002.
[4] Abramson, Mark A., Charles Audet, J. E. Dennis, Jr., and Sebastien
Le Digabel. “ORTHOMADS: A deterministic MADS instance with
orthogonal directions.” SIAM Journal on Optimization, Volume 20,
Number 2, 2009, pp. 948–966.
[5] Kolda, Tamara G., Robert Michael Lewis, and Virginia Torczon.
“Optimization by direct search: new perspectives on some classical and
modern methods.” SIAM Review, Volume 45, Issue 3, 2003, pp. 385–482.
[6] Kolda, Tamara G., Robert Michael Lewis, and Virginia Torczon.
“A generating set direct search augmented Lagrangian algorithm for
optimization with a combination of general and linear constraints.”
Technical Report SAND2006-5315, Sandia National Laboratories,
August 2006.
[7] Lewis, Robert Michael, Anne Shepherd, and Virginia Torczon.
“Implementing generating set search methods for linearly constrained
minimization.” SIAM Journal on Scientific Computing, Volume 29,
Issue 6, 2007, pp. 2507–2530.
See Also
optimtool | psoptimset | ga
How To
• “Direct Search”
• “Optimization Problem Setup”
10-56
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psoptimget
Purpose
Obtain values of pattern search options structure
Syntax
val = psoptimget(options,'name')
val = psoptimget(options,'name',default)
Description
val = psoptimget(options,'name') returns the value of the
parameter name from the pattern search options structure options.
psoptimget(options,'name') returns an empty matrix [] if the value
of name is not specified in options. It is only necessary to type enough
leading characters of name to uniquely identify it. psoptimget ignores
case in parameter names.
val = psoptimget(options,'name',default) returns the value of the
parameter name from the pattern search options structure options, but
returns default if the parameter is not specified (as in []) in options.
Examples
val = psoptimget(opts,'TolX',1e-4);
returns val = 1e-4 if the TolX property is not specified in opts.
See Also
psoptimset | patternsearch
How To
• “Pattern Search Options” on page 9-9
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10-57
psoptimset
Purpose
Create pattern search options structure
Syntax
psoptimset
options = psoptimset
options = psoptimset(@patternsearch)
options = psoptimset('param1',value1,'param2',value2,...)
options = psoptimset(oldopts,'param1',value1,...)
options = psoptimset(oldopts,newopts)
Description
psoptimset with no input or output arguments displays a complete list
of parameters with their valid values.
options = psoptimset (with no input arguments) creates a
structure called options that contains the options, or parameters, for
patternsearch, and sets parameters to [], indicating patternsearch
uses the default values.
options = psoptimset(@patternsearch) creates a structure called
options that contains the default values for patternsearch.
options = psoptimset('param1',value1,'param2',value2,...)
creates a structure options and sets the value of 'param1' to value1,
'param2' to value2, and so on. Any unspecified parameters are set
to their default values. It is sufficient to type only enough leading
characters to define the parameter name uniquely. Case is ignored
for parameter names.
options = psoptimset(oldopts,'param1',value1,...) creates a
copy of oldopts, modifying the specified parameters with the specified
values.
options = psoptimset(oldopts,newopts) combines an existing
options structure, oldopts, with a new options structure, newopts.
Any parameters in newopts with nonempty values overwrite the
corresponding old parameters in oldopts.
Options
10-58
The following table lists the options you can set with psoptimset. See
“Pattern Search Options” on page 9-9 for a complete description of
the options and their values. Values in {} denote the default value.
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psoptimset
You can also view the optimization parameters and defaults by typing
psoptimset at the command line.
Option
Description
Values
Cache
With Cache set to 'on',
patternsearch keeps
a history of the mesh
points it polls and does
not poll points close to
them again at subsequent
iterations. Use this option
if patternsearch runs
slowly because it is taking
a long time to compute
the objective function. If
the objective function is
stochastic, it is advised not
to use this option.
'on' | {'off'}
CacheSize
Size of the history
Positive scalar | {1e4}
CacheTol
Positive scalar specifying
how close the current mesh
point must be to a point
in the history in order for
patternsearch to avoid
polling it. Use if 'Cache'
option is set to 'on'.
Positive scalar | {eps}
CompletePoll
Complete poll around
current iterate
'on' | {'off'}
CompleteSearch
Complete search around
current iterate when the
search method is a poll
method
'on' | {'off'}
Display
Level of display
'off' | 'iter' | 'diagnose' |
{'final'}
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10-59
psoptimset
Option
Description
Values
InitialMeshSize
Initial mesh size for
pattern algorithm
Positive scalar | {1.0}
InitialPenalty
Initial value of the penalty
parameter
Positive scalar | {10}
MaxFunEvals
Maximum number
of objective function
evaluations
Positive integer |
Maximum number of
iterations
Positive integer |
MaxMeshSize
Maximum mesh size used
in a poll/search step
Positive scalar | {Inf}
MeshAccelerator
Accelerate convergence
near a minimum
'on'| {'off'}
MeshContraction
Mesh contraction factor,
used when iteration is
unsuccessful
Positive scalar | {0.5}
MeshExpansion
Mesh expansion factor,
expands mesh when
iteration is successful
Positive scalar | {2.0}
MeshRotate
Rotate the pattern before
declaring a point to be
optimum
'off' | {'on'}
OutputFcns
Specifies a user-defined
function that an
optimization function
calls at each iteration
Function handle or cell array of function
handles | {[]}
PenaltyFactor
Penalty update parameter
Positive scalar| {100}
PlotFcns
Specifies plots of output
from the pattern search
@psplotbestf | @psplotmeshsize |
@psplotfuncount | @psplotbestx |
{[]}
MaxIter
10-60
{2000*numberOfVariables}
{100*numberOfVariables}
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psoptimset
Option
Description
Values
PlotInterval
Specifies that plot
functions will be called
at every interval
{1}
PollingOrder
Order of poll directions in
pattern search
'Random'| 'Success'|
{'Consecutive'}
PollMethod
Polling strategy used in
pattern search
{'GPSPositiveBasis2N'} |
'GPSPositiveBasisNp1'|
'GSSPositiveBasis2N'|
'GSSPositiveBasisNp1'|
'MADSPositiveBasis2N'|
'MADSPositiveBasisNp1'
ScaleMesh
Automatic scaling of
variables
{'on'} | 'off'
SearchMethod
Type of search used in
pattern search
@GPSPositiveBasis2N |
@GPSPositiveBasisNp1 |
@GSSPositiveBasis2N |
@GSSPositiveBasisNp1 |
@MADSPositiveBasis2N |
@MADSPositiveBasisNp1 | @searchga
| @searchlhs | @searchneldermead
| {[]}
TimeLimit
Total time (in seconds)
allowed for optimization
Positive scalar | {Inf}
TolBind
Binding tolerance
Positive scalar | {1e-3}
TolCon
Tolerance on constraints
Positive scalar | {1e-6}
TolFun
Tolerance on function,
stop if both the change
in function value and the
mesh size are less than
Positive scalar | {1e-6}
TolFun
TolMesh
Tolerance on mesh size
Positive scalar | {1e-6}
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psoptimset
Option
Description
Values
TolX
Tolerance on variable,
stop if both the change in
position and the mesh size
are less than TolX
Positive scalar | {1e-6}
UseParallel
Compute objective and
nonlinear constraint
functions in parallel, see
“Vectorize and Parallel
Options (User Function
Evaluation)” on page 9-25
and “How to Use Parallel
Processing” on page 8-12.
true | {false}
Vectorized
Specifies whether
functions are vectorized,
see “Vectorize and Parallel
Options (User Function
Evaluation)” on page
9-25 and “Vectorize the
Objective and Constraint
Functions” on page 4-85
'on' | {'off'}
See Also
10-62
patternsearch | psoptimget
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GlobalSearch.run
Purpose
Find global minimum
Syntax
x = run(gs,problem)
[x,fval] = run(gs,problem)
[x,fval,exitflag] = run(gs,problem)
[x,fval,exitflag,output] = run(gs,problem)
[x,fval,exitflag,output,solutions] = run(gs,problem)
Description
x = run(gs,problem) finds a point x that solves the optimization
problem described in the problem structure.
[x,fval] = run(gs,problem) returns the value of the objective
function in problem at the point x.
[x,fval,exitflag] = run(gs,problem) returns an exit flag
describing the results of the multiple local searches.
[x,fval,exitflag,output] = run(gs,problem) returns an output
structure describing the iterations of the run.
[x,fval,exitflag,output,solutions] = run(gs,problem) returns
a vector of solutions containing the distinct local minima found during
the run.
Input
Arguments
gs
A GlobalSearch object.
problem
Problem structure. Create problem with createOptimProblem
or by exporting a problem structure from the Optimization app.
problem must contain at least the following fields:
• solver
• objective
• x0
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GlobalSearch.run
• options — Both createOptimProblem and the Optimization
app always include an options field in the problem structure.
Output
Arguments
x
Minimizing point of the objective function.
fval
Objective function value at the minimizer x.
exitflag
Describes the results of the multiple local searches. Values are:
2
At least one local minimum found. Some runs of
the local solver converged.
1
At least one local minimum found. All runs of
the local solver converged.
0
No local minimum found. Local solver called at
least once, and at least one local solver exceeded
the MaxIter or MaxFunEvals tolerances.
-1
One or more local solver runs stopped by the
local solver output or plot function.
-2
No feasible local minimum found.
-5
MaxTime limit exceeded.
-8
No solution found. All runs had local solver exit
flag -2 or smaller, not all equal -2.
-10
Failures encountered in user-provided functions.
output
A structure describing the iterations of the run. Fields in the
structure:
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GlobalSearch.run
funcCount
Number of function evaluations.
localSolverIncomplete
Number of local solver runs with 0 exit flag.
localSolverNoSolution
Number of local solver runs with negative exit flag.
localSolverSuccess
Number of local solver runs with positive exit flag.
localSolverTotal
Total number of local solver runs.
message
Exit message.
solutions
A vector of GlobalOptimSolution objects containing the distinct
local solutions found during the run. The vector is sorted by
objective function value; the first element is best (smallest value).
The object contains:
Examples
X
Solution point returned by the local solver.
Fval
Objective function value at the solution.
Exitflag
Integer describing the result of the local solver
run.
Output
Output structure returned by the local solver.
X0
Cell array of start points that led to the solution.
Use a default GlobalSearch object to solve the six-hump camel back
problem (see “Run the Solver” on page 3-24):
gs = GlobalSearch;
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3]);
[xmin,fmin,flag,outpt,allmins] = run(gs,problem);
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GlobalSearch.run
Algorithms
A detailed description of the algorithm appears in “GlobalSearch
Algorithm” on page 3-54. Ugray et al. [1] describes both the algorithm
and the scatter-search method of generating trial points.
References
[1] Ugray, Zsolt, Leon Lasdon, John Plummer, Fred Glover, James
Kelly, and Rafael Martí. Scatter Search and Local NLP Solvers: A
Multistart Framework for Global Optimization. INFORMS Journal on
Computing, Vol. 19, No. 3, 2007, pp. 328–340.
See Also
GlobalSearch | createOptimProblem | GlobalOptimSolution
Tutorials
• “Run the Solver” on page 3-24
• “Optimization Workflow” on page 3-4
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MultiStart.run
Purpose
Run local solver from multiple points
Syntax
x = run(ms,problem,k)
x = run(ms,problem,startpts)
[x,fval] = run(...)
[x,fval,exitflag] = run(...)
[x,fval,exitflag,output] = run(...)
[x,fval,exitflag,output,solutions] = run(...)
Description
x = run(ms,problem,k) runs the ms MultiStart object on the
optimization problem specified in problem for a total of k runs. x is the
point where the lowest function value was found. For the lsqcurvefit
and lsqnonlin solvers, MultiStart minimizes the sum of squares at x,
also known as the residual.
x = run(ms,problem,startpts) runs the ms MultiStart object on
the optimization problem specified in problem using the start points
described in startpts.
[x,fval] = run(...) returns the lowest objective function value fval
at x. For the lsqcurvefit and lsqnonlin solvers, fval contains the
residual.
[x,fval,exitflag] = run(...) returns an exit flag describing the
return condition.
[x,fval,exitflag,output] = run(...) returns an output structure
containing statistics of the run.
[x,fval,exitflag,output,solutions] = run(...) returns a vector
of solutions containing the distinct local minima found during the run.
Input
Arguments
ms
MultiStart object.
problem
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MultiStart.run
Problem structure. Create problem with createOptimProblem
or by exporting a problem structure from the Optimization app.
problem must contain the following fields:
• solver
• objective
• x0
• options — Both createOptimProblem and the Optimization
app always include an options field in the problem structure.
k
Number of start points to run. MultiStart generates
k - 1 start points using the same algorithm as list for a
RandomStartPointSet object. MultiStart also uses the x0 point
from the problem structure.
startpts
A CustomStartPointSet or RandomStartPointSet object.
startpts can also be a cell array of these objects.
Output
Arguments
x
Point at which the objective function attained the lowest value
during the run. For lsqcurvefit and lsqnonlin, the objective
function is the sum of squares, also known as the residual.
fval
Smallest objective function value attained during the run. For
lsqcurvefit and lsqnonlin, the objective function is the sum of
squares, also known as the residual.
exitflag
Integer describing the return condition:
10-68
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MultiStart.run
2
At least one local minimum found. Some runs of
the local solver converged.
1
At least one local minimum found. All runs of
the local solver converged.
0
No local minimum found. Local solver called at
least once, and at least one local solver exceeded
the MaxIter or MaxFunEvals tolerances.
-1
One or more local solver runs stopped by the
local solver output or plot function.
-2
No feasible local minimum found.
-5
MaxTime limit exceeded.
-8
No solution found. All runs had local solver exit
flag -2 or smaller, not all equal -2.
-10
Failures encountered in user-provided functions.
output
Structure containing statistics of the optimization. Fields in the
structure:
funcCount
Number of function evaluations.
localSolverIncomplete
Number of local solver runs with 0 exit flag.
localSolverNoSolution
Number of local solver runs with negative exit flag.
localSolverSuccess
Number of local solver runs with positive exit flag.
localSolverTotal
Total number of local solver runs.
message
Exit message.
solutions
A vector of GlobalOptimSolution objects containing the distinct
local solutions found during the run. The vector is sorted by
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MultiStart.run
objective function value; the first element is best (smallest value).
The object contains:
Examples
X
Solution point returned by the local solver.
Fval
Objective function value at the solution.
Exitflag
Integer describing the result of the local solver
run.
Output
Output structure returned by the local solver.
X0
Cell array of start points that led to the solution.
Use a default MultiStart object to solve the six-hump camel back
problem (see “Run the Solver” on page 3-24):
ms = MultiStart;
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
+ x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
'objective',sixmin,'lb',[-3,-3],'ub',[3,3]);
[xmin,fmin,flag,outpt,allmins] = run(ms,problem,30);
Algorithms
A detailed description of the algorithm appears in “MultiStart
Algorithm” on page 3-59.
See Also
MultiStart | createOptimProblem | CustomStartPointSet |
RandomStartPointSet | GlobalOptimSolution
Tutorials
• “Run the Solver” on page 3-24
• “Optimization Workflow” on page 3-4
10-70
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RandomStartPointSet
Purpose
Random start points
Description
Describes how to generate a set of pseudorandom points for use with
MultiStart. A RandomStartPointSet object does not contain points. It
contains parameters used in generating the points when MultiStart
runs, or by the list method.
Construction
RS = RandomStartPointSet constructs a default RandomStartPointSet
object.
RS = RandomStartPointSet('PropertyName',PropertyValue,...)
constructs the object using options, specified as property name and
value pairs.
RS =
RandomStartPointSet(OLDRS,'PropertyName',PropertyValue,...)
creates a copy of the OLDRS RandomStartPointSet object, with the
named properties altered with the specified values.
Properties
ArtificialBound
Absolute value of default bounds to use for unbounded problems
(positive scalar).
If a component has no bounds, list uses a lower bound of
-ArtificialBound, and an upper bound of ArtificialBound.
If a component has a lower bound lb, but no upper bound, list
uses an upper bound of lb + 2*ArtificialBound. Similarly, if a
component has an upper bound ub, but no lower bound, list uses
a lower bound of ub - 2*ArtificialBound.
Default: 1000
NumStartPoints
Number of start points to generate (positive integer)
Default: 10
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RandomStartPointSet
Methods
list
Generate start points
Copy
Semantics
Value. To learn how value classes affect copy operations, see Copying
Objects in the MATLAB Programming Fundamentals documentation.
Examples
Create a RandomStartPointSet object for 40 points, and use list to
generate a point matrix for a seven-dimensional problem:
rs = RandomStartPointSet('NumStartPoints',40); % 40 points
problem = createOptimProblem('fminunc','x0',ones(7,1),...
'objective',@rosenbrock);
ptmatrix = list(rs,problem); % 'list' generates the matrix
See Also
MultiStart | CustomStartPointSet | list
Tutorials
• “RandomStartPointSet Object for Start Points” on page 3-21
How To
• Class Attributes
• Property Attributes
10-72
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saoptimget
Purpose
Values of simulated annealing options structure
Syntax
val = saoptimget(options, 'name')
val = saoptimget(options, 'name', default)
Description
val = saoptimget(options, 'name') returns the value of the
parameter name from the simulated annealing options structure
options. saoptimget(options, 'name') returns an empty matrix []
if the value of name is not specified in options. It is only necessary
to type enough leading characters of name to uniquely identify the
parameter. saoptimget ignores case in parameter names.
val = saoptimget(options, 'name', default) returns the 'name'
parameter, but returns the default value if the 'name' parameter is
not specified (or is []) in options.
Examples
opts = saoptimset('TolFun',1e-4);
val = saoptimget(opts,'TolFun');
returns val = 1e-4 for TolFun.
See Also
saoptimset | simulannealbnd
How To
• “Simulated Annealing Options” on page 9-58
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saoptimset
Purpose
Create simulated annealing options structure
Syntax
saoptimset
options = saoptimset
options = saoptimset('param1',value1,'param2',value2,...)
options = saoptimset(oldopts,'param1',value1,...)
options = saoptimset(oldopts,newopts)
options = saoptimset('simulannealbnd')
Description
saoptimset with no input or output arguments displays a complete list
of parameters with their valid values.
options = saoptimset (with no input arguments) creates a structure
called options that contains the options, or parameters, for the
simulated annealing algorithm, with all parameters set to [].
options = saoptimset('param1',value1,'param2',value2,...)
creates a structure options and sets the value of 'param1' to value1,
'param2' to value2, and so on. Any unspecified parameters are set to
[]. It is sufficient to type only enough leading characters to define the
parameter name uniquely. Case is ignored for parameter names. Note
that for string values, correct case and the complete string are required.
options = saoptimset(oldopts,'param1',value1,...) creates a
copy of oldopts, modifying the specified parameters with the specified
values.
options = saoptimset(oldopts,newopts) combines an existing
options structure, oldopts, with a new options structure, newopts.
Any parameters in newopts with nonempty values overwrite the
corresponding old parameters in oldopts.
options = saoptimset('simulannealbnd') creates an options
structure with all the parameter names and default values relevant to
'simulannealbnd'. For example,
saoptimset('simulannealbnd')
ans =
10-74
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saoptimset
AnnealingFcn:
TemperatureFcn:
AcceptanceFcn:
TolFun:
StallIterLimit:
MaxFunEvals:
TimeLimit:
MaxIter:
ObjectiveLimit:
Display:
DisplayInterval:
HybridFcn:
HybridInterval:
PlotFcns:
PlotInterval:
OutputFcns:
InitialTemperature:
ReannealInterval:
DataType:
Options
@annealingfast
@temperatureexp
@acceptancesa
1.0000e-006
'500*numberofvariables'
'3000*numberofvariables'
Inf
Inf
-Inf
'final'
10
[]
'end'
[]
1
[]
100
100
'double'
The following table lists the options you can set with saoptimset. See
“Simulated Annealing Options” on page 9-58 for a complete description
of these options and their values. Values in {} denote the default value.
You can also view the options parameters by typing saoptimset at
the command line.
Option
Description
Values
AcceptanceFcn
Handle to the function the
algorithm uses to determine
if a new point is accepted
Function handle
|{@acceptancesa}
AnnealingFcn
Handle to the function the
algorithm uses to generate
new points
Function handle |
@annealingboltz |
Type of decision variable
'custom' | {'double'}
DataType
{@annealingfast}
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10-75
saoptimset
Option
Description
Values
Display
Level of display
'off' | 'iter' | 'diagnose' |
{'final'}
DisplayInterval
Interval for iterative display
Positive integer | {10}
HybridFcn
Automatically run HybridFcn
(another optimization
function) during or at the end
of iterations of the solver
@fminsearch | @patternsearch
| @fminunc | @fmincon | {[]}
1-by-2 cell array | {@solver,
hybridoptions}, where solver
= fminsearch, patternsearch,
fminunc, or fmincon {[]}
Interval (if not 'end' or
'never') at which HybridFcn
is called
Positive integer | 'never' |
InitialTemperature
Initial value of temperature
Positive scalar |{100}
MaxFunEvals
Maximum number of
objective function evaluations
allowed
Positive integer |
MaxIter
Maximum number of
iterations allowed
Positive integer | {Inf}
ObjectiveLimit
Minimum objective function
value desired
Scalar | {-Inf}
OutputFcns
Function(s) get(s) iterative
data and can change options
at run time
Function handle or cell array of
function handles | {[]}
PlotFcns
Plot function(s) called during
iterations
Function handle or cell
array of function handles |
@saplotbestf | @saplotbestx |
@saplotf | @saplotstopping |
@saplottemperature | {[]}
HybridInterval
10-76
or
{'end'}
{3000*numberOfVariables}
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saoptimset
Option
Description
Values
PlotInterval
Plot functions are called at
every interval
Positive integer | {1}
ReannealInterval
Reannealing interval
Positive integer | {100}
StallIterLimit
Number of iterations over
which average change in
fitness function value at
current point is less than
Positive integer |
{500*numberOfVariables}
options.TolFun
Function used to update
temperature schedule
Function handle |
TimeLimit
The algorithm stops after
running for TimeLimit
seconds
Positive scalar | {Inf}
TolFun
Termination tolerance on
function value
Positive scalar | {1e-6}
TemperatureFcn
@temperatureboltz |
@temperaturefast |
{@temperatureexp}
See Also
saoptimget | simulannealbnd
How To
• “Simulated Annealing Options” on page 9-58
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10-77
simulannealbnd
Purpose
Find minimum of function using simulated annealing algorithm
Syntax
x = simulannealbnd(fun,x0)
x = simulannealbnd(fun,x0,lb,ub)
x = simulannealbnd(fun,x0,lb,ub,options)
x = simulannealbnd(problem)
[x,fval] = simulannealbnd(...)
[x,fval,exitflag] = simulannealbnd(...)
[x,fval,exitflag,output] = simulannealbnd(fun,...)
Description
x = simulannealbnd(fun,x0) starts at x0 and finds a local minimum
x to the objective function specified by the function handle fun. The
objective function accepts input x and returns a scalar function value
evaluated at x. x0 may be a scalar or a vector.
x = simulannealbnd(fun,x0,lb,ub) defines a set of lower and upper
bounds on the design variables, x, so that a solution is found in the
range lb ≤ x ≤ ub. Use empty matrices for lb and ub if no bounds exist.
Set lb(i) to -Inf if x(i) is unbounded below; set ub(i) to Inf if x(i)
is unbounded above.
x = simulannealbnd(fun,x0,lb,ub,options) minimizes with the
default optimization parameters replaced by values in the structure
options, which can be created using the saoptimset function. See the
saoptimset reference page for details.
x = simulannealbnd(problem) finds the minimum for problem, where
problem is a structure containing the following fields:
10-78
objective
Objective function
x0
Initial point of the search
lb
Lower bound on x
ub
Upper bound on x
rngstate
Optional field to reset the state of the
random number generator
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simulannealbnd
solver
'simulannealbnd'
options
Options structure created using saoptimset
Create the structure problem by exporting a problem from Optimization
app, as described in “Importing and Exporting Your Work” in the
Optimization Toolbox documentation.
[x,fval] = simulannealbnd(...) returns fval, the value of the
objective function at x.
[x,fval,exitflag] = simulannealbnd(...) returns exitflag,
an integer identifying the reason the algorithm terminated. The
following lists the values of exitflag and the corresponding reasons
the algorithm terminated:
• 1 — Average change in the value of the objective function over
options.StallIterLimit iterations is less than options.TolFun.
• 5 — options.ObjectiveLimit limit reached.
• 0 — Maximum number of function evaluations or iterations exceeded.
• -1 — Optimization terminated by an output function or plot function.
• -2 — No feasible point found.
• -5 — Time limit exceeded.
[x,fval,exitflag,output] = simulannealbnd(fun,...) returns
output, a structure that contains information about the problem and
the performance of the algorithm. The output structure contains the
following fields:
• problemtype — Type of problem: unconstrained or bound
constrained.
• iterations — The number of iterations computed.
• funccount — The number of evaluations of the objective function.
• message — The reason the algorithm terminated.
• temperature — Temperature when the solver terminated.
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10-79
simulannealbnd
• totaltime — Total time for the solver to run.
• rngstate — State of the MATLAB random number generator, just
before the algorithm started. You can use the values in rngstate
to reproduce the output of simulannealbnd. See “Reproduce Your
Results” on page 6-18.
Examples
Minimization of De Jong’s fifth function, a two-dimensional function
with many local minima. Enter the command dejong5fcn to generate
the following plot.
x0 = [0 0];
[x,fval] = simulannealbnd(@dejong5fcn,x0)
Optimization terminated: change in best function value
10-80
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simulannealbnd
less than options.TolFun.
x =
0.0392
-31.9700
fval =
2.9821
Minimization of De Jong’s fifth function subject to lower and upper
bounds:
x0 = [0 0];
lb = [-64 -64];
ub = [64 64];
[x,fval] = simulannealbnd(@dejong5fcn,x0,lb,ub)
Optimization terminated: change in best function value
less than options.TolFun.
x =
-31.9652
-32.0286
fval =
0.9980
The objective can also be an anonymous function:
fun = @(x) 3*sin(x(1))+exp(x(2));
x = simulannealbnd(fun,[1;1],[0 0])
Optimization terminated: change in best function value
less than options.TolFun.
x =
457.1045
0.0000
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10-81
simulannealbnd
Minimization of De Jong’s fifth function while displaying plots:
x0 = [0 0];
options = saoptimset('PlotFcns',{@saplotbestx,...
@saplotbestf,@saplotx,@saplotf});
simulannealbnd(@dejong5fcn,x0,[],[],options)
Optimization terminated: change in best function value
less than options.TolFun.
ans =
0.0230
-31.9806
The plots displayed are shown below.
10-82
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simulannealbnd
See Also
ga | patternsearch | saoptimget | saoptimset
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‫ﻣﺮﺟﻊ آﻣﻮزش ﺑﺮﻧﺎﻣﻪﻧﻮﯾﺴﯽ ﻣﺘﻠﺐ در اﯾﺮان‬
10-83
‫معرفی چند منبع در زمینه آموزش برنامه نویسی ‪ MATLAB‬یا متلب‬
‫کتاب های به زبان انگلیسی‬
‫عنوان‪Matlab, Third Edition: A Practical Introduction to :‬‬
‫‪Programming and Problem Solving‬‬
‫ترجمه عنوان‪ :‬متلب‪ :‬مقدمه ای عملی بر برنامه نويسی و حل مساله‪ ،‬چاپ سوم‬
‫مولفین‪Stormy Attaway :‬‬
‫سال چاپ‪2013 :‬‬
‫انتشارات‪Butterworth-Heinemann :‬‬
‫کتاب های به زبان فارسی‬
‫عنوان‪ :‬اصول و مبانی متلب برای علوم مهندسی‬
‫مولفین‪ :‬برايان هان‪ ،‬دانیل تی‪ ،‬والنتین‬
‫مترجمین‪ :‬رامین موالنا پور‪ ،‬سارا موالناپور‪ ،‬نینا اسدی پور‬
‫انتشارات‪ :‬سها دانش‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪MATLAB For Dummies :‬‬
‫ترجمه عنوان‪ :‬تلب به زبان ساده‬
‫مولفین‪Jim Sizemore, John Paul Mueller :‬‬
‫سال چاپ‪2014 :‬‬
‫انتشارات‪For Dummies :‬‬
‫عنوان‪ :‬کاربرد ‪ MATLAB‬در علوم مهندسی‬
‫مولفین‪ :‬حیدرعلی شايانفر‪ ،‬حسین شايقی‬
‫انتشارات‪ :‬ياوريان‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Essential MATLAB for Engineers and Scientists :‬‬
‫عنوان‪ :‬برنامه نويسی ‪ MATLAB‬برای مهندسان‬
‫ترجمه عنوان‪ :‬آنچه بايد مهندسین و دانشمندان از متلب بدانند‬
‫مولفین‪ :‬محمود کشاورز مهر‪ ،‬بهزاد عبدی‬
‫مولفین‪Brian Hahn, Daniel Valentine:‬‬
‫سال چاپ‪2013 :‬‬
‫انتشارات‪Academic Press :‬‬
‫انتشارات‪ :‬نوپردازان‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪MATLAB: An Introduction with Applications :‬‬
‫عنوان‪ :‬آموزش کاربردی مباحث پیشرفته با ‪MATLAB‬‬
‫ترجمه عنوان‪ :‬مقدمه ای بر متلب و کاربردهای آن‬
‫مولفین‪ :‬نیما جمشیدی‪ ،‬علی ابويی مهريزی‪ ،‬رسول مواليی‬
‫مولف‪Amos Gilat :‬‬
‫انتشارات‪ :‬عابد‬
‫سال چاپ‪2014 :‬‬
‫انتشارات‪Wiley :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪MATLAB For Beginners: A Gentle Approach:‬‬
‫عنوان‪ :‬کاملترين مرجع آموزشی و کاربردی ‪MATLAB‬‬
‫ترجمه عنوان‪ :‬متلب برای افراد مبتدی با يک رويکرد تدريجی‬
‫مولفین‪ :‬علی اکبر علمداری‪ ،‬نسرين علمداری‬
‫مولف‪Peter I. Kattan:‬‬
‫انتشارات‪ :‬نگارنده دانش‬
‫سال چاپ‪2008 :‬‬
‫انتشارات‪CreateSpace Independent Publishing Platform :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪MATLAB for Engineers :‬‬
‫عنوان‪ :‬برنامه نويسی ‪ MATLAB‬برای مهندسین‬
‫ترجمه عنوان‪ :‬متلب برای مهندسین‬
‫مولف‪ :‬استفن چاپمن‬
‫مولف‪Holly Moore :‬‬
‫سال چاپ‪2011 :‬‬
‫انتشارات‪Prentice Hall :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Mastering MATLAB :‬‬
‫ترجمه عنوان‪ :‬تسلط بر متلب‬
‫مولفین‪Duane C. Hanselman, Bruce L. Littlefield :‬‬
‫سال چاپ‪2011 :‬‬
‫انتشارات‪Prentice Hall :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫مترجم‪ :‬سعدان زکائی‬
‫انتشارات‪ :‬دانشگاه صنعتی خواجه نصیرالدين طوسی‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪ :‬آموزش گام به گام محاسبات عددی با متلب‬
‫مولف‪ :‬کلیو مولر‬
‫مترجم‪ :‬رسول نصیری‬
‫انتشارات‪ :‬نشر گستر‬
‫لینک دسترسی‪ :‬لینک‬
‫منابع آموزشی آنالین‬
‫عنوان‪ :‬مجموعه فرادرسهای برنامهنويسی متلب‬
‫مدرس‪ :‬دکتر سید مصطفی کالمی هريس‬
‫مدت زمان‪ ۹ :‬ساعت و ‪ ۳‬دقیقه‬
‫زبان‪ :‬فارسی‬
‫ارائه دهنده‪ :‬فرادرس‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪ :‬مجموعه فرادرسهای متلب برای علوم و مهندسی‬
‫مدرس‪ :‬دکتر سید مصطفی کالمی هريس‬
‫مدت زمان‪ 14 :‬ساعت و ‪ 2۲‬دقیقه‬
‫زبان‪ :‬فارسی‬
‫ارائه دهنده‪ :‬فرادرس‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪ :‬مجموعه فرادرسهای برنامه نويسی متلب پیشرفته‬
‫مدرس‪ :‬دکتر سید مصطفی کالمی هريس‬
‫مدت زمان‪ ۲ :‬ساعت و ‪ 12‬دقیقه‬
‫زبان‪ :‬فارسی‬
‫ارائه دهنده‪ :‬فرادرس‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Introduction to Programming with MATLAB :‬‬
‫ترجمه عنوان‪ :‬آشنايی با برنامهنويسی متلب‬
‫مدرسین‪Akos Ledeczi, Michael Fitzpatrick, Robert Tairas :‬‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪Vanderbilt University :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Introduction to MATLAB :‬‬
‫ترجمه عنوان‪ :‬مقدمهای بر متلب‬
‫مدرس‪Danilo Šćepanović :‬‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪MIT OCW :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Up and Running with MATLAB :‬‬
‫ترجمه عنوان‪ :‬شروع سريع کار با متلب‬
‫مدرس‪Patrick Royal :‬‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪lynda.com :‬‬
‫لینک دسترسی‪ :‬لینک‬
‫عنوان‪Modelling and Simulation using MATLAB :‬‬
‫ترجمه عنوان‪ :‬مدلسازی و شبیهسازی با استفاده از متلب‬
‫مدرسین‪ Prof. Dr.-Ing. Georg Fries :‬و دیگران‬
‫زبان‪ :‬انگلیسی‬
‫ارائه دهنده‪iversity.org :‬‬
‫لینک دسترسی‪ :‬لینک‬
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