gpkit Documentation
gpkit Documentation
Release 0.5.3
MIT Department of Aeronautics and Astronautics
Aug 28, 2017
Contents
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What are Signomials / Signomial Programs?
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Bleeding-edge / developer installations
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Creating Monomials and Posynomials
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Sensitivities and dual variables
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6 Visualization and Interaction
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10.1 iPython Notebook Examples
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10.3 Maximizing the Volume of a Box
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11.6 gpkit.modified_ctypesgen module
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11.7 gpkit.repr_conventions module
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11.8 gpkit.small_classes module
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11.9 gpkit.small_scripts module
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GPkit is a Python package for defining and manipulating geometric programming (GP) models.
Our hopes are to bring the mathematics of Geometric Programming into the engineering design process in a disciplined and collaborative way, and to encourage research with and on GPs by providing an easily extensible object-oriented framework.
GPkit abstracts away the backend solver so that users can work directly with engineering equations and optimization concepts. Supported solvers are MOSEK and CVXOPT .
Join our mailing list and/or chatroom for support and examples.
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2 Contents
CHAPTER
1
Geometric Programming 101
What is a GP?
A Geometric Program (GP) is a type of non-linear optimization problem whose objective and constraints have a particular form.
The decision variables must be strictly positive (non-zero, non-negative) quantities. This is a good fit for engineering design equations (which are often constructed to have only positive quantities), but any model with variables of unknown sign (such as forces and velocities without a predefined direction) may be difficult to express in a GP. Such models might be better expressed as
More precisely, GP objectives and inequalities are formed out of monomials and posynomials. In the context of GP, a monomial is defined as: 𝑓 (𝑥) = 𝑐𝑥 𝑎
1
1 𝑥 𝑎
2
2
...𝑥 𝑎 𝑛 𝑛 where 𝑐 is a positive constant, 𝑥
1..𝑛 are decision variables, and 𝑎 taking 𝑥, 𝑦 and 𝑧 to be positive variables, the expressions
1..𝑛 are real exponents. For example,
7𝑥 4𝑥𝑦
2 𝑧
2𝑥 𝑦
2 𝑧
0.3
√︀
2𝑥𝑦 are all monomials. Building on this, a posynomial is defined as a sum of monomials: 𝑔(𝑥) =
𝐾
∑︁ 𝑐 𝑘 𝑥 𝑎
1
1 𝑘 𝑥 𝑎
2
2 𝑘
...𝑥 𝑎 𝑛 𝑛 𝑘 𝑘=1
For example, the expressions 𝑥
2
+ 2𝑥𝑦 + 1 7𝑥𝑦 + 0.4(𝑦𝑧)
−1/3
0.56 + 𝑥
0.7
𝑦𝑧 are all posynomials. Alternatively, monomials can be defined as the subset of posynomials having only one term. Using 𝑓 𝑖 to represent a monomial and 𝑔 𝑖 to represent a posynomial, a GP in standard form is
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written as: minimize 𝑔
0
(𝑥) subject to 𝑓 𝑖
(𝑥) = 1, 𝑖 = 1, ...., 𝑚 𝑔 𝑖
(𝑥) ≤ 1, 𝑖 = 1, ...., 𝑛
Boyd et. al. give the following example of a GP in standard form: minimize 𝑥
−1 𝑦
−1/2 𝑧
−1 subject to (1/3)𝑥
−2 𝑦
−2
+ 2.3𝑥𝑧 + 4𝑥𝑦𝑧
+ (4/3)𝑦
1/2 𝑧
−1
≤ 1 𝑥 + 2𝑦 + 3𝑧 ≤ 1
(1/2)𝑥𝑦 = 1
Why are GPs special?
Geometric programs have several powerful properties:
1. Unlike most non-linear optimization problems, large GPs can be solved extremely quickly.
2. If there exists an optimal solution to a GP, it is guaranteed to be globally optimal.
3. Modern GP solvers require no initial guesses or tuning of solver parameters.
These properties arise because GPs become convex optimization problems via a logarithmic transformation. In addition to their mathematical benefits, recent research has shown that many practical problems can be formulated as GPs or closely approximated as GPs.
What are Signomials / Signomial Programs?
When the coefficients in a posynomial are allowed to be negative (but the variables stay strictly positive), that is called a Signomial.
A Signomial Program has signomial constraints. While they cannot be solved as quickly or to global optima, because they build on the structure of a GP they can often be solved more quickly than a generic nonlinear program. More information can be found under
Where can I learn more?
To learn more about GPs, refer to the following resources:
• A tutorial on geometric programming , by S. Boyd, S.J. Kim, L. Vandenberghe, and A. Hassibi.
• Convex optimization , by S. Boyd and L. Vandenberghe.
• Geometric Programming for Aircraft Design Optimization , Hoburg, Abbeel 2014
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CHAPTER
2
GPkit Overview
GPkit is a Python package for defining and manipulating geometric programming (GP) models, abstracting away the backend solver.
Our hopes are to bring the mathematics of Geometric Programming into the engineering design process in a disciplined and collaborative way, and to encourage research with and on GPs by providing an easily extensible object-oriented framework.
Symbolic expressions
GPkit is a limited symbolic algebra language, allowing only for the creation of geometric program compatible equations (or signomial program compatible ones, if signomial programming is enabled). As mentioned in
Geometric Programming 101 , one can view monomials as posynomials with a single term,
and posynomials as signomials that have only positive coefficients. The inheritance structure of these objects in GPkit follows this mathematical basis.
Substitution
The Varkey object in the graph above is not a algebraic expression, but what GPkit uses as a variable’s
“name”. It carries the LaTeX representation of a variable and its units, as well as any other information the user wishes to associate with a variable. The use of VarKeys as opposed to numeric indexing is an important part of the GPkit framework, because it allows a user to keep variable information local and modular.
GPkit keeps its internal representation of objects entirely symbolic until it solves. This means that any expression or Model object can replace any instance of a variable (as represented by a VarKey) with a number, new VarKey, or even an entire Monomial at any time with the .sub() method.
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Model objects
In GPkit, a Model object represents a symbolic problem declaration. That problem may be either GPcompatible or SP-compatible. To avoid confusion, calling the solve() method on a model will either attempt to solve it for a global optimum (if it’s a GP) or return an error immediately (if it’s an SP). Similarly, calling localsolve() will either start the process of SP-solving (stepping through a sequence of
GP-approximations) or return an error for GP-compatible Models. This framework is illustrated below.
2.3. Model objects 7
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CHAPTER
3
Installation Instructions
If you encounter bugs during installation, please email [email protected] or raise a GitHub issue .
Installation dependencies
To install GPkit, you’ll need to have the following python packages already installed on your system:
• pip
• numpy version 1.8.1 or newer
• scipy
• pint and at least one solver, which we’ll choose and install in a later step.
There are many ways to install these dependencies, but here’s our suggestion:
Get pip
Mac OS X Run easy_install pip at a terminal window.
Linux
Use your package manager to install pip Ubuntu: python-pip
Windows Install the Python 2.7 64-bit version of Anaconda .
Get python packages
Mac OS X
Run the following commands: sudo apt-get install
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• pip install pip --upgrade
• pip install numpy
• pip install scipy
• pip install pint
Linux
Use your package manager to install numpy and scipy Ubuntu: install python-numpy python-scipy
Run pip install pint (for system python installs, use sudo pip)
Windows Do nothing at this step; Anaconda already has the needed packages.
sudo apt-get
Install a GP solver
GPkit interfaces with two off the shelf solvers: cvxopt, and mosek. Cvxopt is open source; mosek requires a commercial licence or (free) academic license.
At least one solver is required.
Installing cvxopt
Mac OSX Run pip install cvxopt
Linux Run sudo apt-get install libblas-dev liblapack-dev libsuitesparse-dev or otherwise install those libraries
Run pip install cvxopt (for system python installs, use sudo pip)
If experiencing issues with wheel in Ubuntu 16.04, try the official installer.
Windows Run conda install -c omnia cvxopt in an Anaconda Command Prompt.
Installing mosek
Dependency note: GPkit uses the python package ctypesgen to interface with the MOSEK C bindings.
Licensing note: if you do not have a paid license, you will need an academic or trial license to proceed.
Mac OS X
• If which gcc does not return anything, install XCode and the Apple Command Line Tools .
• Install ctypesgen with pip install ctypesgen --pre.
• Download MOSEK 7 , then:
– Move the mosek folder to your home directory
– Follow these steps for Mac .
– Request an academic license file and put it in ~/mosek/
Linux
• Install ctypesgen with pip install ctypesgen --pre (for system python installs, use sudo pip
)
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• Download MOSEK , then:
– Move the mosek folder to your home directory
– Follow these steps for Linux .
– Request an academic license file and put it in ~/mosek/
Windows
• Install ctypesgen by running pip install ctypesgen --pre in an Anaconda Command Prompt .
• Download MOSEK , then:
– Follow these steps for Windows .
– Request an academic license
C:\Users\(your_username)\mosek\
– Make sure gcc is on your system path.
file and put it in
* To do this, type gcc into a command prompt.
* If you get executable not found, then install the 64-bit version (x86_64 installer architecture dropdown option) of mingw .
* Make sure the mingw bin directory is on your system path (you may have to add it manually).
Install GPkit
• Run pip install gpkit at the command line (for system python installs, use sudo pip)
• Run pip install jupyter to install jupyter notebook (recommended)
• Run jupyter nbextension enable --py widgetsnbextension for interactive control of models in jupyter (recommended)
• Run python -c "import gpkit.tests; gpkit.tests.run()" to run the tests; if any tests do not pass, please email [email protected] or raise a GitHub issue .
• Join our mailing list and/or chatroom for support and examples.
Debugging installation
You may need to rebuild GPkit if any of the following occur:
• You install a new solver (mosek or cvxopt) after installing GPkit
• You delete the .gpkit folder from your home directory
• You see Could not load settings file. when importing GPkit, or
• Could not load MOSEK library:
ImportError('$HOME/.gpkit/ expopt.so not found.')
To rebuild GPkit, first try running python -c "from gpkit.build import rebuild; rebuild()". If that doesn’t work then try the following:
• Run pip uninstall gpkit
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• Run pip install --no-cache-dir --no-deps gpkit
• Run python -c "import gpkit.tests; gpkit.tests.run()"
• If any tests fail, please email [email protected] or raise a GitHub issue .
Bleeding-edge / developer installations
Active developers may wish to install the latest GPkit directly from the source code on Github. To do so,
1. Run pip uninstall gpkit to uninstall your existing GPkit.
2. Run git clone https://github.com/hoburg/gpkit.git to clone the GPkit repository.
3. Run pip install -e gpkit to install that directory as your environment-wide GPkit.
4. Run cd ..; python -c "import gpkit.tests; gpkit.tests.run()" to test your installation from a non-local directory.
12 Chapter 3. Installation Instructions
CHAPTER
4
Getting Started
GPkit is a Python package, so we assume basic familiarity with Python: if you’re new to Python we recommend you take a look at Learn Python .
Otherwise,
and import away:
from gpkit import
Variable, VectorVariable, Model
Declaring Variables
Instances of the Variable class represent scalar variables. They create a VarKey to store the variable’s name, units, a description, and value (if the Variable is to be held constant), as well as other metadata.
Free Variables
# Declare a variable, x x = Variable( "x" )
# Declare a variable, y, with units of meters y = Variable( "y" , "m" )
# Declare a variable, z, with units of meters, and a description z = Variable( "z" , "m" , "A variable called z with units of meters" )
Fixed Variables
To declare a variable with a constant value, use the Variable class, as above, but put a number before the units:
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# Declare \rho equal to 1.225 kg/m^3.
# NOTE: in python string literals, backslashes must be doubled rho = Variable( "\\rho" , 1.225
, "kg/m^3" , "Density of air at sea level" )
In the example above, the key name "\rho" is for LaTeX printing (described later). The unit and description arguments are optional.
#Declare pi equal to 3.14
pi = Variable( "\\pi" , 3.14
)
Vector Variables
Vector variables are represented by the VectorVariable class. The first argument is the length of the vector. All other inputs follow those of the Variable class.
# Declare a 3-element vector variable "x" with units of "m" x = VectorVariable( 3 , "x" , "m" , "Cube corner coordinates" ) x_min = VectorVariable( 3 , "x" , [ 1 , 2 , 3 ], "m" , "Cube corner minimum" )
Creating Monomials and Posynomials
Monomial and posynomial expressions can be created using mathematical operations on variables.
# create a Monomial term xy^2/z x = Variable( "x" ) y = Variable( "y" ) z = Variable( "z" ) m = x
* y
**
2 / z type (m) # gpkit.nomials.Monomial
# create a Posynomial expression x + xy^2 x = Variable( "x" ) y = Variable( "y" ) p = x + x
* y
**
2 type (p) # gpkit.nomials.Posynomial
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Declaring Constraints
Constraint objects represent constraints of the form Monomial >= Posynomial or Monomial
== Monomial
(which are the forms required for GP-compatibility).
Note that constraints must be formed using <=, >=, or == operators, not < or >.
# consider a block with dimensions x, y, z less than 1
# constrain surface area less than 1.0 m^2 x = Variable( "x" , "m" ) y = Variable( "y" , "m" ) z = Variable( "z" , "m" )
S = Variable( "S" , 1.0
, "m^2" ) c = ( 2
* x
* y + 2
* x
* z + 2
* y
* z <= S) type (c) # gpkit.nomials.PosynomialInequality
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Formulating a Model
The Model class represents an optimization problem. To create one, pass an objective and list of Constraints.
By convention, the objective is the function to be minimized. If you wish to maximize a function, take its reciprocal. For example, the code below creates an objective which, when minimized, will maximize x*y*z.
objective = 1 / (x
* y
* z) constraints = [ 2
* x
* y + 2
* x
* z + 2
* y
* z <= S, x >= 2
* y] m = Model(objective, constraints)
Solving the Model
When solving the model you can change the level of information that gets printed to the screen with the verbosity setting. A verbosity of 1 (the default) prints warnings and timing; a verbosity of 2 prints solver output, and a verbosity of 0 prints nothing.
sol = m .
solve(verbosity = 0 )
Printing Results
The solution object can represent itself as a table:
sol .
table()
Cost
----
15.59
[ 1 / m
**
3 ]
Free Variables
-------------x : 0.5774
[m] y : 0.2887
[m] z : 0.3849
[m]
Constants
---------
S : 1 [m
**
2 ]
Sensitivities
-------------
S : 1.5
We can also print the optimal value and solved variables individually.
"The optimal value is %s ." % sol[ "cost" ]
"The x dimension is %s ." % sol(x)
"The y dimension is %s ." % sol[ "variables" ][ "y" ]
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The optimal value
is
15.5884619886
.
The x dimension
is
0.5774
meter .
The y dimension
is
0.2887
meter .
Sensitivities and dual variables
When a GP is solved, the solver returns not just the optimal value for the problem’s variables (known as the “primal solution”) but also the effect that relaxing each constraint would have on the overall objective
(the “dual solution”).
From the dual solution GPkit computes the sensitivities for every fixed variable in the problem. This can be quite useful for seeing which constraints are most crucial, and prioritizing remodeling and assumptionchecking.
Using variable sensitivities
Fixed variable sensitivities can be accessed
["sensitivities"]["constants"] dict, as in this example: from
import gpkit
x = gpkit .
Variable( "x" ) x_min = gpkit .
Variable( "x_{min}" , 2 ) sol = gpkit .
Model(x, [x_min <= x]) .
solve()
assert
sol[ "sensitivities" ][ "constants" ][x_min] == 1 a SolutionArray’s
These sensitivities are actually log derivatives ( 𝑑log(𝑦) 𝑑log(𝑥)
); whereas a regular derivative is a tangent line, these are tangent monomials, so the 1 above indicates that x_min has a linear relation with the objective.
This is confirmed by a further example:
import gpkit
x = gpkit .
Variable( "x" ) x_squared_min = gpkit .
Variable( "x^2_{min}" , 2 ) sol = gpkit .
Model(x, [x_squared_min <= x
**
2 ]) .
solve()
assert
sol[ "sensitivities" ][ "constants" ][x_squared_min] == 2
16 Chapter 4. Getting Started
CHAPTER
5
Debugging Models
A number of errors and warnings may be raised when attempting to solve a model. A model may be primal infeasible: there is no possible solution that satisfies all constraints. A model may be dual infeasible: the optimal value of one or more variables is 0 or infinity (negative and positive infinity in logspace).
For a GP model that does not solve, solvers may be able to prove its primal or dual infeasibility, or may return an unknown status.
GPkit contains several tools for diagnosing which constraints and variables might be causing infeasibility.
The first thing to do with a model m that won’t solve is to run m.debug(), which will search for changes that would make the model feasible:
"Debug examples"
from gpkit import
Variable, Model x = Variable( "x" , "ft" ) x_min = Variable( "x_min" , 2 , "ft" ) x_max = Variable( "x_max" , 1 , "ft" ) y = Variable( "y" , "volts" ) m = Model(x / y, [x <= x_max, x >= x_min]) m .
debug(verbosity = 0 )
> Trying to solve with bounded variables and relaxed constants
Solves with these variables bounded: value near upper bound: [y] sensitive to upper bound: [y] and these constants relaxed: x_min [ft]: relaxed from 2 to 1
> ...success!
> Trying to solve with relaxed constraints
> ...does not solve with relaxed constraints.
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Note that certain modeling errors (such as omitting or forgetting a constraint) may be difficult to diagnose from this output.
Potential errors and warnings
• RuntimeWarning:
final status of solver 'mosek' was 'DUAL_INFEAS_CER', not 'optimal’
– The solver found a certificate of dual infeasibility: the optimal value of one or more variables is 0 or infinity. See Dual Infeasibility below for debugging advice.
• RuntimeWarning:
final status of solver 'mosek' was 'PRIM_INFEAS_CER', not 'optimal’
– The solver found a certificate of primal infeasibility: no possible solution satisfies all constraints. See Primal Infeasibility below for debugging advice.
• RuntimeWarning:
final status of solver 'cvxopt' was 'unknown', not 'optimal’
or RuntimeWarning:
final status of solver 'mosek' was ‘UNKNOWN’, not 'optimal’.
– The solver could not solve the model or find a certificate of infeasibility. This may indicate a dual infeasible model, a primal infeasible model, or other numerical issues. Try debugging with the techniques in Dual and Primal Infeasibility below.
• RuntimeWarning:
Primal solution violates constraint: 1.0000149786 is greater than 1
– this warning indicates that the solver-returned solution violates a constraint of the model, likely because the solver’s tolerance for a final solution exceeds GPkit’s tolerance during solution checking. This is sometimes seen in dual infeasible models, see Dual Infeasibility below. If you run into this, please note on this GitHub issue your solver and operating system.
• RuntimeWarning:
Dual cost nan does not match primal cost 1.00122315152
– Similarly to the above, this warning may be seen in dual infeasible models, see Dual
Infeasibility below.
Dual Infeasibility
In some cases a model will not solve because the optimal value of one or more variables is 0 or infinity
(negative or positive infinity in logspace). Such a problem is dual infeasible because the GP’s dual problem, which determines the optimal values of the sensitivites, does not have any feasible solution. If the solver can prove that the dual is infeasible, it will return a dual infeasibility certificate. Otherwise, it may finish with a solution status of unknown.
gpkit.constraints.bounded.Bounded
is a simple tool that can be used to detect unbounded variables and get dual infeasible models to solve by adding extremely large upper bounds and extremely small lower bounds to all variables in a ConstraintSet.
When a model with a Bounded ConstraintSet is solved, it checks whether any variables slid off to the bounds, notes this in the solution dictionary and prints a warning (if verbosity is greater than 0).
For example, Mosek returns DUAL_INFEAS_CER when attempting to solve the following model:
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"Demonstrate a trivial unbounded variable"
from gpkit import
Variable, Model
from gpkit.constraints.bounded
import
Bounded x = Variable( "x" ) constraints = [x >= 1 ] m = Model( 1 / x, constraints) # MOSEK returns DUAL_INFEAS_CER on .solve() m = Model( 1 / x, Bounded(constraints))
# by default, prints bounds warning during solve sol = m .
solve(verbosity = 0 ) print sol .
summary() print "sol['boundedness'] is:" , sol[ "boundedness" ]
Upon viewing the printed output,
Solves
with
these variables bounded: value near upper bound: [x] sensitive to upper bound: [x]
Cost
----
1e-30
Free Variables
-------------x : 1e+30 sol[ 'boundedness' ]
is
: { 'value near upper bound' : array([x], dtype = object ),
˓→
'sensitive to upper bound' : array([x], dtype = object )}
The problem, unsurprisingly, is that the cost 1/x has no lower bound because x has no upper bound.
For details read the Bounded docstring.
Primal Infeasibility
A model is primal infeasible when there is no possible solution that satisfies all constraints. A simple example is presented below.
"A simple primal infeasible example"
from gpkit import
Variable, Model x = Variable( "x" ) y = Variable( "y" ) m = Model(x
* y, [ x >= 1 , y >= 2 , x
* y >= 0.5
, x
* y <= 1.5
])
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20
# m.solve() # raises uknown on cvxopt
# and PRIM_INFEAS_CER on mosek
It is not possible for x*y to be less than 1.5 while x is greater than 1 and y is greater than 2.
A common bug in large models that use substitutions is to substitute overly constraining values in for variables that make the model primal infeasible. An example of this is given below.
"Another simple primal infeasible example"
from gpkit import
Variable, Model
#Make the necessary Variables x = Variable( "x" ) y = Variable( "y" , 2 )
#make the constraints constraints = [ x >= 1 ,
0.5
<= x
* y, x
* y <= 1.5
]
#declare the objective objective = x
* y
#construct the model m = Model(objective, constraints)
#solve the model
#raises RuntimeWarning uknown on cvxopt and RuntimeWarning
#PRIM_INFES_CER with mosek
#m.solve()
Since y is now set to 2 and x can be no less than 1, it is again impossible for x*y to be less than 1.5 and the model is primal infeasible. If y was instead set to 1, the model would be feasible and the cost would be 1.
Relaxation
If you suspect your model is primal infeasible, you can find the nearest primal feasible version of it by relaxing constraints: either relaxing all constraints by the smallest number possible (that is, dividing the less-than side of every constraint by the same number), relaxing each constraint by its own number and minimizing the product of those numbers, or changing each constant by the smallest total percentage possible.
"Relaxation examples"
from gpkit import
Variable, Model x = Variable( "x" ) x_min = Variable( "x_min" , 2 ) x_max = Variable( "x_max" , 1 ) m = Model(x, [x <= x_max, x >= x_min]) print "Original model" print "==============" print m print
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# m.solve() # raises a RuntimeWarning!
print "With constraints relaxed equally" print "================================"
from gpkit.constraints.relax
import
ConstraintsRelaxedEqually allrelaxed = ConstraintsRelaxedEqually(m) mr1 = Model(allrelaxed .
relaxvar, allrelaxed) print mr1 print mr1 .
solve(verbosity = 0 ) .
table() # solves with an x of 1.414
print print "With constraints relaxed individually" print "====================================="
from gpkit.constraints.relax
import
ConstraintsRelaxed constraintsrelaxed = ConstraintsRelaxed(m) mr2 = Model(constraintsrelaxed .
relaxvars .
prod()
* m .
cost
**
0.01
,
# add a bit of the original cost in constraintsrelaxed) print mr2 print mr2 .
solve(verbosity = 0 ) .
table() # solves with an x of 1.0
print print "With constants relaxed individually" print "==================================="
from gpkit.constraints.relax
import
ConstantsRelaxed constantsrelaxed = ConstantsRelaxed(m) mr3 = Model(constantsrelaxed .
relaxvars .
prod()
* m .
cost
**
0.01
,
# add a bit of the original cost in constantsrelaxed) print mr3 print mr3 .
solve(verbosity = 0 ) .
table() # brings x_min down to 1.0
Original model
==============
# minimize x
# subject to x <= x_max x >= x_min
With constraints relaxed equally
================================
# minimize
C_Relax
# subject to
C_Relax >= x
* x_max
**-
1
C_Relax >= x
**-
1
* x_min
C_Relax >= 1
Cost
----
1.414
Free Variables
--------------
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22
x : 1.414
| Relax
C : 1.414
Constants
--------x_max : 1 x_min : 2
Sensitivities
------------x_min : + 0.5
x_max : 0.5
With constraints relaxed individually
=====================================
# minimize
C_Relax .
1 _( 0 ,)
*
C_Relax .
1 _( 1 ,)
* x
**
0.01
# subject to
C_Relax .
1 >= [gpkit .
Monomial(x
* x_max
**-
1 ), gpkit .
Monomial(x
**-
1
* x_
˓→ min)]
C_Relax .
1 >= 1
Cost
----
2
Free Variables
-------------x : 1
| Relax .
1
C : [ 1
Constants
--------x_max : 1 x_min : 2
Sensitivities
------------x_min : + 1 x_max : 0.99
2 ]
With constants relaxed individually
===================================
# minimize x
**
0.01
* x_max_Relax .
2
* x_min_Relax .
2
# subject to x <= x_max x >= x_min x_min_Relax .
2 >= 1 x_min >= x_min_Relax .
2
**-
1
* x_min_{before}_Relax .
2
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x_min <= x_min_Relax .
2
* x_min_{before}_Relax .
2 x_max_Relax .
2 >= 1 x_max >= x_max_Relax .
2
**-
1
* x_max_{before}_Relax .
2 x_max <= x_max_Relax .
2
* x_max_{before}_Relax .
2
Cost
----
2
Free Variables
-------------x : 1 x_max : 1 x_min : 1
| Relax .
2 x_max : 1 x_min : 2
Constants
--------x_max_{before} : 1 x_min_{before} : 2
Sensitivities
------------x_min : + 1 x_max : 0.99
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6
Visualization and Interaction
Interactive Control Panel
A model can be manipulated and visualized in Jupyter Notebook by calling model.controlpanel().
By default this creates a slider for every constant in the model and gives them automatic ranges, but variables and/or ranges can be changed in the Settings tab or specified in the first argument to controlpanel()
.
Besides the default behaviour shown above, the control panel can also display custom analyses and plots via the fn_of_sol argument, which accepts a function (or list of functions) that take the solution as their input.
Plotting a 1D Sweep
Methods exist to facilitate creating, solving, and plotting the results of a single-variable sweep (see
for details). Example usage is as follows:
"Demonstrates manual and auto sweeping and plotting"
import matplotlib as mpl
mpl .
use( 'Agg' )
# comment out the lines above to show figures in a window
import numpy as np from gpkit import
Model, Variable, units x = Variable( "x" , "m" , "Swept Variable" ) y = Variable( "y" , "m^2" , "Cost" ) m = Model(y, [y >= (x / 2 )
**-
0.5
* units .
m
**
2.5
+ 1
* units .
m
**
2 , y >= (x / 2 )
**
2 ])
# arguments are: model, swept: values, posnomial for y-axis sol = m .
sweep({x: np .
linspace( 1 , 3 , 20 )}, verbosity = 0 )
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f, ax = sol .
plot(y) ax .
set_title( "Manually swept (20 points)" ) f .
show() f .
savefig( "plot_sweep1d.png" )
# arguments are: model, swept: (min, max, optional logtol), posnomial for y-
˓→ axis sol = m .
autosweep({x: ( 1 , 3 )}, tol = 0.001
, verbosity = 0 ) f, ax = sol .
plot(y) ax .
set_title( "Autoswept (7 points)\nGuaranteed to be in blue region" ) f .
show() f .
savefig( "plot_autosweep1d.png" )
Which results in:
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28 Chapter 6. Visualization and Interaction
CHAPTER
7
Building Complex Models
Inheriting from Model
GPkit encourages an object-oriented modeling approach, where the modeler creates objects that inherit from Model to break large systems down into subsystems and analysis domains. The benefits of this approach include modularity, reusability, and the ability to more closely follow mental models of system hierarchy. For example: two different models for a simple beam, designed by different modelers, should be able to be used interchangeably inside another subsystem (such as an aircraft wing) without either modeler having to write specifically with that use in mind.
When you create a class that inherits from Model, write a .setup() method to create the model’s variables and return its constraints. GPkit.Model.__init__ will call that method and automatically add your model’s name and unique ID to any created variables.
Variables created in a setup method are added to the model even if they are not present in any constraints.
This allows for simplistic ‘template’ models, which assume constant values for parameters and can grow incrementally in complexity as those variables are freed.
At the end of this page a detailed example shows this technique in practice.
Accessing Variables in Models
GPkit provides several ways to access a Variable in a Model (or ConstraintSet):
• using Model.variables_byname(key). This returns all Variables in the Model, as well as in any submodels, that match the key.
• using Model.topvar(key). This returns the top-level Variable that matches the key. The
Variable must appear at the top level, not in a submodel.
• using Model.__getitem__. Model[key] returns the only variable matching the key, even if the match occurs in a submodel. If multiple variables match the key, an error is raised.
These methods are illustrated in the following example.
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"Demo of accessing variables in models"
from gpkit import
Model, Variable
class Battery
(Model):
"A simple battery"
def
setup ( self ): h = Variable( "h" , 200 , "Wh/kg" , "specific energy" )
E = Variable( "E" , "MJ" , "stored energy" ) m = Variable( "m" , "lb" , "battery mass" )
return
[E <= m
* h]
class Motor
(Model):
"Electric motor"
def
setup ( self ): m = Variable( "m" , "lb" , "motor mass" ) f = Variable( "f" , 20 , "lb/hp" , "mass per unit power" )
Pmax = Variable( "P_ {max} " , "hp" , "max output power" )
return
[m >= f
*
Pmax]
class PowerSystem
(Model):
"A battery powering a motor"
def
setup ( self ): components = [Battery(), Motor()] m = Variable( "m" , "lb" , "mass" )
return
[components, m >= sum (comp .
topvar( "m" )
for
comp
in
components)]
PS = PowerSystem() print "Getting the only var 'E': " , PS[ "E" ] print "The top-level var 'm': " , PS .
topvar( "m" ) print "All the variables 'm': " , PS .
variables_byname( "m" )
Getting the only var 'E' : E_PowerSystem / Battery [MJ]
The top level var 'm' : m_PowerSystem [lb]
All the variables 'm' : [gpkit .
Variable(m_PowerSystem [lb]), gpkit .
˓→
Variable(m_PowerSystem / Battery [lb]), gpkit .
Variable(m_PowerSystem / Motor
˓→
[lb])]
Vectorization
gpkit.Vectorize
creates an environment in which Variables are created with an additional dimension:
"from gpkit/tests/t_vars.py"
def
test_shapes ( self ):
with
gpkit .
Vectorize( 3 ):
with
gpkit .
Vectorize( 5 ): y = gpkit .
Variable( "y" ) x = gpkit .
VectorVariable( 2 , "x" ) z = gpkit .
VectorVariable( 7 , "z" )
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self .
assertEqual(y .
shape, ( 5 , 3 )) self .
assertEqual(x .
shape, ( 2 , 5 , 3 )) self .
assertEqual(z .
shape, ( 7 , 3 ))
This allows models written with scalar constraints to be created with vector constraints:
"Vectorization demonstration"
from gpkit import
Model, Variable, Vectorize
class Test
(Model):
"A simple scalar model"
def
setup ( self ): x = Variable( "x" )
return
[x >= 1 ] print "SCALAR" m = Test() m .
cost = m[ "x" ] print m .
solve(verbosity = 0 ) .
summary() print "__________\n" print "VECTORIZED"
with
Vectorize( 3 ): m = Test() m .
cost = m[ "x" ] .
prod() m .
append(m[ "x" ][ 1 ] >= 2 ) print m .
solve(verbosity = 0 ) .
summary()
SCALAR
Cost
----
1
Free Variables
-------------x : 1
__________
VECTORIZED
Cost
----
2
Free Variables
-------------x : [ 1 2 1 ]
Multipoint analysis modeling
In many engineering models, there is a physical object that is operated in multiple conditions. Some variables correspond to the design of the object (size, weight, construction) while others are vectorized over
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the different conditions (speed, temperature, altitude). By combining named models and vectorization we can create intuitive representations of these systems while maintaining modularity and interoperability.
In the example below, the models Aircraft and Wing have a .dynamic() method which creates instances of AircraftPerformance and WingAero, respectively. The Aircraft and Wing models create variables, such as size and weight without fuel, that represent a physical object. The dynamic models create properties that change based on the flight conditions, such as drag and fuel weight.
This means that when an aircraft is being optimized for a mission, you can create the aircraft (AC in this example) and then pass it to a Mission model which can create vectorized aircraft performance models for each flight segment and/or flight condition.
"""Modular aircraft concept"""
import numpy as np from gpkit import
Model, Variable, Vectorize
class Aircraft
(Model):
"The vehicle model"
def
setup ( self ): self .
fuse = Fuselage() self .
wing = Wing() self .
components = [ self .
fuse, self .
wing]
W = Variable( "W" , "lbf" , "weight" )
return
self .
components, [
W >= sum (c .
topvar( "W" )
for
c
in
self .
components)
]
def
dynamic ( self , state):
"This component's performance model for a given state."
return
AircraftP( self , state)
class AircraftP
(Model):
"Aircraft flight physics: weight <= lift, fuel burn"
def
setup ( self , aircraft, state): self .
aircraft = aircraft self .
wing_aero = aircraft .
wing .
dynamic(state) self .
perf_models = [ self .
wing_aero]
Wfuel = Variable( "W_ {fuel} " , "lbf" , "fuel weight" )
Wburn = Variable( "W_ {burn} " , "lbf" , "segment fuel burn" )
return
self .
perf_models, [ aircraft .
topvar( "W" ) + Wfuel <= ( 0.5
* state[ "\\rho" ]
* state[ "V" ]
**
2
* self .
wing_aero[ "C_L" ]
* aircraft .
wing[ "S" ]),
Wburn >= 0.1
* self .
wing_aero[ "D" ]
]
class FlightState
(Model):
"Context for evaluating flight physics"
def
setup ( self ):
Variable( "V" , 40 , "knots" , "true airspeed" )
Variable( "\\mu" , 1.628e-5 , "N*s/m^2" , "dynamic viscosity" )
Variable( "\\rho" , 0.74
, "kg/m^3" , "air density" )
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class FlightSegment
(Model):
"Combines a context (flight state) and a component (the aircraft)"
def
setup ( self , aircraft): self .
flightstate = FlightState() self .
aircraftp = aircraft .
dynamic( self .
flightstate)
return
self .
flightstate, self .
aircraftp
class Mission
(Model):
"A sequence of flight segments"
def
setup ( self , aircraft):
with
Vectorize( 4 ): # four flight segments self .
fs = FlightSegment(aircraft)
Wburn = self .
fs .
aircraftp[ "W_ {burn} " ]
Wfuel = self .
fs .
aircraftp[ "W_ {fuel} " ] self .
takeoff_fuel = Wfuel[ 0 ]
return
self .
fs, [Wfuel[: 1 ] >= Wfuel[ 1 :] + Wburn[: 1 ],
Wfuel[ 1 ] >= Wburn[ 1 ]]
class Wing
(Model):
"Aircraft wing model"
def
dynamic ( self , state):
"Returns this component's performance model for a given state."
return
WingAero( self , state)
def
setup ( self ):
W = Variable( "W" , "lbf" , "weight" )
S = Variable( "S" , 190 , "ft^2" , "surface area" ) rho = Variable( "\\rho" , 1 , "lbf/ft^2" , "areal density" )
A = Variable( "A" , 27 , "-" , "aspect ratio" ) c = Variable( "c" , "ft" , "mean chord" )
return
[W >= S
* rho, c == (S / A)
**
0.5
]
class WingAero
(Model):
"Wing aerodynamics"
def
setup ( self , wing, state):
CD = Variable( "C_D" , "-" , "drag coefficient" )
CL = Variable( "C_L" , "-" , "lift coefficient" ) e = Variable( "e" , 0.9
, "-" , "Oswald efficiency" )
Re = Variable( "Re" , "-" , "Reynold's number" )
D = Variable( "D" , "lbf" , "drag force" )
return
[
CD >= ( 0.074
/ Re
**
0.2
+ CL
**
2 / np .
pi / wing[ "A" ] / e),
Re == state[ "\\rho" ]
* state[ "V" ]
* wing[ "c" ] / state[ "\\mu" ],
D >= 0.5
* state[ "\\rho" ]
* state[ "V" ]
**
2
*
CD
* wing[ "S" ],
]
class Fuselage
(Model):
"The thing that carries the fuel, engine, and payload"
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34 def
setup ( self ):
# fuselage needs an external dynamic drag model,
# left as an exercise for the reader
# V = Variable("V", 16, "gal", "volume")
# d = Variable("d", 12, "in", "diameter")
# S = Variable("S", "ft^2", "wetted area")
# cd = Variable("c_d", .0047, "-", "drag coefficient")
# CDA = Variable("CDA", "ft^2", "drag area")
Variable( "W" , 100 , "lbf" , "weight" )
AC = Aircraft()
MISSION = Mission(AC)
M = Model(MISSION .
takeoff_fuel, [MISSION, AC]) sol = M .
solve(verbosity = 0 ) vars_of_interest = set (AC .
varkeys) vars_of_interest .
update(MISSION .
fs .
aircraftp .
unique_varkeys) vars_of_interest .
add( "D" ) print sol .
summary(vars_of_interest)
Note that the output table can be filtered with a list of variables to show.
Cost
----
1.943
[lbf]
Free Variables
--------------
| Aircraft
W : 290
| Aircraft / Wing
W : 190 c : 2.653
[lbf] weight
[lbf] weight
[ft] mean chord
W_{burn} : [
˓→ burn
| Mission / FlightSegment / AircraftP
0.487
W_{fuel} : [ 1.94
0.486
1.46
0.485
0.97
0.485
0.485
]
]
[lbf] segment fuel
[lbf] fuel weight
| Mission / FlightSegment / AircraftP / WingAero
D : [ 4.87
4.86
4.85
4.85
] [lbf] drag force
Sensitivities
-------------
| Aircraft / Fuselage
W : + 0.25
weight
| Aircraft / Wing
S : + 0.68
\rho : + 0.48
A : 0.31
surface area areal density aspect ratio
Next Largest Sensitivities
--------------------------
| Mission / FlightSegment / AircraftP / WingAero e : [ 0.093
0.092
0.092
0.092
] Oswald efficiency
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| Mission / FlightSegment / FlightState
V : [ + 0.1
+ 0.1
+ 0.1
+ 0.1
\rho : [ + 0.034
+ 0.034
+ 0.035
+ 0.035
] true airspeed
] air density
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8
Advanced Commands
Derived Variables
Evaluated Fixed Variables
Some fixed variables may be derived from the values of other fixed variables. For example, air density, viscosity, and temperature are functions of altitude. These can be represented by a substitution or value that is a one-argument function accepting model.substitutions (for details, see
below).
# code from t_GPSubs.test_calcconst in tests/t_sub.py
x = Variable( "x" , "hours" ) t_day = Variable( "t_{day}" , 12 , "hours" ) t_night = Variable( "t_{night}" ,
lambda
c: 24 c[t_day], "hours" )
# note that t_night has a function as its value m = Model(x, [x >= t_day, x >= t_night]) sol = m .
solve(verbosity = 0 ) self .
assertAlmostEqual(sol(t_night) / gpkit .
ureg .
hours, 12 ) m .
substitutions .
update({t_day: ( "sweep" , [ 8 , 12 , 16 ])}) sol = m .
solve(verbosity = 0 ) self .
assertEqual( len (sol[ "cost" ]), 3 ) npt .
assert_allclose(sol(t_day) + sol(t_night), 24 )
Evaluated Free Variables
Some free variables may be evaluated from the values of other (non-evaluated) free variables after the optimization is performed. For example, if the efficiency 𝜈 of a motor is not a GP-compatible variable, but (1 − 𝜈) is a valid GP variable, then 𝜈 can be calculated after solving. These evaluated free variables can be represented by a Variable with evalfn metadata. Note that this variable should not be used in constructing your model!
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# code from t_constraints.test_evalfn in tests/t_sub.py
x = Variable( "x" ) x2 = Variable( "x^2" , evalfn =
lambda
v: v[x]
**
2 ) m = Model(x, [x >= 2 ]) m .
unique_varkeys = set ([x2 .
key]) sol = m .
solve(verbosity = 0 ) self .
assertAlmostEqual(sol(x2), sol(x)
**
2 )
For evaluated variables that can be used during a solution, see externalfn under
Sweeps
Sweeps are useful for analyzing tradeoff surfaces.
A sweep “value” is an Iterable of numbers, e.g. [1, 2, 3]. The simplest way to sweep a model is to call model.sweep({sweepvar: sweepvalues}) , which will return a solution array but not change the model’s substitutions dictionary. If multiple sweepvars are given, the method will run them all as independent one-dimensional sweeps and return a list of one solution per sweep. The method model.autosweep({sweepvar:
(start, end)}, tol=0.01) behaves very similarly, except that only the bounds of the sweep need be specified and the region in betwen will be swept to a maximum possible error of tol in the log of the cost. For details see
below.
Sweep Substitutions
Alternatively, or to sweep a higher-dimensional grid, Variables can swept with a substitution value takes the form ('sweep', Iterable), such as ('sweep', np.linspace(1e6, 1e7, 100)).
During variable declaration, giving an Iterable value for a Variable is assumed to be giving it a sweep value: for example, x = Variable("x", [1, 2, 3]) will sweep x over three values.
Vector variables may also be substituted for: y = VectorVariable(3, "y", ("sweep" ,
[[1, 2], [1, 2], [1, 2]]) will sweep 𝑦 ∀ 𝑦 𝑖
∈ {1, 2}.
A Model with sweep substitutions will solve for all possible combinations: e.g., if there’s a variable x with value ('sweep', [1, 3]) and a variable y with value ('sweep', [14, 17]) then the gp will be solved four times, for (𝑥, 𝑦) ∈ {(1, 14), (1, 17), (3, 14), (3, 17)}. The returned solutions will be a one-dimensional array (or 2-D for vector variables), accessed in the usual way.
Parallel Sweeps
During a normal sweep, each result is independent, so they can be run in parallel. To use this feature, run
$ ipcluster start at a terminal: it will automatically start a number of iPython parallel computing engines equal to the number of cores on your machine, and when you next import gpkit you should see a note like Using parallel execution of sweeps on 4 clients. If you do, then all sweeps performed with that import of gpkit will be parallelized.
This parallelization sets the stage for gpkit solves to be outsourced to a server, which may be valuable for faster results; alternately, it could allow the use of gpkit without installing a solver.
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1D Autosweeps
If you’re only sweeping over a single variable, autosweeping lets you specify a tolerance for cost error instead of a number of exact positions to solve at. GPkit will then search the sweep segment for a locally optimal number of sweeps that can guarantee a max absolute error on the log of the cost.
Accessing variable and cost values from an autosweep is slightly different, as can be seen in this example:
"Show autosweep_1d functionality"
import numpy as np import gpkit from gpkit import
units, Variable, Model
from gpkit.tools.autosweep
import
autosweep_1d
from gpkit.small_scripts
import
mag
A = Variable( "A" , "m**2" ) l = Variable( "l" , "m" ) m1 = Model(A
**
2 , [A >= l
**
2 + units .
m
**
2 ]) tol1 = 1e-3 bst1 = autosweep_1d(m1, tol1, l, [ 1 , 10 ], verbosity = 0 ) print "Solved after %2i passes, cost logtol +/%.3g
" % (bst1 .
nsols, bst1 .
tol)
# autosweep solution accessing l_vals = np .
linspace( 1 , 10 , 10 ) sol1 = bst1 .
sample_at(l_vals) print "values of l:" , l_vals print "values of A:" , sol1( "A" ) cost_estimate = sol1[ "cost" ] cost_lb, cost_ub = sol1 .
cost_lb(), sol1 .
cost_ub() print "cost lower bound:" , cost_lb print "cost estimate: " , cost_estimate print "cost upper bound:" , cost_ub
# you can evaluate arbitrary posynomials np .
testing .
assert_allclose(mag( 2
* sol1(A)), mag(sol1( 2
*
A)))
assert
(sol1[ "cost" ] == sol1(A
**
2 )) .
all()
# the cost estimate is the logspace mean of its upper and lower bounds np .
testing .
assert_allclose((np .
log(mag(cost_lb)) + np .
log(mag(cost_ub))) / 2 , np .
log(mag(cost_estimate)))
# this problem is two intersecting lines in logspace m2 = Model(A
**
2 , [A >= (l / 3 )
**
2 , A >= (l / 3 )
**
0.5
* units .
m
**
1.5
]) tol2 = { "mosek" : 1e-12 , "cvxopt" : 1e-7 ,
"mosek_cli" : 1e-6 }[gpkit .
settings[ "default_solver" ]] bst2 = autosweep_1d(m2, tol2, l, [ 1 , 10 ], verbosity = 0 ) print "Solved after %2i passes, cost logtol +/%.3g
" % (bst2 .
nsols, bst2 .
tol) print "Table of solutions used in the autosweep:" print bst2 .
solarray .
table()
If you need access to the raw solutions arrays, the smallest simplex tree containing any given point can be gotten with min_bst = bst.min_bst(val), the extents of that tree with bst.bounds and solutions of that tree with bst.sols. More information is in help(bst).
Tight ConstraintSets
Tight ConstraintSets will warn if any inequalities they contain are not tight (that is, the right side does not equal the left side) after solving. This is useful when you know that a constraint _should_ be tight for a
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given model, but reprenting it as an equality would be non-convex.
from gpkit import
Variable, Model
from gpkit.constraints.tight
import
Tight
Tight .
reltol = 1e-2 # set the global tolerance of Tight x = Variable( 'x' ) x_min = Variable( 'x_{min}' , 2 ) m = Model(x, [Tight([x >= 1 ], reltol = 1e-3 ), # set the specific tolerance x >= x_min]) m .
solve(verbosity = 0 ) # prints warning
Substitutions
Substitutions are a general-purpose way to change every instance of one variable into either a number or another variable.
Substituting into Posynomials, NomialArrays, and GPs
The examples below all use Posynomials and NomialArrays, but the syntax is identical for GPs (except when it comes to sweep variables).
# adapted from t_sub.py / t_NomialSubs / test_Basic
from gpkit import
Variable x = Variable( "x" ) p = x
**
2
assert
p .
sub(x, 3 ) == 9
assert
p .
sub(x .
varkeys[ "x" ], 3 ) == 9
assert
p .
sub( "x" , 3 ) == 9
Here the variable x is being replaced with 3 in three ways: first by substituting for x directly, then by substituting for the VarKey("x"), then by substituting the string “x”. In all cases the substitution is understood as being with the VarKey: when a variable is passed in the VarKey is pulled out of it, and when a string is passed in it is used as an argument to the Posynomial’s varkeys dictionary.
Substituting multiple values
# adapted from t_sub.py / t_NomialSubs / test_Vector
from gpkit import
Variable, VectorVariable x = Variable( "x" ) y = Variable( "y" ) z = VectorVariable( 2 , "z" ) p = x
* y
* z
assert
all (p .
sub({x: 1 , "y" : 2 }) == 2
* z)
assert
all (p .
sub({x: 1 , y: 2 , "z" : [ 1 , 2 ]}) == z .
sub(z, [ 2 , 4 ]))
To substitute in multiple variables, pass them in as a dictionary where the keys are what will be replaced and values are what it will be replaced with. Note that you can also substitute for VectorVariables by their name or by their NomialArray.
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Substituting with nonnumeric values
You can also substitute in sweep variables (see
Sweeps ), strings, and monomials:
# adapted from t_sub.py / t_NomialSubs
from gpkit import
Variable
from gpkit.small_scripts
import
mag x = Variable( "x" , "m" ) xvk = x .
varkeys .
values()[ 0 ] descr_before = x .
exp .
keys()[ 0 ] .
descr y = Variable( "y" , "km" ) yvk = y .
varkeys .
values()[ 0 ]
for
x_
in
[ "x" , xvk, x]:
for
y_
in
[ "y" , yvk, y]:
if not
isinstance (y_, str )
and
type (xvk .
units) != str : expected = 0.001
else
: expected = 1.0
assert
abs (expected mag(x .
sub(x_, y_) .
c)) < 1e-6
if
type (xvk .
units) != str :
# this means units are enabled z = Variable( "z" , "s" )
# y.sub(y, z) will raise ValueError due to unit mismatch
Note that units are preserved, and that the value can be either a string (in which case it just renames the variable), a varkey (in which case it changes its description, including the name) or a Monomial (in which case it substitutes for the variable with a new monomial).
Updating ConstraintSet substitutions
ConstraintSets have a .substitutions KeyDict attribute which will be substituted before solving.
This KeyDict accepts variable names, VarKeys, and Variable objects as keys, and can be updated (or deleted from) like a regular Python dictionary to change the substitutions that will be used at solvetime. If a ConstraintSet itself contains ConstraintSets, it and all its elements share pointers to the same substitutions dictionary object, so that updating any one of them will update all of them.
Substituting with replacement
Any of the substitutions above can be run with p.subinplace(*args) to substitute directly into the object in question.
Fixed Variables
When a Model is created, any fixed Variables are used to form a dictionary:
{var: var.
descr["value"] for var in self.varlocs if "value" in var.descr}
. This dictionary in then substituted into the Model’s cost and constraints before the substitutions argument is (and hence values are supplanted by any later substitutions).
solution.subinto(p) will substitute the solution(s) for variables into the posynomial p, returning a NomialArray. For a non-swept solution, this is equivalent to p.sub(solution["variables"]).
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You can also substitute by just calling the solution, i.e. solution(p). This returns a numpy array of just the coefficients (c) of the posynomial after substitution, and will raise a‘ ValueError` if some of the variables in p were not found in solution.
Freeing Fixed Variables
After creating a Model, it may be useful to “free” a fixed variable and resolve. This can be done using the command del m.substitutions["x"], where m is a Model. An example of how to do this is shown below.
from gpkit import
Variable, Model x = Variable( "x" ) y = Variable( "y" , 3 ) # fix value to 3 m = Model(x, [x >= 1 + y, y >= 1 ])
_ = m .
solve() # optimal cost is 4; y appears in sol["constants"]
del
m .
substitutions[ "y" ]
_ = m .
solve() # optimal cost is 2; y appears in Free Variables
Note that del m.substitutions["y"] affects m but not y.key. y.value will still be 3, and if y is used in a new model, it will still carry the value of 3.
42
Composite Objectives
Given 𝑛 posynomial objectives 𝑔 𝑖
, you can sweep out the problem’s Pareto frontier with the composite objective: 𝑔
0 𝑤
0
∏︀ 𝑖̸=0 𝑣 𝑖
+ 𝑔
1 𝑤
1
∏︀ 𝑖̸=1 𝑣 𝑖
+ ... + 𝑔 𝑛
∏︀ 𝑖 𝑣 𝑖 where 𝑖 ∈ 0...𝑛 − 1 and 𝑣 𝑖
= 1 − 𝑤 𝑖 and 𝑤 𝑖
∈ [0, 1]
GPkit has the helper function composite_objective for constructing these.
import numpy as np import gpkit
L, W = gpkit .
Variable( "L" ), gpkit .
Variable( "W" ) eqns = [L >= 1 , W >= 1 , L
*
W == 10 ] co_sweep = [ 0 ] + np .
logspace( 6 , 0 , 10 ) .
tolist() obj = gpkit .
tools .
composite_objective(L + W, W
**-
1
*
L
**-
3 , normsub = {L: 10 , W: 10 }, sweep = co_sweep) m = gpkit .
Model(obj, eqns) m .
solve()
The normsub argument specifies an expected value for your solution to normalize the different 𝑔 𝑖
(you can also do this by hand). The feasibility of the problem should not depend on the normalization, but the spacing of the sweep will.
The sweep argument specifies what points between 0 and 1 you wish to sample the weights at. If you want different resolutions or spacings for different weights, the sweeps argument accepts a list of sweep arrays.
Chapter 8. Advanced Commands
CHAPTER
9
Signomial Programming
Signomial programming finds a local solution to a problem of the form: minimize 𝑔
0
(𝑥) subject to 𝑓 𝑖
(𝑥) = 1, 𝑖 = 1, ...., 𝑚 𝑔 𝑖
(𝑥) − ℎ 𝑖
(𝑥) ≤ 1, 𝑖 = 1, ...., 𝑛 where each 𝑓 is monomial while each 𝑔 and ℎ is a posynomial.
This requires multiple solutions of geometric programs, and so will take longer to solve than an equivalent geometric programming formulation.
In general, when given the choice of which variables to include in the positive-posynomial / 𝑔 side of the constraint, the modeler should:
1. maximize the number of variables in 𝑔,
2. prioritize variables that are in the objective,
3. then prioritize variables that are present in other constraints.
The .localsolve syntax was chosen to emphasize that signomial programming returns a local optimum. For the same reason, calling .solve on an SP will raise an error.
By default, signomial programs are first solved conservatively (by assuming each ℎ is equal only to its constant portion) and then become less conservative on each iteration.
Example Usage
"""Adapted from t_SP in tests/t_geometric_program.py"""
import gpkit
# Decision variables x = gpkit .
Variable( 'x' ) y = gpkit .
Variable( 'y' )
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44
# must enable signomials for subtraction
with
gpkit .
SignomialsEnabled(): constraints = [x >= 1 y, y <= 0.1
]
# create and solve the SP m = gpkit .
Model(x, constraints) print m .
localsolve(verbosity = 0 ) .
summary()
assert
abs (m .
solution(x) 0.9
) < 1e-6
When using the localsolve method, the reltol argument specifies the relative tolerance of the solver: that is, by what percent does the solution have to improve between iterations? If any iteration improves less than that amount, the solver stops and returns its value.
If you wish to start the local optimization at a particular point 𝑥 𝑘
, however, you may do so by putting that position (a dictionary formatted as you would a substitution) as the xk argument.
Sequential Geometric Programs
The method of solving local GP approximations of a non-GP compatible model can be generalized, at the cost of the general smoothness and lack of a need for trust regions that SPs guarantee.
For some applications, it is useful to call external codes which may not be GP compatible. Imagine we wished to solve the following optimization problem: minimize 𝑦 subject to 𝑦 ≥ sin(𝑥) 𝜋
4
≤ 𝑥 ≤ 𝜋
2
This problem is not GP compatible due to the sin(x) constraint. One approach might be to take the first term of the Taylor expansion of sin(x) and attempt to solve:
"Can be found in gpkit/docs/source/examples/sin_approx_example.py"
import numpy as np from gpkit import
Variable, Model x = Variable( "x" ) y = Variable( "y" ) objective = y constraints = [y >= x, x <= np .
pi / 2.
,
] x >= np .
pi / 4.
, m = Model(objective, constraints) print m .
solve(verbosity = 0 ) .
summary()
Cost
----
0.7854
Free Variables
--------------
Chapter 9. Signomial Programming
gpkit Documentation, Release 0.5.3
x : 0.7854
y : 0.7854
We can do better, however, by utilizing some built in functionality of GPkit. For simple cases with a single
Variable, GPkit looks for externalfn metadata:
"Can be found in gpkit/docs/source/examples/external_sp2.py"
import numpy as np from gpkit import
Variable, Model x = Variable( "x" )
def
y_ext ( self , x0):
"Returns constraints on y derived from x0"
if
x
not in
x0:
return
self >= x
return
self >= x / x0[x]
* np .
sin(x0[x]) y = Variable( "y" , externalfn = y_ext) m = Model(y, [np .
pi / 4 <= x, x <= np .
pi / 2 ]) print m .
localsolve(verbosity = 0 ) .
summary()
Cost
----
0.7854
Free Variables
-------------x : 0.7854
y : 0.7854
However, for external functions not intrinsically tied to a single variable it’s best to use the full ConstraintSet API, as follows:
Assume we have some external code which is capable of evaluating our incompatible function:
"""External function for GPkit to call.
Can be found in gpkit/docs/source/examples/external_function.py"""
import numpy as np def
external_code (x):
"Returns sin(x)"
return
np .
sin(x)
Now, we can create a ConstraintSet that allows GPkit to treat the incompatible constraint as though it were a signomial programming constraint:
"Can be found in gpkit/docs/source/examples/external_constraint.py"
from gpkit.exceptions
import
InvalidGPConstraint
from external_function import
external_code
class ExternalConstraint
( object ):
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"Class for external calling" varkeys = {}
def
__init__ ( self , x, y):
# We need a GPkit variable defined to return in our constraint.
The
# easiest way to do this is to read in the parameters of interest in
# the initiation of the class and store them here.
self .
x = x self .
y = y
def
as_posyslt1 ( self , _):
"Ensures this is treated as an SGP constraint"
raise
InvalidGPConstraint( "ExternalConstraint cannot solve as a GP." )
def
as_gpconstr ( self , x0, _):
"Returns locally-approximating GP constraint"
# Unpacking the GPkit variables x = self .
x y = self .
y
# Creating a default constraint for the first solve
if not
x0:
return
(y >= x)
# Returns constraint updated with new call to the external code
else
:
# Unpack Design Variables at the current point x_star = x0[ "x" ]
# Call external code res = external_code(x_star)
# Return linearized constraint
return
(y >= res
* x / x_star) and replace the incompatible constraint in our GP:
"Can be found in gpkit/docs/source/examples/external_sp.py"
import numpy as np from gpkit import
Variable, Model
from external_constraint import
ExternalConstraint x = Variable( "x" ) y = Variable( "y" ) objective = y constraints = [ExternalConstraint(x, y), x <= np .
pi / 2.
,
] x >= np .
pi / 4.
, m = Model(objective, constraints) print m .
localsolve(verbosity = 0 ) .
summary()
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Cost
----
0.7854
Free Variables
-------------x : 0.7854
y : 0.7854
which is the expected result. This method has been generalized to larger problems, such as calling XFOIL and AVL.
If you wish to start the local optimization at a particular point 𝑥
0
, however, you may do so by putting that position (a dictionary formatted as you would a substitution) as the x0 argument
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48 Chapter 9. Signomial Programming
CHAPTER
10
Examples
iPython Notebook Examples
More examples, including some with in-depth explanations and interactive visualizations, can be seen on nbviewer .
A Trivial GP
The most trivial GP we can think of: minimize 𝑥 subject to the constraint 𝑥 ≥ 1.
"Very simple problem: minimize x while keeping x greater than 1."
from gpkit import
Variable, Model
# Decision variable x = Variable( "x" )
# Constraint constraints = [x >= 1 ]
# Objective (to minimize) objective = x
# Formulate the Model m = Model(objective, constraints)
# Solve the Model sol = m .
solve(verbosity = 0 )
# print selected results print ( "Optimal cost: %s " % sol[ "cost" ]) print ( "Optimal x val: %s " % sol(x))
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Of course, the optimal value is 1. Output:
Optimal cost: 1.0
Optimal x val: 1.0
Maximizing the Volume of a Box
This example comes from Section 2.4 of the GP tutorial , by S. Boyd et. al.
"Maximizes box volume given area and aspect ratio constraints."
from gpkit import
Variable, Model
# Parameters alpha = Variable( "alpha" , 2 , "-" , "lower limit, wall aspect ratio" ) beta = Variable( "beta" , 10 , "-" , "upper limit, wall aspect ratio" ) gamma = Variable( "gamma" , 2 , "-" , "lower limit, floor aspect ratio" ) delta = Variable( "delta" , 10 , "-" , "upper limit, floor aspect ratio" )
A_wall = Variable( "A_ {wall} " , 200 , "m^2" , "upper limit, wall area" )
A_floor = Variable( "A_ {floor} " , 50 , "m^2" , "upper limit, floor area" )
# Decision variables h = Variable( "h" , "m" , "height" ) w = Variable( "w" , "m" , "width" ) d = Variable( "d" , "m" , "depth" )
# Constraints constraints = [A_wall >= 2
* h
* w + 2
* h
* d,
A_floor >= w
* d, h / w >= alpha, h / w <= beta, d / w >= gamma, d / w <= delta]
# Objective function
V = h
* w
* d objective = 1 / V # To maximize V, we minimize its reciprocal
# Formulate the Model m = Model(objective, constraints)
# Solve the Model and print the results table print m .
solve(verbosity = 0 ) .
table()
The output is
Cost
----
0.003674
[ 1 / m
**
3 ]
Free Variables
-------------d : 8.17
[m] depth h : 8.163
[m] height w : 4.081
[m] width
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Constants
---------
A_{floor} : 50 [m
**
2 ] upper limit, floor area
A_{wall} : 200 [m
**
2 ] upper limit, wall area alpha : 2 lower limit, wall aspect ratio beta : 10 upper limit, wall aspect ratio delta : 10 gamma : 2 upper limit, floor aspect ratio lower limit, floor aspect ratio
Sensitivities
-------------
A_{wall} : alpha :
-
+
1.5
0.5
upper limit, wall area lower limit, wall aspect ratio gamma : + 0.0003
lower limit, floor aspect ratio
A_{floor} : 5.7e-09 upper limit, floor area beta : 1.4e-09 upper limit, wall aspect ratio delta : 1.4e-09 upper limit, floor aspect ratio
Water Tank
Say we had a fixed mass of water we wanted to contain within a tank, but also wanted to minimize the cost of the material we had to purchase (i.e. the surface area of the tank):
"Minimizes cylindrical tank surface area for a particular volume."
from gpkit import
Variable, VectorVariable, Model
M = Variable( "M" , 100 , "kg" , "Mass of Water in the Tank" ) rho = Variable( "\\rho" , 1000 , "kg/m^3" , "Density of Water in the Tank" )
A = Variable( "A" , "m^2" , "Surface Area of the Tank" )
V = Variable( "V" , "m^3" , "Volume of the Tank" ) d = VectorVariable( 3 , "d" , "m" , "Dimension Vector" ) constraints = (A >= 2
*
(d[ 0 ]
* d[ 1 ] + d[ 0 ]
* d[ 2 ] + d[ 1 ]
* d[ 2 ]),
V == d[ 0 ]
* d[ 1 ]
* d[ 2 ],
M == V
* rho) m = Model(A, constraints) sol = m .
solve(verbosity = 0 ) print sol .
summary()
The output is
Cost
----
1.293
[m
**
2 ]
Free Variables
--------------
A : 1.293
V : 0.1
d : [ 0.464
0.464
Sensitivities
0.464
[m
**
2 ] Surface Area of the Tank
[m
**
3 ] Volume of the Tank
] [m] Dimension Vector
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-------------
M : + 0.67
Mass of Water
in
the Tank
\rho : 0.67
Density of Water
in
the Tank
52
Simple Wing
This example comes from Section 3 of Geometric Programming for Aircraft Design Optimization , by W.
Hoburg and P. Abbeel.
"Minimizes airplane drag for a simple drag and structure model."
import numpy as np from gpkit import
Variable, Model pi = np .
pi
# Constants k = Variable( "k" , 1.2
, "-" , "form factor" ) e = Variable( "e" , 0.95
, "-" , "Oswald efficiency factor" ) mu = Variable( "\\mu" , 1.78e-5 , "kg/m/s" , "viscosity of air" ) rho = Variable( "\\rho" , 1.23
, "kg/m^3" , "density of air" ) tau = Variable( "\\tau" , 0.12
, "-" , "airfoil thickness to chord ratio" )
N_ult = Variable( "N_ {ult} " , 3.8
, "-" , "ultimate load factor" )
V_min = Variable( "V_ {min} " , 22 , "m/s" , "takeoff speed" )
C_Lmax = Variable( "C_{L,max}" , 1.5
, "-" , "max CL with flaps down" )
S_wetratio = Variable( "(\\frac {S} {S_ {wet} })" , 2.05
, "-" , "wetted area ratio" )
W_W_coeff1 = Variable( "W_{W_ {coeff1} }" , 8.71e-5 , "1/m" ,
"Wing Weight Coefficent 1" )
W_W_coeff2 = Variable( "W_{W_ {coeff2} }" , 45.24
, "Pa" ,
"Wing Weight Coefficent 2" )
CDA0 = Variable( "(CDA0)" , 0.031
, "m^2" , "fuselage drag area" )
W_0 = Variable( "W_0" , 4940.0
, "N" , "aircraft weight excluding wing" )
# Free Variables
D = Variable( "D" , "N" , "total drag force" )
A = Variable( "A" , "-" , "aspect ratio" )
S = Variable( "S" , "m^2" , "total wing area" )
V = Variable( "V" , "m/s" , "cruising speed" )
W = Variable( "W" , "N" , "total aircraft weight" )
Re = Variable( "Re" , "-" , "Reynold's number" )
C_D = Variable( "C_D" , "-" , "Drag coefficient of wing" )
C_L = Variable( "C_L" , "-" , "Lift coefficent of wing" )
C_f = Variable( "C_f" , "-" , "skin friction coefficient" )
W_w = Variable( "W_w" , "N" , "wing weight" ) constraints = []
# Drag model
C_D_fuse = CDA0 / S
C_D_wpar = k
*
C_f
*
S_wetratio
C_D_ind = C_L
**
2 / (pi
*
A
* e) constraints += [C_D >= C_D_fuse + C_D_wpar + C_D_ind]
# Wing weight model
W_w_strc = W_W_coeff1
*
(N_ult
*
A
**
1.5
*
(W_0
*
W
*
S)
**
0.5
) / tau
Chapter 10. Examples
gpkit Documentation, Release 0.5.3
W_w_surf = W_W_coeff2
*
S constraints += [W_w >= W_w_surf + W_w_strc]
# and the rest of the models constraints += [D >= 0.5
* rho
*
S
*
C_D
*
V
**
2 ,
Re <= (rho / mu)
*
V
*
(S / A)
**
0.5
,
C_f >= 0.074
/ Re
**
0.2
,
W <= 0.5
* rho
*
S
*
C_L
*
V
**
2 ,
W <= 0.5
* rho
*
S
*
C_Lmax
*
V_min
**
2 ,
W >= W_0 + W_w] print ( "SINGLE\n======" ) m = Model(D, constraints) sol = m .
solve(verbosity = 0 ) print (sol .
summary()) print ( "SWEEP\n=====" )
N = 2 sweeps = {V_min: ( "sweep" , np .
linspace( 20 , 25 , N)),
V: ( "sweep" , np .
linspace( 45 , 55 , N)), } m .
substitutions .
update(sweeps) sweepsol = m .
solve(verbosity = 0 ) print (sweepsol .
summary())
The output is
SINGLE
======
Cost
----
303.1
[N]
Free Variables
--------------
A : 8.46
C_D : 0.02059
C_L : 0.4988
C_f : 0.003599
D : 303.1
Re : 3.675e+06
S : 16.44
V : 38.15
W : 7341
W_w : 2401 aspect ratio
Drag coefficient of wing
Lift coefficent of wing skin friction coefficient total drag force [N]
Reynold 's number
[m
**
2 ] total wing area
[m / s] cruising speed
[N] total aircraft weight
[N] wing weight
Most Sensitive
--------------
W_0 : + 1 aircraft weight excluding wing e : 0.48
Oswald efficiency factor
(\frac{S}{S_{wet}}) : + 0.43
wetted area ratio k : + 0.43
form factor
V_{ min } : 0.37
takeoff speed
SWEEP
=====
Cost
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----
[ 338 294
Sweep Variables
---------------
V : [ 45
V_{ min } : [ 20
396 326 ] [N]
45
25
55
20
55
25
] [m / s] cruising speed
] [m / s] takeoff speed
Free Variables
--------------
A : [ 6.2
C_D : [ 0.0146
˓→ wing
C_L : [ 0.296
˓→ wing
C_f : [ 0.00333
˓→ coefficient
8.84
0.0196
0.463
0.00361
4.77
0.0123
0.198
0.00314
7.16
0.0157
0.31
0.00342
]
]
]
] aspect ratio
Drag coefficient of
Lift coefficent of skin friction
D : [ 338 294 396 326 ] [N]
Re : [ 5.38e+06 3.63e+06 7.24e+06 4.75e+06 ] total drag force
Reynold 's number
S : [
˓→ weight
18.6
12.1
17.3
W : [ 6.85e+03 6.97e+03 6.4e+03
11.2
6.44e+03
] [m
**
2 ] total wing area
] [N]
W_w : [ 1.91e+03 2.03e+03 1.46e+03 1.5e+03 ] [N] total aircraft wing weight
Most Sensitive
--------------
W_0 : [ + 0.92
˓→ weight excluding wing
V_{ min } : [ 0.82
˓→ speed
V : [ + 0.59
˓→ speed
(\frac{S}{S_{wet}}) : [ + 0.56
˓→ ratio k : [ + 0.56
+
-
+
+
+
0.95
0.41
0.25
0.45
0.45
+
-
+
+
+
0.85
1
0.97
0.63
0.63
+
-
+
+
+
0.85
0.71
0.75
0.54
0.54
] aircraft
] takeoff
] cruising
] wetted area
] form factor
54
Simple Beam
In this example we consider a beam subjected to a uniformly distributed transverse force along its length.
The beam has fixed geometry so we are not optimizing its shape, rather we are simply solving a discretization of the Euler-Bernoulli beam bending equations using GP.
"""
A simple beam example with fixed geometry. Solves the discretized
Euler-Bernoulli beam equations for a constant distributed load
"""
import numpy as np from gpkit import
Variable, VectorVariable, Model, ureg
from gpkit.small_scripts
import
mag
class Beam
(Model):
"""Discretization of the Euler beam equations for a distributed load.
Chapter 10. Examples
gpkit Documentation, Release 0.5.3
Arguments
---------
N : int
Number of finite elements that compose the beam.
L : float
[m] Length of beam.
EI : float
[N m^2] Elastic modulus times cross-section's area moment of inertia.
q : float or N-vector of floats
[N/m] Loading density: can be specified as constants or as an array.
"""
def
setup ( self , N = 4 ):
EI = Variable( "EI" , 1e4 , "N*m^2" ) dx = Variable( "dx" , "m" , "Length of an element" )
L = Variable( "L" , 5 , "m" , "Overall beam length" ) q = VectorVariable(N, "q" , 100
* np .
ones(N), "N/m" ,
"Distributed load at each point" )
V = VectorVariable(N, "V" , "N" , "Internal shear" )
V_tip = Variable( "V_ {tip} " , 0 , "N" , "Tip loading" )
M = VectorVariable(N, "M" , "N*m" , "Internal moment" )
M_tip = Variable( "M_ {tip} " , 0 , "N*m" , "Tip moment" ) th = VectorVariable(N, "\\theta" , "-" , "Slope" ) th_base = Variable( "\\theta_ {base} " , 0 , "-" , "Base angle" ) w = VectorVariable(N, "w" , "m" , "Displacement" ) w_base = Variable( "w_ {base} " , 0 , "m" , "Base deflection" )
# below: trapezoidal integration to form a piecewise-linear
# approximation of loading, shear, and so on
# shear and moment increase from tip to base (left > right) shear_eq = (V >= V .
right + 0.5
* dx
*
(q + q .
right)) shear_eq[ 1 ] = (V[ 1 ] >= V_tip) # tip boundary condition moment_eq = (M >= M .
right + 0.5
* dx
*
(V + V .
right)) moment_eq[ 1 ] = (M[ 1 ] >= M_tip)
# slope and displacement increase from base to tip (right > left) theta_eq = (th >= th .
left + 0.5
* dx
*
(M + M .
left) / EI) theta_eq[ 0 ] = (th[ 0 ] >= th_base) # base boundary condition displ_eq = (w >= w .
left + 0.5
* dx
*
(th + th .
left)) displ_eq[ 0 ] = (w[ 0 ] >= w_base)
# minimize tip displacement (the last w) self .
cost = w[ 1 ]
return
[shear_eq, moment_eq, theta_eq, displ_eq,
L == (N 1 )
* dx] b = Beam(N = 6 , substitutions = { "L" : 6 , "EI" : 1.1e4
, "q" : 110
* np .
ones( 6 )}) b .
zero_lower_unbounded_variables() sol = b .
solve(verbosity = 0 ) print sol .
summary() w_gp = sol( "w" ) # deflection along beam
L, EI, q = sol( "L" ), sol( "EI" ), sol( "q" ) x = np .
linspace( 0 , mag(L), len (q))
* ureg .
m # position along beam q = q[ 0 ] # assume uniform loading for the check below w_exact = q / ( 24.
*
EI)
* x
**
2
*
(x
**
2 4
*
L
* x + 6
*
L
**
2 ) # analytic soln
assert
max ( abs (w_gp w_exact)) <= 1.1
* ureg .
cm
PLOT =
False if
PLOT:
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import matplotlib.pyplot
as plt
x_exact = np .
linspace( 0 , L, 1000 ) w_exact = q / ( 24.
*
EI)
* x_exact
**
2
*
(x_exact
**
2 4
*
L
* x_exact + 6
*
L
**
2 ) plt .
plot(x, w_gp, color = 'red' , linestyle = 'solid' , marker = '^' , markersize = 8 ) plt .
plot(x_exact, w_exact, color = 'blue' , linestyle = 'dashed' ) plt .
xlabel( 'x [m]' ) plt .
ylabel( 'Deflection [m]' ) plt .
axis( 'equal' ) plt .
legend([ 'GP solution' , 'Analytical solution' ]) plt .
show()
The output is
Cost
----
1.62
[m]
Free Variables
-------------dx : 1.2
˓→ element
M : [ 1.98e+03 1.27e+03 713
˓→ moment
V : [ 660
\theta : [ w : [ -
528
0.177
0.106
396
0.285
0.384
Most Sensitive
--------------
L : + 4
EI : 1 q : [ + 0.0072
˓→ point
+ 0.042
+ 0.12
+ 0.23
317
264
0.341
0.759
[m] Length of an
...
] [N
* m] Internal
...
] [N]
...
]
...
] [m]
Internal shear
Slope
Displacement
Overall beam length
...
] Distributed load at each
By plotting the deflection, we can see that the agreement between the analytical solution and the GP solution is good.
56 Chapter 10. Examples
CHAPTER
11
Glossary
For an alphabetical listing of all commands, check out the genindex
Subpackages gpkit.constraints package
Submodules gpkit.constraints.array module
Implements ArrayConstraint class gpkit.constraints.array.ArrayConstraint(constraints, left, oper, right)
Bases:
gpkit.constraints.single_equation.SingleEquationConstraint
gpkit.constraints.set.ConstraintSet
A ConstraintSet for prettier array-constraint printing.
ArrayConstraint gets its sub method from ConstrainSet, and so left and right are only used for printing.
When created by NomialArray left and right are likely to be be either NomialArrays or Varkeys of
VectorVariables.
subinplace
( subs)
Substitutes in place, updating self.substitutions accordingly.
Keys substituted with subinplace are no longer present, so if such a key is also in self.substitutions that substitution is now orphaned. If subs[key] describes some key in the
ConstraintSet (i.e. one key has been substituted for another), then a substitution is added, mapping the orphaned value to this new key; otherwise, an error is raised.
57
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gpkit.constraints.bounded module
Implements Bounded class gpkit.constraints.bounded.Bounded(constraints, verbosity=1 , eps=1e-30 , lower=None , upper=None)
Bases:
gpkit.constraints.set.ConstraintSet
Bounds contained variables so as to ensure dual feasibility.
constraints [iterable] constraints whose varkeys will be bounded substitutions [dict] as in ConstraintSet.__init__
verbosity [int] how detailed of a warning to print 0: nothing 1: print warnings eps [float] default lower bound is eps, upper bound is 1/eps lower [float] lower bound for all varkeys, replaces eps upper [float] upper bound for all varkeys, replaces 1/eps
process_result
( result)
Creates (and potentially prints) a dictionary of unbounded variables.
sens_from_dual
( las
, nus)
Return sensitivities while capturing the relevant lambdas gpkit.constraints.bounded.varkey_bounds( varkeys , lower, upper)
Returns constraints list bounding all varkeys.
varkeys [iterable] list of varkeys to create bounds for lower [float] lower bound for all varkeys upper [float] upper bound for all varkeys
gpkit.constraints.costed module
Implement CostedConstraintSet class gpkit.constraints.costed.CostedConstraintSet(cost, constraints substitutions=None ,
, add_cost_values_to_substitutions=True)
Bases:
gpkit.constraints.set.ConstraintSet
A ConstraintSet with a cost cost : gpkit.Posynomial constraints : Iterable substitutions : dict
controlpanel
(
*args , **kwargs)
Easy model control in IPython / Jupyter
Like interact(), but with the ability to control sliders and their ranges live. args and kwargs are passed on to interact()
interact
( ranges=None , fn_of_sol=None, **solvekwargs)
Easy model interaction in IPython / Jupyter
By default, this creates a model with sliders for every constant which prints a new solution table whenever the sliders are changed.
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fn_of_sol [function] The function called with the solution after each solve that displays the result. By default prints a table.
ranges [dictionary {str: Slider object or tuple}] Determines which sliders get created. Tuple values may contain two or three floats: two correspond to (min, max), while three correspond to (min, step, max)
** solvekwargs kwargs which get passed to the solve()/localsolve() method.
reset_varkeys
()
Resets varkeys to what is in the cost and constraints
rootconstr_latex
( excluded=None)
Latex showing cost, to be used when this is the top constraint
rootconstr_str
( excluded=None)
String showing cost, to be used when this is the top constraint
subinplace
( subs)
Substitutes in place.
gpkit.constraints.geometric_program module
Implement the GeometricProgram class class gpkit.constraints.geometric_program.GeometricProgram(cost, straints , consubstitutions=None , verbosity=1)
Bases:
gpkit.constraints.costed.CostedConstraintSet
Standard mathematical representation of a GP.
cost [Constraint] Posynomial to minimize when solving constraints [list of Posynomials] Constraints to maintain when solving (implicitly Posynomials
<= 1) GeometricProgram does not accept equality constraints (e.g. x == 1); instead use two inequality constraints (e.g. x <= 1, 1/x <= 1) verbosity [int (optional)] If verbosity is greater than zero, warns about missing bounds on creation.
solver_out and solver_log are set during a solve result is set at the end of a solve if solution status is optimal
>>>
gp = gpkit .
geometric_program .
GeometricProgram(
# minimize x,
[ # subject to
1/x # <= 1, implicitly
])
>>>
gp .
solve()
check_solution
( cost , primal, nu, la, tol=0.001, abstol=1e-20)
Run a series of checks to mathematically confirm sol solves this GP cost: float cost returned by solver primal: list primal solution returned by solver
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60
nu: numpy.ndarray monomial lagrange multiplier la: numpy.ndarray posynomial lagrange multiplier
RuntimeWarning, if any problems are found
gen
( verbosity=1)
Generates nomial and solve data (A, p_idxs) from self.posynomials
solve
( solver=None , verbosity=1, warn_on_check=False, *args, **kwargs)
Solves a GeometricProgram and returns the solution.
solver [str or function (optional)] By default uses one of the solvers found during installation.
If set to “mosek”, “mosek_cli”, or “cvxopt”, uses that solver. If set to a function, passes that function cs, A, p_idxs, and k.
verbosity [int (optional)] If greater than 0, prints solver name and solve time.
* args, ** kwargs : Passed to solver constructor and solver function.
result [dict] A dictionary containing the translated solver result; keys below.
cost [float] The value of the objective at the solution.
variables [dict] The value of each variable at the solution.
sensitivities [dict] monomials [array of floats] Each monomial’s dual variable value at the solution.
posynomials [array of floats] Each posynomials’s dual variable value at the solution.
gpkit.constraints.geometric_program.genA( exps , varlocs)
Generates A matrix from exps and varlocs exps [list of Hashvectors] Exponents for each monomial in a GP varlocs [dict] Locations of each variable in exps
A [sparse Cootmatrix] Exponents of the various free variables for each monomial: rows of A are monomials, columns of A are variables.
missingbounds [dict] Keys: variables that lack bounds. Values: which bounds are missed.
gpkit.constraints.linked module gpkit.constraints.model module
Implements Model class gpkit.constraints.model.Model(cost=None, constraints=None
**kwargs)
Bases:
gpkit.constraints.costed.CostedConstraintSet
,
Symbolic representation of an optimization problem.
*args ,
The Model class is used both directly to create models with constants and sweeps, and indirectly inherited to create custom model classes.
cost [Posynomial (optional)] Defaults to Monomial(1).
constraints [ConstraintSet or list of constraints (optional)] Defaults to an empty list.
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substitutions [dict (optional)] This dictionary will be substituted into the problem before solving, and also allows the declaration of sweeps and linked sweeps.
name [str (optional)] Allows “naming” a model in a way similar to inherited instances, and overrides the inherited name if there is one.
program is set during a solve solution is set at the end of a solve
autosweep
( sweeps
, tol=0.01, samplepoints=100, **solveargs)
Autosweeps {var: (start, end)} pairs in sweeps to tol.
Returns swept and sampled solutions. The original simplex tree can be accessed at sol.bst
debug
( solver=None , verbosity=1, **solveargs)
Attempts to diagnose infeasible models.
gp
( verbosity=1 , constants=None, **kwargs)
Return program version of self program: NomialData Class to return, e.g. GeometricProgram or SignomialProgram return_attr: string attribute to return in addition to the program
localsolve
( solver=None , verbosity=1, skipsweepfailures=False, *args, **kwargs)
Forms a mathematical program and attempts to solve it.
solver [string or function (optional)] If None, uses the default solver found in installation.
verbosity [int (optional)] If greater than 0 prints runtime messages. Is decremented by one and then passed to programs.
skipsweepfailures [bool (optional)] If True, when a solve errors during a sweep, skip it.
* args, ** kwargs : Passed to solver sol [SolutionArray] See the SolutionArray documentation for details.
ValueError if the program is invalid. RuntimeWarning if an error occurs in solving or parsing the solution.
name
= None
naming
= None
num
= None
program
= None
solution
= None
solve
( solver=None , verbosity=1, skipsweepfailures=False, *args, **kwargs)
Forms a mathematical program and attempts to solve it.
solver [string or function (optional)] If None, uses the default solver found in installation.
verbosity [int (optional)] If greater than 0 prints runtime messages. Is decremented by one and then passed to programs.
skipsweepfailures [bool (optional)] If True, when a solve errors during a sweep, skip it.
* args,
** kwargs : Passed to solver sol [SolutionArray] See the SolutionArray documentation for details.
ValueError if the program is invalid. RuntimeWarning if an error occurs in solving or parsing the solution.
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sp
( verbosity=1 , constants=None, **kwargs)
Return program version of self program: NomialData Class to return, e.g. GeometricProgram or SignomialProgram return_attr: string attribute to return in addition to the program
subconstr_latex
( excluded=None)
The collapsed appearance of a ConstraintBase
subconstr_str
( excluded=None)
The collapsed appearance of a ConstraintBase
sweep
( sweeps , **solveargs)
Sweeps {var: values} pairs in sweeps. Returns swept solutions.
zero_lower_unbounded_variables
()
Recursively substitutes 0 for variables that lack a lower bound
gpkit.constraints.prog_factories module
Scripts for generating, solving and sweeping programs gpkit.constraints.prog_factories.run_sweep( genfunction , self, solution, skipsweepfailures , constants, sweep, linkedsweep , solver , verbosity ,
*args , **kwargs)
Runs through a sweep.
gpkit.constraints.relax module
Models for assessing primal feasibility class gpkit.constraints.relax.ConstantsRelaxed(constraints, clude_only=None , clude=None)
Bases:
gpkit.constraints.set.ConstraintSet
Relax constants in a constraintset.
constraints [iterable] Constraints which will be relaxed (made easier).
include_only [set] if declared, variable names must be on this list to be relaxed exclude [set] if declared, variable names on this list will never be relaxed inexrelaxvars [Variable] The variables controlling the relaxation. A solved value of 1 means no relaxation was necessary or optimal for a particular constant. Higher values indicate the amount by which that constant has been made easier: e.g., a value of 1.5 means it was made 50 percent easier in the final solution than in the original problem. Of course, this can also be determined by looking at the constant’s new value directly.
process_result
( result) class gpkit.constraints.relax.ConstraintsRelaxed(constraints)
Bases:
gpkit.constraints.set.ConstraintSet
Relax constraints, as in Eqn. 11 of [Boyd2007].
constraints [iterable] Constraints which will be relaxed (made easier).
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relaxvars [Variable] The variables controlling the relaxation. A solved value of 1 means no relaxation was necessary or optimal for a particular constraint. Higher values indicate the amount by which that constraint has been made easier: e.g., a value of 1.5 means it was made 50 percent easier in the final solution than in the original problem.
[Boyd2007] : “A tutorial on geometric programming”, Optim Eng 8:67-122 class gpkit.constraints.relax.ConstraintsRelaxedEqually(constraints)
Bases:
gpkit.constraints.set.ConstraintSet
Relax constraints the same amount, as in Eqn. 10 of [Boyd2007].
constraints [iterable] Constraints which will be relaxed (made easier).
relaxvar [Variable] The variable controlling the relaxation. A solved value of 1 means no relaxation. Higher values indicate the amount by which all constraints have been made easier: e.g., a value of 1.5 means all constraints were 50 percent easier in the final solution than in the original problem.
[Boyd2007] : “A tutorial on geometric programming”, Optim Eng 8:67-122
gpkit.constraints.set module
Implements ConstraintSet class gpkit.constraints.set.ConstraintSet(constraints, substitutions=None)
Bases: list
Recursive container for ConstraintSets and Inequalities
append
( value)
as_gpconstr
( x0 , substitutions=None)
Returns GPConstraint approximating this constraint at x0
When x0 is none, may return a default guess.
as_posyslt1
( substitutions=None)
Returns list of posynomials which must be kept <= 1
flat
( constraintsets=True)
Yields contained constraints, optionally including constraintsets.
latex
( excluded=None)
LaTeX representation of a ConstraintSet.
process_result
( result)
Does arbitrary computation / manipulation of a program’s result
There’s no guarantee what order different constraints will process results in, so any changes made to the program’s result should be careful not to step on other constraint’s toes.
•check that an inequality was tight
•add values computed from solved variables
reset_varkeys
()
Goes through constraints and collects their varkeys.
rootconstr_latex
( excluded=None)
The appearance of a ConstraintSet in addition to its contents
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rootconstr_str
( excluded=None)
The appearance of a ConstraintSet in addition to its contents
sens_from_dual
( las , nus)
Computes constraint and variable sensitivities from dual solution las [list] Sensitivity of each posynomial returned by self.as_posyslt1
nus: list of lists Each posynomial’s monomial sensitivities constraint_sens [dict] The interesting and computable sensitivities of this constraint var_senss [dict] The variable sensitivities of this constraint
str_without
( excluded=None)
String representation of a ConstraintSet.
subconstr_latex
( excluded=None)
The collapsed appearance of a ConstraintSet
subconstr_str
( excluded=None)
The collapsed appearance of a ConstraintSet
subinplace
( subs)
Substitutes in place, updating self.substitutions accordingly.
Keys substituted with subinplace are no longer present, so if such a key is also in self.substitutions that substitution is now orphaned. If subs[key] describes some key in the
ConstraintSet (i.e. one key has been substituted for another), then a substitution is added, mapping the orphaned value to this new key; otherwise, an error is raised.
topvar
( key)
If a variable by a given name exists in the top model, return it
unique_varkeys
= frozenset([])
variables_byname
( key)
Get all variables with a given name
varkeys
= None gpkit.constraints.set.raise_badelement( cns
, i, constraint)
Identify the bad element and raise a ValueError gpkit.constraints.set.raise_elementhasnumpybools( constraint)
Identify the bad subconstraint array and raise a ValueError
gpkit.constraints.sigeq module
Implements SignomialEquality class gpkit.constraints.sigeq.SignomialEquality(left, right)
Bases:
gpkit.constraints.set.ConstraintSet
A constraint of the general form posynomial == posynomial
gpkit.constraints.signomial_program module
Implement the SignomialProgram class
64 Chapter 11. Glossary
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class gpkit.constraints.signomial_program.SignomialProgram(cost, straints , consubstitutions=None , verbosity=1)
Bases:
gpkit.constraints.costed.CostedConstraintSet
Prepares a collection of signomials for a SP solve.
cost [Posynomial] Objective to minimize when solving constraints [list of Constraint or SignomialConstraint objects] Constraints to maintain when solving (implicitly Signomials <= 1) verbosity [int (optional)] Currently has no effect: SignomialPrograms don’t know anything new after being created, unlike GeometricPrograms.
gps is set during a solve result is set at the end of a solve
>>>
gp = gpkit .
geometric_program .
SignomialProgram(
# minimize x,
[ # subject to
1/x - y/x, # <= 1, implicitly y/10 # <= 1
])
>>>
gp .
solve()
firstgp
( x0 , substitutions)
Generates a simplified GP representation for later modification
gp
( x0=None , verbosity=1, modifylastgp=False)
The GP approximation of this SP at x0.
localsolve
( solver=None , verbosity=1, x0=None, reltol=0.0001, iteration_limit=50, modifylastgp=True , **kwargs)
Locally solves a SignomialProgram and returns the solution.
solver [str or function (optional)] By default uses one of the solvers found during installation.
If set to “mosek”, “mosek_cli”, or “cvxopt”, uses that solver. If set to a function, passes that function cs, A, p_idxs, and k.
verbosity [int (optional)] If greater than 0, prints solve time and number of iterations. Each
GP is created and solved with verbosity one less than this, so if greater than 1, prints solver name and time for each GP.
x0 [dict (optional)] Initial location to approximate signomials about.
reltol [float] Iteration ends when this is greater than the distance between two consecutive solve’s objective values.
iteration_limit [int] Maximum GP iterations allowed.
* args, ** kwargs : Passed to solver function.
result [dict] A dictionary containing the translated solver result.
gpkit.constraints.single_equation module
Implements SingleEquationConstraint
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class gpkit.constraints.single_equation.SingleEquationConstraint(left, oper , right)
Bases: object
Constraint expressible in a single equation.
func_opers
= {‘<=’: <built-in function le>, ‘=’: <built-in function eq>, ‘>=’: <built-in function ge>}
latex
( excluded=None)
Latex representation without attributes in excluded list
latex_opers
= {‘<=’: ‘\\leq’, ‘=’: ‘=’, ‘>=’: ‘\\geq’}
process_result
( result)
Process solver results
str_without
( excluded=None)
String representation without attributes in excluded list
sub
( subs)
Returns a substituted version of this constraint.
subconstr_latex
( excluded)
The collapsed latex of a constraint
subconstr_str
( excluded)
The collapsed string of a constraint gpkit.constraints.single_equation.trycall( obj , attr, arg, default)
Try to call method of an object, returning default if it does not exist
gpkit.constraints.tight module
Implements Tight class gpkit.constraints.tight.Tight(constraints, substitutions=None, reltol=None, raiseerror=False)
Bases:
gpkit.constraints.set.ConstraintSet
ConstraintSet whose inequalities must result in an equality.
process_result
( result)
Checks that all constraints are satisfied with equality
reltol
= 1e-06
Module contents
Contains ConstraintSet and related classes and objects
gpkit.interactive package
Submodules gpkit.interactive.chartjs module
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gpkit.interactive.linking_diagram module
Module for creating diagrams illustrating variables shared between submodels.
gpkit.interactive.linking_diagram.linking_diagram( topmodel , subsystems, filename)
Method to create a latex diagram illustrating how variables are linked between a parent model and its submodels topmodel - a model object, the parent model of all submodels susbystems - a list of model objects, each a submodel of the topmodel filename - a string which is the name of the file latex output will be written to note: the following packages must be used in the latex file usepackage{tikz} usetikzlibrary{backgrounds}
gpkit.interactive.plot_sweep module
Implements plot_sweep1d function gpkit.interactive.plot_sweep.assign_axes( var , posys, axes)
Assigns axes to posys, creating and formatting if necessary gpkit.interactive.plot_sweep.format_and_label_axes( var
, posys, axes, ylabel=True)
Formats and labels axes gpkit.interactive.plot_sweep.plot_1dsweepgrid( model , sweeps , posys , origsol=None , tol=0.01
,
**solveargs)
Creates and plots a sweep from an existing model
Example usage: f, _ = plot_sweep_1d(m, {‘x’: np.linspace(1, 2, 5)}, ‘y’) f.savefig(‘mysweep.png’)
gpkit.interactive.plotting module
Plotting methods gpkit.interactive.plotting.compare( models , sweeps, posys, tol=0.001)
Compares the values of posys over a sweep of several models.
If posys is of the same length as models, this will plot different variables from different models.
Currently only supports a single sweepvar.
Example Usage: compare([aec, fbc], {“R”: (160, 300)},
[”cost”, (“W_{rm batt}”, “W_{rm fuel}”)], tol=0.001) gpkit.interactive.plotting.plot_convergence( model)
Plots the convergence of a signomial programming model model: Model Signomial programming model that has already been solved matplotlib.pyplot Figure Plot of cost as functions of SP iteration #
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gpkit.interactive.ractor module
Implements Ractor-based interactive CADtoons gpkit.interactive.ractor.ractorjs( title , model , update_py , ranges , constraint_js=’‘)
Creates Javascript/HTML for CADtoon interaction without installing GPkit.
gpkit.interactive.ractor.ractorpy( model , update_py, ranges, constraint_js=’‘, showtables=(‘cost’ , ‘sensitivities’))
Creates interactive iPython widget for controlling a CADtoon gpkit.interactive.ractor.showcadtoon( title , css=’‘)
Displays cadtoon as iPython HTML
gpkit.interactive.sensitivity_map module
Implements heatmapped equations to highlight sensitivities.
class gpkit.interactive.sensitivity_map.SensitivityMap(model, paintby=’constants’)
Bases: object
Latex representations of a model heatmapped by its latest sensitivities model [Model] The Model object that the Map will be based on paintby [string] The unit of colouring. Must be one of “constants”, “monomials”, or “posynomials”.
from IPython.display import display for key in m.solution[”sensitivities”]: print key display(SensitivityMap(m, paintby=key))
constraint_latex_list
( paintby)
Generates LaTeX for constraints.
latex
LaTeX representation.
solution
Gets solution, indexing into a sweep if necessary.
gpkit.interactive.sensitivity_map.colorfn_gen( scale , power=0.66)
Generates color gradient of a given power law.
gpkit.interactive.sensitivity_map.signomial_print( sig , sol , colorfn , paintby=’constants’ , idx=None)
For pretty printing with Sympy
gpkit.interactive.widgets module
Module contents
Module for the interactive and plotting functions of GPkit
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gpkit.nomials package
Submodules gpkit.nomials.array module
Module for creating NomialArray instances.
Example
>>>
x = gpkit .
Monomial( 'x' )
>>>
px = gpkit .
NomialArray([ 1 , x, x
**
2 ]) class gpkit.nomials.array.NomialArray
Bases: numpy.ndarray
A Numpy array with elementwise inequalities and substitutions.
input_array : array-like
>>>
px = gpkit .
NomialArray([ 1 , x, x
**
2 ])
c
The coefficient vector in the GP input data sense
latex
( matwrap=True)
Returns 1D latex list of contents.
left
Returns (0, self[0], self[1] ... self[N-1])
outer
( other)
Returns the array and argument’s outer product.
padleft
( padding)
Returns ({padding}, self[0], self[1] ... self[N])
padright
( padding)
Returns (self[0], self[1] ... self[N], {padding})
prod
( *args , **kwargs)
Returns a product. O(N) if no arguments and only contains monomials.
right
Returns (self[1], self[2] ... self[N], 0)
str_without
( excluded=None)
Returns string without certain fields (such as ‘models’).
sub
( subs , require_positive=True)
Substitutes into the array
sum
(
*args , **kwargs)
Returns a sum. O(N) if no arguments are given.
units
units must have same dimensions across the entire nomial array
vectorize
( function , *args, **kwargs)
Apply a function to each terminal constraint, returning the array
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gpkit.nomials.array.array_constraint( symbol , func)
Return function which creates constraints of the given operator.
gpkit.nomials.data module
Machinery for exps, cs, varlocs data – common to nomials and programs class gpkit.nomials.data.NomialData(exps=None, cs=None, simplify=True)
Bases: object
Object for holding cs, exps, and other basic ‘nomial’ properties.
cs: array (coefficient of each monomial term) exps: tuple of {VarKey: float} (exponents of each monomial term) varlocs: {VarKey: list} (terms each variable appears in) units: pint.UnitsContainer
diff
( var)
Derivative of this with respect to a Variable var (Variable): Variable to take derivative with respect to
NomialData
init_from_nomials
( nomials)
Way to initialize from nomials. Calls __init__. Used by subclass __init__ methods.
values
The NomialData’s values, created when necessary.
varkeys
The NomialData’s varkeys, created when necessary for a substitution.
gpkit.nomials.data.simplify_exps_and_cs( exps , cs, return_map=False)
Reduces the number of monomials, and casts them to a sorted form.
exps [list of Hashvectors] The exponents of each monomial cs [array of floats or Quantities] The coefficients of each monomial return_map [bool (optional)] Whether to return the map of which monomials combined to form a simpler monomial, and their fractions of that monomial’s final c.
exps [list of Hashvectors] Exponents of simplified monomials.
cs [array of floats or Quantities] Coefficients of simplified monomials.
mmap [list of HashVectors] List for each new monomial of {originating indexes: fractions}
gpkit.nomials.nomial_core module
The shared non-mathematical backbone of all Nomials class gpkit.nomials.nomial_core.Nomial(exps=None, cs=None, simplify=True)
Bases:
Shared non-mathematical properties of all nomials
c
= None
convert_to
( arg)
Convert this signomial to new units
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latex
( excluded=None)
For pretty printing with Sympy
prod
()
Return self for compatibility with NomialArray
str_without
( excluded=None)
String representation excluding fields (‘units’, varkey attributes)
sub
= None
sum
()
Return self for compatibility with NomialArray
to
( arg)
Create new Signomial converted to new units
unitstr
( units , into=’%s’, options=’~’, dimless=’‘)
Returns the string corresponding to an object’s units.
value
Self, with values substituted for variables that have values float, if no symbolic variables remain after substitution (Monomial, Posynomial, or Nomial), otherwise.
gpkit.nomials.nomial_math module
Signomial, Posynomial, Monomial, Constraint, & MonoEQCOnstraint classes class gpkit.nomials.nomial_math.Monomial(exps=None, quire_positive=True ,
**descr)
Bases:
gpkit.nomials.nomial_math.Posynomial
cs=1 , resimplify=True ,
A Posynomial with only one term
Same as Signomial. Note: Monomial historically supported several different init formats
These will be deprecated in the future, replaced with a single __init__ syntax, same as
Signomial.
mono_approximation
( x0) class gpkit.nomials.nomial_math.MonomialEquality(left, oper, right)
Bases:
gpkit.nomials.nomial_math.PosynomialInequality
A Constraint of the form Monomial == Monomial.
sens_from_dual
( la , nu)
Returns the variable/constraint sensitivities from lambda/nu class gpkit.nomials.nomial_math.Posynomial(exps=None, cs=1
, require_positive=True
, simplify=True,
**descr)
Bases:
gpkit.nomials.nomial_math.Signomial
A Signomial with strictly positive cs
Same as Signomial. Note: Posynomial historically supported several different init formats
These will be deprecated in the future, replaced with a single __init__ syntax, same as
Signomial.
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mono_lower_bound
( x0)
Monomial lower bound at a point x0 x0 (dict): point to make lower bound exact
Monomial class gpkit.nomials.nomial_math.PosynomialInequality(left, oper, right)
Bases:
gpkit.nomials.nomial_math.ScalarSingleEquationConstraint
A constraint of the general form monomial >= posynomial Stored in the posylt1_rep attribute as a single Posynomial (self <= 1) Usually initialized via operator overloading, e.g. cc = (y**2 >= 1 + x)
as_gpconstr
( x0 , substitutions)
GP version of a Posynomial constraint is itself
as_posyslt1
( substitutions=None)
Returns the posys <= 1 representation of this constraint.
sens_from_dual
( la , nu)
Returns the variable/constraint sensitivities from lambda/nu class gpkit.nomials.nomial_math.ScalarSingleEquationConstraint(left, oper
, right)
Bases:
gpkit.constraints.single_equation.SingleEquationConstraint
A SingleEquationConstraint with scalar left and right sides.
nomials
= []
subinplace
( substitutions)
Modifies the constraint in place with substitutions.
class gpkit.nomials.nomial_math.Signomial(exps=None, cs=1 , require_positive=True , simplify=True ,
**descr)
Bases:
gpkit.nomials.nomial_core.Nomial
A representation of a Signomial.
exps: tuple of dicts Exponent dicts for each monomial term cs: tuple Coefficient values for each monomial term require_positive: bool If True and Signomials not enabled, c <= 0 will raise ValueError
Signomial Posynomial (if the input has only positive cs) Monomial (if the input has one term and only positive cs)
diff
( var)
Derivative of this with respect to a Variable var (Variable): Variable to take derivative with respect to
Signomial (or Posynomial or Monomial)
mono_approximation
( x0)
Monomial approximation about a point x0 x0 (dict): point to monomialize about
Monomial (unless self(x0) < 0, in which case a Signomial is returned)
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posy_negy
()
Get the positive and negative parts, both as Posynomials
Posynomial, Posynomial: p_pos and p_neg in (self = p_pos - p_neg) decomposition,
sub
( substitutions , require_positive=True)
Returns a nomial with substitued values.
3 == (x**2 + y).sub({‘x’: 1, y: 2}) 3 == (x).gp.sub(x, 3) substitutions [dict or key] Either a dictionary whose keys are strings, Variables, or VarKeys, and whose values are numbers, or a string, Variable or Varkey.
val [number (optional)] If the substitutions entry is a single key, val holds the value require_positive [boolean (optional, default is True)] Controls whether the returned value can be a Signomial.
Returns substituted nomial.
subinplace
( substitutions)
Substitutes in place.
class gpkit.nomials.nomial_math.SignomialInequality(left, oper, right)
Bases:
gpkit.nomials.nomial_math.ScalarSingleEquationConstraint
A constraint of the general form posynomial >= posynomial Stored internally (exps, cs) as a single
Signomial (0 >= self) Usually initialized via operator overloading, e.g. cc = (y**2 >= 1 + x - y)
Additionally retains input format (lhs vs rhs) in self.left and self.right Form is self.left >= self.right.
as_approxsgt
( x0)
Returns monomial-greater-than sides, to be called after as_approxlt1
as_approxslt
()
Returns posynomial-less-than sides of a signomial constraint
as_gpconstr
( x0 , substitutions=None)
Returns GP approximation of an SP constraint at x0
as_posyslt1
( substitutions=None)
Returns the posys <= 1 representation of this constraint.
class gpkit.nomials.nomial_math.SingleSignomialEquality(left, right)
Bases:
gpkit.nomials.nomial_math.SignomialInequality
A constraint of the general form posynomial == posynomial
as_approxsgt
( x0)
Returns monomial-greater-than sides, to be called after as_approxlt1
as_approxslt
()
Returns posynomial-less-than sides of a signomial constraint
as_gpconstr
( x0 , substitutions=None)
Returns GP approximation of an SP constraint at x0
as_posyslt1
( substitutions=None)
Returns the posys <= 1 representation of this constraint.
gpkit.nomials.nomial_math.non_dimensionalize( posy)
Non-dimensionalize a posy (warning: mutates posy)
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gpkit.nomials.substitution module
Module containing the substitution function gpkit.nomials.substitution.append_sub( sub , keys, constants, sweep, linkedsweep)
Appends sub to constants, sweep, or linkedsweep.
gpkit.nomials.substitution.parse_subs( varkeys , substitutions)
Seperates subs into constants, sweeps linkedsweeps actually present.
gpkit.nomials.substitution.substitution( nomial , substitutions)
Efficient substituton into a list of monomials.
varlocs [dict] Dictionary mapping variables to lists of monomial indices.
exps [Iterable of dicts] Dictionary mapping variables to exponents, for each monomial.
cs [list] Coefficient for each monomial.
substitutions [dict] Substitutions to apply to the above.
val [number (optional)] Used to substitute singlet variables.
varlocs_ [dict] Dictionary of monomial indexes for each variable.
exps_ [dict] Dictionary of variable exponents for each monomial.
cs_ [list] Coefficients each monomial.
subs_ [dict] Substitutions to apply to the above.
74 gpkit.nomials.variables module
Implement Variable and ArrayVariable classes class gpkit.nomials.variables.ArrayVariable
Bases:
gpkit.nomials.array.NomialArray
A described vector of singlet Monomials.
shape [int or tuple] length or shape of resulting array
* args : may contain “name” (Strings)
“value” (Iterable) “units” (Strings + Quantity) and/or “label” (Strings)
** descr : VarKey description
NomialArray of Monomials, each containing a VarKey with name ‘$name_{i}’, where $name is the vector’s name and i is the VarKey’s index.
class gpkit.nomials.variables.Variable(*args, **descr)
Bases:
gpkit.nomials.nomial_math.Monomial
A described singlet Monomial.
* args [list] may contain “name” (Strings)
“value” (Numbers + Quantity) or (Iterable) for a sweep “units” (Strings + Quantity)
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and/or “label” (Strings)
** descr [dict] VarKey description
Monomials containing a VarKey with the name ‘$name’, where $name is the vector’s name and i is the VarKey’s index.
key
Get the VarKey associated with this Variable
sub
(
*args , **kwargs)
Same as nomial substitution, but also allows single-argument calls x = Variable(‘x’) assert x.sub(3) == Variable(‘x’, value=3)
to
( arg)
Create new Signomial converted to new units class gpkit.nomials.variables.VectorizableVariable(*args, **descr)
Bases:
gpkit.nomials.variables.Variable
A Variable outside a vectorized environment, an ArrayVariable within.
Module contents
Contains nomials, inequalities, and arrays
gpkit.tests package
Submodules gpkit.tests.diff_output module
Function to diff example output and allow small numerical errors gpkit.tests.diff_output.diff( output1 , output2, tol=0.001) check that output1 and output2 are same up to small errors in numbers
gpkit.tests.from_paths module
Runs each file listed in pwd/TESTS as a test class gpkit.tests.from_paths.TestFiles(methodName=’runTest’)
Bases: unittest.case.TestCase
Stub to be filled with files in $pwd/TESTS gpkit.tests.from_paths.add_filetest( testclass , path)
Add test that imports the given path and runs its test() function gpkit.tests.from_paths.clean( string)
Parses string into valid python variable name https://stackoverflow.com/questions/3303312/ how-do-i-convert-a-string-to-a-valid-variable-namein-python gpkit.tests.from_paths.newtest_fn( name , solver, import_dict, path)
Doubly nested callbacks to run the test with getattr(self, name)()
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gpkit.tests.from_paths.run( filename=’TESTS’ , xmloutput=False, skipsolvers=None)
Parse and run paths from a given file for each solver
gpkit.tests.helpers module
Convenience classes and functions for unit testing class gpkit.tests.helpers.NewDefaultSolver(solver)
Bases: object
Creates an environment with a different default solver class gpkit.tests.helpers.NullFile
Bases: object
A fake file interface that does nothing
close
()
Having not written, cease.
write
( string)
Do not write, do not pass go.
class gpkit.tests.helpers.StdoutCaptured(logfilepath=None)
Bases: object
Puts everything that would have printed to stdout in a log file instead gpkit.tests.helpers.generate_example_tests( path , testclasses, solvers=None, newtest_fn=None)
Mutate TestCase class so it behaves as described in TestExamples docstring path [str] directory containing example modules to test testclass [class] class that inherits from unittest.TestCase
newtest_fn [function] function that returns new tests. defaults to import_test_and_log_output solvers [iterable] solvers to run for; or only for default if solvers is None gpkit.tests.helpers.logged_example_testcase( name , imported, path)
Returns a method for attaching to a unittest.TestCase that imports or reloads module ‘name’ and stores in imported[name]. Runs top-level code, which is typically a docs example, in the process.
Returns a method.
gpkit.tests.helpers.new_test( name , solver, import_dict, path, testfn=None) logged_example_testcase with a NewDefaultSolver gpkit.tests.helpers.run_tests( tests
, xmloutput=None, verbosity=2)
Default way to run tests, to be used in __main__.
tests: iterable of unittest.TestCase xmloutput: string or None if not None, generate xml output for continuous integration, with name given by the input string verbosity: int verbosity level for unittest.TextTestRunner
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gpkit.tests.run_tests module
Script for running all gpkit unit tests gpkit.tests.run_tests.import_tests()
Get a list of all GPkit unit test TestCases gpkit.tests.run_tests.run( xmloutput=False , tests=None, unitless=True)
Run all gpkit unit tests.
xmloutput: bool If true, generate xml output files for continuous integration
gpkit.tests.t_constraints module
Unit tests for Constraint, MonomialEquality and SignomialInequality class gpkit.tests.t_constraints.TestBounded(methodName=’runTest’)
Bases: unittest.case.TestCase
Test bounded constraint set
test_substitution_issue905
() class gpkit.tests.t_constraints.TestConstraint(methodName=’runTest’)
Bases: unittest.case.TestCase
Tests for Constraint class
test_additive_scalar
()
Make sure additive scalars simplify properly
test_additive_scalar_gt1
()
1 can’t be greater than (1 + something positive)
test_bad_elements
()
test_constraintget
()
test_equality_relaxation
()
test_evalfn
()
test_init
()
Test Constraint __init__
test_oper_overload
()
Test Constraint initialization by operator overloading class gpkit.tests.t_constraints.TestMonomialEquality(methodName=’runTest’)
Bases: unittest.case.TestCase
Test monomial equality constraint class
test_inheritance
()
Make sure MonomialEquality inherits from the right things
test_init
()
Test initialization via both operator overloading and __init__
test_non_monomial
()
Try to initialize a MonomialEquality with non-monomial args
test_str
()
Test that MonomialEquality.__str__ returns a string
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class gpkit.tests.t_constraints.TestSignomialInequality(methodName=’runTest’)
Bases: unittest.case.TestCase
Test Signomial constraints
test_init
()
Test initialization and types
test_posyslt1
() class gpkit.tests.t_constraints.TestTight(methodName=’runTest’)
Bases: unittest.case.TestCase
Test tight constraint set
test_posyconstr_in_gp
()
Tests tight constraint set with solve()
test_posyconstr_in_sp
()
test_sigconstr_in_sp
()
Tests tight constraint set with localsolve()
gpkit.tests.t_examples module
Unit testing of tests in docs/source/examples class gpkit.tests.t_examples.TestExamples(methodName=’runTest’)
Bases: unittest.case.TestCase
To test a new example, add a function called test_$EXAMPLENAME, where $EXAMPLENAME is the name of your example in docs/source/examples without the file extension.
This function should accept two arguments (e.g. ‘self’ and ‘example’). The imported example script will be passed to the second: anything that was a global variable (e.g, “sol”) in the original script is available as an attribute (e.g., “example.sol”)
If you don’t want to perform any checks on the example besides making sure it runs, just put “pass” as the function’s body, e.g.: def test_dummy_example(self, example): pass
But it’s good practice to ensure the example’s solution as well, e.g.: def test_dummy_example(self, example): self.assertAlmostEqual(example.sol[”cost”],
3.121)
test_autosweep
( example)
test_beam
( example)
test_debug
( example)
test_external_sp
( example)
test_external_sp2
( example)
test_model_var_access
( example)
test_performance_modeling
( example)
test_primal_infeasible_ex1
( example)
test_primal_infeasible_ex2
( example)
test_relaxation
( example)
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test_simple_box
( example)
test_simple_sp
( example)
test_simpleflight
( example)
test_sin_approx_example
( example)
test_unbounded
( example)
test_vectorize
( example)
test_water_tank
( example)
test_x_greaterthan_1
( example) gpkit.tests.t_examples.assert_logtol( first , second, logtol=1e-06)
Asserts that the logs of two arrays have a given abstol
gpkit.tests.t_keydict module
Test KeyDict class class gpkit.tests.t_keydict.TestKeyDict(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the KeyDict class
test_dictlike
()
test_failed_getattr
()
test_getattr
()
test_setattr
()
test_vector
()
gpkit.tests.t_model module
Tests for GP and SP classes class gpkit.tests.t_model.Box(cost=None, constraints=None, *args, **kwargs)
Bases:
simple box for model testing
setup
() class gpkit.tests.t_model.BoxAreaBounds(cost=None, constraints=None
,
*args
,
Bases:
**kwargs) for testing functionality of separate analysis models
setup
( box) class gpkit.tests.t_model.TestGP(methodName=’runTest’)
Bases: unittest.case.TestCase
Test GeometricPrograms. This TestCase gets run once for each installed solver.
name
= ‘TestGP_’
ndig
= None
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solver
= None
test_601
()
test_additive_constants
()
test_constants_in_objective_1
()
Issue 296
test_constants_in_objective_2
()
Issue 296
test_cost_freeing
()
Test freeing a variable that’s in the cost.
test_exps_is_tuple
() issue 407
test_mdd_example
()
test_posy_simplification
() issue 525
test_sensitivities
()
test_sigeq
()
test_simple_united_gp
()
test_singular
()
Create and solve GP with a singular A matrix
test_terminating_constant_
()
test_trivial_gp
()
Create and solve a trivial GP: minimize x + 2y subject to xy >= 1
The global optimum is (x, y) = (sqrt(2), 1/sqrt(2)).
test_trivial_vector_gp
()
Create and solve a trivial GP with VectorVariables
test_zero_lower_unbounded
()
test_zeroing
() class gpkit.tests.t_model.TestModelNoSolve(methodName=’runTest’)
Bases: unittest.case.TestCase
model tests that don’t require a solver
test_modelname_added
()
test_no_naming_on_var_access
() class gpkit.tests.t_model.TestModelSolverSpecific(methodName=’runTest’)
Bases: unittest.case.TestCase
test cases run only for specific solvers
test_cvxopt_kwargs
() class gpkit.tests.t_model.TestSP(methodName=’runTest’)
Bases: unittest.case.TestCase
test case for SP class – gets run for each installed solver
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name
= ‘TestSP_’
ndig
= None
solver
= None
test_initially_infeasible
()
test_issue180
()
test_partial_sub_signomial
()
Test SP partial x0 initialization
test_relaxation
()
test_sigs_not_allowed_in_cost
()
test_small_named_signomial
()
test_sp_bounded
()
test_sp_initial_guess_sub
()
test_sp_substitutions
()
test_trivial_sp
()
test_trivial_sp2
()
test_unbounded_debugging
()
Test nearly-dual-feasible problems
test_values_vs_subs
() class gpkit.tests.t_model.Thing(cost=None, constraints=None, *args, **kwargs)
Bases:
a thing, for model testing
setup
( length) gpkit.tests.t_model.test alias of TestSP_cvxopt gpkit.tests.t_model.testcase alias of
gpkit.tests.t_nomial_array module
Tests for NomialArray class class gpkit.tests.t_nomial_array.TestNomialArray(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the NomialArray class. Also tests VectorVariable, since VectorVariable returns a NomialArray
test_array_mult
()
test_constraint_gen
()
test_elementwise_mult
()
test_empty
()
test_getitem
()
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82 test_left_right
()
test_ndim
()
test_outer
()
test_prod
()
test_shape
()
test_substition
()
test_sum
()
test_units
()
gpkit.tests.t_nomials module
Tests for Monomial, Posynomial, and Signomial classes class gpkit.tests.t_nomials.TestMonomial(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the Monomial class
test_div
()
Test Monomial division
test_eq_ne
()
Test equality and inequality comparators
test_init
()
Test multiple ways to create a Monomial
test_latex
()
Test latex string creation
test_mul
()
Test monomial multiplication
test_numerical_precision
() not sure what to test here, placeholder for now
test_pow
()
Test Monomial exponentiation
test_repr
()
Simple tests for __repr__, which prints more than str
test_str_with_units
()
Make sure __str__() works when units are involved
test_units
() make sure multiplication with units works (issue 492) class gpkit.tests.t_nomials.TestPosynomial(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the Posynomial class
test_constraint_gen
()
Test creation of Constraints via operator overloading
test_diff
()
Test differentiation (!!)
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test_eq
()
Test Posynomial __eq__
test_eq_units
()
test_init
()
Test Posynomial construction
test_integer_division
()
Make sure division by integer doesn’t use Python integer division
test_mono_lower_bound
()
Test monomial approximation
test_posyposy_mult
()
Test multiplication of Posynomial with Posynomial
test_simplification
()
Make sure like monomial terms get automatically combined class gpkit.tests.t_nomials.TestSignomial(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the Signomial class
test_eq_ne
()
Test Signomial equality and inequality operators
test_init
()
Test Signomial construction
gpkit.tests.t_small module
Tests for small_classes.py and small_scripts.py
class gpkit.tests.t_small.TestCootMatrix(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the CootMatrix class
test_shape
() class gpkit.tests.t_small.TestHashVector(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the HashVector class
test_init
()
Make sure HashVector acts like a dict
test_mul_add
()
Test multiplication and addition
test_neg
()
Test negation
test_pow
()
Test exponentiation class gpkit.tests.t_small.TestSmallScripts(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for gpkit.small_scripts
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test_pint_366
()
test_unitstr
()
gpkit.tests.t_solution_array module
Tests for SolutionArray class class gpkit.tests.t_solution_array.TestResultsTable(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for results_table()
test_nan_printing
()
Test that solution prints when it contains nans
test_result_access
() class gpkit.tests.t_solution_array.TestSolutionArray(methodName=’runTest’)
Bases: unittest.case.TestCase
Unit tests for the SolutionArray class
test_call
()
test_call_units
()
test_call_vector
()
test_key_options
()
test_subinto
()
test_table
()
test_units_sub
()
gpkit.tests.t_sub module
Test substitution capability across gpkit class gpkit.tests.t_sub.TestModelSubs(methodName=’runTest’)
Bases: unittest.case.TestCase
Test substitution for Model objects
test_bad_gp_sub
()
test_bad_subinplace
()
test_calcconst
()
test_getkey
()
test_model_composition_units
()
test_model_recursion
()
test_persistence
()
test_phantoms
()
test_quantity_sub
()
test_skipfailures
()
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test_united_sub_sweep
()
test_vector_init
()
test_vector_sub
()
test_vector_sweep
()
Test sweep involving VectorVariables class gpkit.tests.t_sub.TestNomialSubs(methodName=’runTest’)
Bases: unittest.case.TestCase
Test substitution for nomial-family objects
test_basic
()
Basic substitution, symbolic
test_dimensionless_units
()
test_numeric
()
Basic substitution of numeric value
test_scalar_units
()
test_signomial
()
Test Signomial substitution
test_string_mutation
()
test_unitless_monomial_sub
()
Tests that dimensionless and undimensioned subs can interact.
test_variable
()
Test special single-argument substitution for Variable
test_vector
()
gpkit.tests.t_tools module
Tests for tools module class gpkit.tests.t_tools.TestTools(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for math models
test_binary_sweep_tree
()
test_composite_objective
()
test_fmincon_generator
()
Test fmincon comparison tool
test_fmincon_generator_logspace
()
Test fmincon comparison tool (logspace)
test_te_exp_minus1
()
Test Taylor expansion of e^x - 1
test_te_secant
()
Test Taylor expansion of secant(var)
test_te_tangent
()
Test Taylor expansion of tangent(var)
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86
gpkit.tests.t_tools.assert_logtol( first , second, logtol=1e-06)
Asserts that the logs of two arrays have a given abstol
gpkit.tests.t_vars module
Test VarKey, Variable, VectorVariable, and ArrayVariable classes class gpkit.tests.t_vars.TestArrayVariable(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the ArrayVariable class
test_is_vector_variable
()
Make sure ArrayVariable is a shortcut to VectorVariable (we want to know if this changes).
test_str
()
Make sure string looks something like a numpy array class gpkit.tests.t_vars.TestVarKey(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the VarKey class
test_dict_key
() make sure variables are well-behaved dict keys
test_eq_neq
()
Test boolean equality operators
test_init
()
Test VarKey initialization
test_repr
()
Test __repr__ method
test_units_attr
()
Make sure VarKey objects have a units attribute class gpkit.tests.t_vars.TestVariable(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the Variable class
test_eq_ne
()
test_hash
()
Hashes should collide independent of units
test_init
()
Test Variable initialization
test_to
()
test_unit_parsing
()
test_value
()
Detailed tests for value kwarg of __init__ class gpkit.tests.t_vars.TestVectorVariable(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for the VectorVariable class. Note: more relevant tests in t_posy_array.
test_constraint_creation_units
()
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test_init
()
Test VectorVariable initialization class gpkit.tests.t_vars.TestVectorize(methodName=’runTest’)
Bases: unittest.case.TestCase
TestCase for gpkit.vectorize
test_shapes
()
gpkit.tests.test_repo module
Implements tests for all external repositories.
gpkit.tests.test_repo.call_and_retry( cmd , max_iterations=5, delay=5)
Tries max_iterations times (waiting d each time) to run a command gpkit.tests.test_repo.get_settings()
Gets settings from a TESTCONFIG file gpkit.tests.test_repo.git_clone( repo , branch=’master’)
Tries several times to clone a given repository gpkit.tests.test_repo.pip_install( package , local=False)
Tries several times to install a pip package gpkit.tests.test_repo.test_repo( repo=’.’ , xmloutput=False)
Test repository.
If no repo name given, runs in current directory. Otherwise, assumes is in directory above the repo with a shared gpkit-models repository.
gpkit.tests.test_repo.test_repos( repos=None , xmloutput=False)
Get the list of external repos to test, and test.
Module contents
GPkit testing module
gpkit.tools package
Submodules gpkit.tools.autosweep module
Tools for optimal fits to GP sweeps class gpkit.tools.autosweep.BinarySweepTree(bounds, sols, sweptvar, costposy)
Bases: object
Spans a line segment. May contain two subtrees that divide the segment.
bounds [two-element list] The left and right boundaries of the segment sols [two-element list] The left and right solutions of the segment costs [array] The left and right logcosts of the segment splits [None or two-element list] If not None, contains the left and right subtrees
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splitval [None or float] The worst-error point, where the split will be if tolerance is too low splitlb [None or float] The cost lower bound at splitval splitub [None or float] The cost upper bound at splitval
add_split
( splitval , splitsol)
Creates subtrees from bounds[0] to splitval and splitval to bounds[1]
add_splitcost
( splitval , splitlb, splitub)
Adds a splitval, lower bound, and upper bound
cost_at
(
_ , value, bound=None)
Logspace interpolates between split and costs. Guaranteed bounded.
min_bst
( value)
Returns smallest bst around value.
posy_at
( posy , value)
Logspace interpolates between sols to get posynomial values.
No guarantees, just like a regular sweep.
sample_at
( values)
Creates a SolutionOracle at a given range of values
solarray
Returns a solution array of all the solutions in an autosweep
sollist
Returns a list of all the solutions in an autosweep class gpkit.tools.autosweep.SolutionOracle(bst, sampled_at)
Bases: object
Acts like a SolutionArray for autosweeps
cost_lb
()
Gets cost lower bounds from the BST and units them
cost_ub
()
Gets cost upper bounds from the BST and units them
plot
( posys=None
, axes=None)
Plots the sweep for each posy
solarray
Returns a solution array of all the solutions in an autosweep gpkit.tools.autosweep.autosweep_1d( model , logtol, sweepvar, bounds, **solvekwargs)
Autosweep a model over one sweepvar gpkit.tools.autosweep.get_tol( costs , bounds, sols, variable)
Gets the intersection point and corresponding bounds from two solutions.
gpkit.tools.autosweep.recurse_splits( model
, bst, variable, logtol, solvekwargs, sols)
Recursively splits a BST until logtol is reached
gpkit.tools.fmincon module
A module to facilitate testing GPkit against fmincon
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gpkit.tools.fmincon.generate_mfiles( model , logspace=False , algorithm=’interior-point’ , guess=’ones’ , gradobj=’on’ , gradconstr=’on’ , writefiles=True)
A method for preparing fmincon input files to run a GPkit program
INPUTS: model [GPkit model] The model to replicate in fmincon logspace [Boolean] Whether to re-produce the model in logspace algorithm: [string] Algorithm used by fmincon ‘interior-point’: uses the interior point solver ‘SQP’: uses the sequential quadratic programming solver guess: [string] The type of initial guess used ‘ones’: One for each variable ‘order-ofmagnitude-floor’: The “log-floor” order of magnitude of the GP/SP optimal solution (i.e. O(99)=10)
‘order-of-magnitude-round’: The “log-nearest” order of magnitude of the GP/SP optimal solution (i.e. O(42)=100)
‘almost-exact-solution’: The GP/SP optimal solution rounded to 1 significant figure
OR [list] The actual values of initial guess to use gradconstr: [string] Include analytical constraint gradients? ‘on’: Yes ‘off’: No gradobj: [string] Include analytical objective gradients? ‘on’: Yes ‘off’: No writefiles: [Boolean] whether or not to actually write the m files gpkit.tools.fmincon.make_initial_guess( model , newlist , logspace=False)
Returns initial guess guess=’ones’ ,
gpkit.tools.spdata module
Implements SPData class class gpkit.tools.spdata.SPData(model)
Bases:
Generates matrices describing an SP.
>>>
spdata = SPData(m)
>>>
spdata .
save( 'example_sp.h5' )
save
( filename)
Save spdata to an h5 file.
gpkit.tools.tools module
Non-application-specific convenience methods for GPkit gpkit.tools.tools.composite_objective(
*objectives , **kwargs)
Creates a cost function that sweeps between multiple objectives.
gpkit.tools.tools.mdmake( filename , make_tex=True)
Make a python file and (optional) a pandoc-ready .tex.md file
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gpkit.tools.tools.mdparse( filename , return_tex=False)
Parse markdown file, returning as strings python and (optionally) .tex.md
gpkit.tools.tools.te_exp_minus1( posy , nterm)
Taylor expansion of e^{posy} - 1 posy [gpkit.Posynomial] Variable or expression to exponentiate nterm [int] Number of non-constant terms in resulting Taylor expansion gpkit.Posynomial Taylor expansion of e^{posy} - 1, carried to nterm terms gpkit.tools.tools.te_secant( var , nterm)
Taylor expansion of secant(var).
var [gpkit.monomial] Variable or expression argument nterm [int] Number of non-constant terms in resulting Taylor expansion gpkit.Posynomial Taylor expansion of secant(x), carried to nterm terms gpkit.tools.tools.te_tangent( var , nterm)
Taylor expansion of tangent(var).
var [gpkit.monomial] Variable or expression argument nterm [int] Number of non-constant terms in resulting Taylor expansion gpkit.Posynomial Taylor expansion of tangent(x), carried to nterm terms
Module contents
Contains miscellaneous tools including fmincon comparison tool
Submodules gpkit.build module
Finds solvers, sets gpkit settings, and builds gpkit class gpkit.build.CVXopt
Bases:
CVXopt finder.
look
()
Attempts to import cvxopt.
name
= ‘cvxopt’ class gpkit.build.Mosek
Bases:
MOSEK finder and builder.
build
()
Builds a dynamic library to GPKITBUILD or $HOME/.gpkit
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look
()
Looks in default install locations for latest mosek version.
name
= ‘mosek’
patches
= {‘dgopt.c’: {‘printf(“Number of Hessian non-zeros: %d\\n”,nlh[0]->numhesnz);’: ‘MSK_echotask(task,MSK_STREAM_MSG,”Number of Hessian non-zeros: %d\\n”,nlh[0]->numhesnz);’}, ‘expopt.c’: {‘printf(“Warning: The variable with index \’%d\’ has only positive coefficients akj.\\n The problem is possibly ill-posed.\\n.\\n”,i);’: ‘MSK_echotask(expopttask,MSK_STREAM_MSG, “Warning: The variable with index \’%d\’ has only positive coefficients akj.\\n The problem is possibly ill-posed.\\n.\\n”,i);’, ‘printf(“Warning: The variable with index \’%d\’ has only negative coefficients akj.\\n The problem is possibly ill-posed.\\n”,i);’: ‘MSK_echotask(expopttask,MSK_STREAM_MSG, “Warning: The variable with index \’%d\’ has only negative coefficients akj.\\n The problem is possibly ill-posed.\\n”,i);’, ‘printf (“solsta = %d, prosta = %d\\n”, (int)*solsta,(int)*prosta);’: ‘MSK_echotask(expopttask,MSK_STREAM_MSG, “solsta = %d, prosta = %d\\n”, (int)*solsta,(int)*prosta);’}} class gpkit.build.MosekCLI
Bases:
MOSEK command line interface finder.
look
()
Attempts to run mskexpopt.
name
= ‘mosek_cli’ class gpkit.build.SolverBackend
Bases: object
Inheritable class for finding solvers. Logs.
build
= None
installed
= False
look
= None
name
= None gpkit.build.build_gpkit()
Builds GPkit gpkit.build.call( cmd)
Calls subprocess. Logs.
gpkit.build.diff( filename , diff_dict)
Applies a simple diff to a file. Logs.
gpkit.build.isfile( path)
Returns true if there’s a file at $path. Logs.
gpkit.build.log( *args)
Print a line and append it to the log string.
gpkit.build.pathjoin( *args)
Join paths, collating multiple arguments.
gpkit.build.rebuild()
Changes to the installed gpkit directory and runs build_gpkit() gpkit.build.replacedir( path)
Replaces directory at $path. Logs.
gpkit.exceptions module
GPkit-specific Exception classes exception gpkit.exceptions.InvalidGPConstraint
Bases: exceptions.Exception
Raised when a non-GP-compatible constraint is used in a GP
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gpkit.keydict module
Implements KeyDict and KeySet classes class gpkit.keydict.KeyDict(*args, **kwargs)
Bases: dict
KeyDicts do two things over a dict: map keys and collapse arrays.
>>>> kd = gpkit.keydict.KeyDict()
If .keymapping is True, a KeyDict keeps an internal list of VarKeys as canonical keys, and their values can be accessed with any object whose key attribute matches one of those VarKeys, or with strings matching any of the multiple possible string interpretations of each key:
For example, after creating the KeyDict kd and setting kd[x] = v (where x is a Variable or VarKey), v can be accessed with by the following keys:
•x
•x.key
•x.name (a string)
•“x_modelname” (x’s name including modelname)
Note that if a item is set using a key that does not have a .key attribute, that key can be set and accessed normally.
If .collapse_arrays is True then VarKeys which have a shape parameter (indicating they are part of an array) are stored as numpy arrays, and automatically de-indexed when a matching VarKey with a particular idx parameter is used as a key.
See also: gpkit/tests/t_keydict.py.
collapse_arrays
= True
get
( key , alternative=<type ‘exceptions.KeyError’>)
keymapping
= True
parse_and_index
( key)
Returns key if key had one, and veckey/idx for indexed veckeys.
update
(
*args , **kwargs)
Iterates through the dictionary created by args and kwargs
update_keymap
()
Updates the keymap with the keys in _unmapped_keys class gpkit.keydict.KeySet(*args, **kwargs)
Bases:
KeyDicts that don’t collapse arrays or store values.
add
( item)
Adds an item to the keyset
collapse_arrays
= False
update
(
*args , **kwargs)
Iterates through the dictionary created by args and kwargs
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gpkit.keydict.clean_value( key , value)
Gets the value of variable-less monomials, so that x.sub({x: gpkit.units.m}) and x.sub({x: gpkit.ureg.m}) are equivalent.
Also converts any quantities to the key’s units, because quantities can’t/shouldn’t be stored as elements of numpy arrays.
gpkit.modified_ctypesgen module gpkit.repr_conventions module
Repository for representation standards gpkit.repr_conventions.unitstr( units , into=’%s’, options=’~’, dimless=’‘)
Returns the string corresponding to an object’s units.
gpkit.small_classes module
Miscellaneous small classes class gpkit.small_classes.CootMatrix(*args, **kwargs)
Bases:
gpkit.small_classes.CootMatrix
A very simple sparse matrix representation.
append
( row , col, data)
Appends entry to matrix.
dot
( arg)
Returns dot product with arg.
tocoo
()
Converts to another type of matrix.
tocsc
()
Converts to another type of matrix.
tocsr
()
Converts to a Scipy sparse csr_matrix
todense
()
Converts to another type of matrix.
todia
()
Converts to another type of matrix.
todok
()
Converts to another type of matrix.
gpkit.small_classes.CootMatrixTuple alias of
class gpkit.small_classes.Count
Bases: object
Like python 2’s itertools.count, for Python 3 compatibility.
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next
()
Increment self.count and return it class gpkit.small_classes.DictOfLists
Bases: dict
A hierarchy of dicionaries, with lists at the bottom.
append
( sol)
Appends a dict (of dicts) of lists to all held lists.
atindex
( i)
Indexes into each list independently.
classify
( cls)
Converts dictionaries whose first key isn’t a string to given class.
to_united_array
( unitless_keys=() , united=False)
Converts all lists into array, potentially grabbing units from keys.
class gpkit.small_classes.HashVector(*args, **kwargs)
Bases: dict
A simple, sparse, string-indexed vector. Inherits from dict.
The HashVector class supports element-wise arithmetic: any undeclared variables are assumed to have a value of zero.
arg : iterable
>>>
x = gpkit .
nomials .
Monomial( 'x' )
>>>
exp = gpkit .
small_classes .
HashVector({x: 2 }) class gpkit.small_classes.SolverLog(verbosity=0, output=None, *args, **kwargs)
Bases: list
Adds a write method to list so it’s file-like and can replace stdout.
write
( writ)
Append and potentially write the new line.
gpkit.small_classes.matrix_converter( name)
Generates conversion function.
gpkit.small_scripts module
Assorted helper methods gpkit.small_scripts.is_sweepvar( sub)
Determines if a given substitution indicates a sweep.
gpkit.small_scripts.latex_num( c)
Returns latex string of numbers, potentially using exponential notation.
gpkit.small_scripts.mag( c)
Return magnitude of a Number or Quantity gpkit.small_scripts.nomial_latex_helper( c
, pos_vars, neg_vars)
Combines (varlatex, exponent) tuples, separated by positive vs negative exponent, into a single latex string
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gpkit.small_scripts.try_str_without( item , excluded)
Try to call item.str_without(excluded); fall back to str(item) gpkit.small_scripts.veckeyed( key)
Return a veckey version of a VarKey
gpkit.solution_array module
Defines SolutionArray class class gpkit.solution_array.SolutionArray
Bases:
gpkit.small_classes.DictOfLists
A dictionary (of dictionaries) of lists, with convenience methods.
cost : array variables: dict of arrays sensitivities: dict containing: monomials : array posynomials : array variables: dict of arrays localmodels [NomialArray] Local power-law fits (small sensitivities are cut off)
>>> import gpkit
>>> import numpy as np
>>>
x = gpkit .
Variable( "x" )
>>>
x_min = gpkit .
Variable( "x_ {min} " , 2 )
>>>
sol = gpkit .
Model(x, [x >= x_min]) .
solve(verbosity = 0 )
>>>
>>>
# VALUES
>>>
values = [sol(x), sol .
subinto(x), sol[ "variables" ][ "x" ]]
>>> assert
all (np .
array(values) == 2 )
>>>
>>>
# SENSITIVITIES
>>>
senss = [sol .
sens(x_min), sol .
sens(x_min)]
>>>
senss .
append(sol[ "sensitivities" ][ "variables" ][ "x_ {min} " ])
>>> assert
all (np .
array(senss) == 1 )
plot
( posys=None
, axes=None)
Plots a sweep for each posy
program
= None
subinto
( posy)
Returns NomialArray of each solution substituted into posy.
summary
( showvars=() , ntopsenss=5)
Print summary table, showing top sensitivities and no constants
table
( showvars=() , tables=(‘cost’, ‘sweepvariables’, ‘freevariables’, ‘constants’, ‘sensitivities’) , **kwargs)
A table representation of this SolutionArray tables: Iterable
Which to print of (“cost”, “sweepvariables”, “freevariables”, “constants”, “sensitivities”) fixedcols: If true, print vectors in fixed-width format latex: int
If > 0, return latex format (options 1-3); otherwise plain text
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included_models: Iterable of strings If specified, the models (by name) to include excluded_models: Iterable of strings If specified, model names to exclude str
table_titles
= {‘variables’: ‘Variables’, ‘freevariables’: ‘Free Variables’, ‘sweepvariables’: ‘Sweep Variables’, ‘constants’: ‘Constants’} gpkit.solution_array.insenss_table( data
, _, maxval=0.1, **kwargs)
Returns insensitivity table lines gpkit.solution_array.results_table( data , title, minval=0, printunits=True, fixedcols=True , varfmt=’%s : ‘, valfmt=’%-.4g
‘ , vecfmt=’%-8.3g’, included_models=None, excluded_models=None , latex=False, sortbyvals=False , **_)
Pretty string representation of a dict of VarKeys Iterable values are handled specially (partial printing) data: dict whose keys are VarKey’s data to represent in table title: string minval: float skip values with all(abs(value)) < minval printunits: bool fixedcols: bool if True, print rhs (val, units, label) in fixed-width cols varfmt: string format for variable names valfmt: string format for scalar values vecfmt: string format for vector values latex: int If > 0, return latex format (options 1-3); otherwise plain text included_models: Iterable of strings If specified, the models (by name) to include excluded_models: Iterable of strings If specified, model names to exclude sortbyvals [boolean] If true, rows are sorted by their average value instead of by name.
gpkit.solution_array.senss_table( data ,
**kwargs) showvars=() ,
Returns sensitivity table lines title=’Sensitivities’ gpkit.solution_array.topsenss_filter( data , showvars, nvars=5)
Filters sensitivities down to top N vars
, gpkit.solution_array.topsenss_table( data , showvars, nvars=5, **kwargs)
Returns top sensitivity table lines
96
gpkit.varkey module
Defines the VarKey class class gpkit.varkey.VarKey(name=None, **kwargs)
Bases: object
An object to correspond to each ‘variable name’.
name [str, VarKey, or Monomial] Name of this Variable, or object to derive this Variable from.
Chapter 11. Glossary
gpkit Documentation, Release 0.5.3
** kwargs : Any additional attributes, which become the descr attribute (a dict).
VarKey with the given name and descr.
eq_ignores
= frozenset([’units’, ‘value’])
latex
( excluded=None)
Returns latex representation.
latex_unitstr
()
Returns latex unitstr
naming
Returns this varkey’s naming tuple
new_unnamed_id
()
Increment self.count and return it
str_without
( excluded=None)
Returns string without certain fields (such as ‘models’).
subscripts
= (‘models’, ‘idx’)
unitstr
( units , into=’%s’, options=’~’, dimless=’‘)
Returns the string corresponding to an object’s units.
Module contents
GP and SP Modeling Package
For examples please see the examples folder.
Requirements
numpy MOSEK or CVXOPT scipy(optional): for complete sparse matrix support sympy(optional): for latex printing in iPython Notebook
Attributes
settings [dict] Contains settings loaded from ./env/settings class gpkit.GPkitUnits
Bases: object
Return monomials instead of Quantitites class gpkit.NamedVariables(model)
Bases: object
Creates an environment in which all variables have a model name and num appended to their varkeys.
class gpkit.SignomialsEnabled
Bases: object
Class to put up and tear down signomial support in an instance of GPkit.
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>>> import gpkit
>>>
x = gpkit .
Variable( "x" )
>>>
y = gpkit .
Variable( "y" , 0.1
)
>>> with
SignomialsEnabled():
>>>
constraints = [x >= 1 y]
>>>
gpkit .
Model(x, constraints) .
localsolve() class gpkit.Vectorize(dimension_length)
Bases: object
Creates an environment in which all variables are exended in an additional dimension.
gpkit.begin_variable_naming( model)
Appends a model name and num to the environment.
gpkit.disable_units()
Disables units support in a particular instance of GPkit.
Posynomials created after calling this are incompatible with those created before.
If gpkit is imported multiple times, this needs to be run each time.
The correct way to call this is: import gpkit gpkit.disable_units()
The following will not have the intended effect: from gpkit import disable_units disable_units() gpkit.enable_units( path=None)
Enables units support in a particular instance of GPkit.
Posynomials created after calling this are incompatible with those created before.
If gpkit is imported multiple times, this needs to be run each time.
gpkit.end_variable_naming()
Pops a model name and num from the environment.
gpkit.load_settings( path=’/home/docs/checkouts/readthedocs.org/user_builds/gpkit/envs/latest/local/lib/python2.7/sitepackages/gpkit/env/settings’)
Load the settings file at SETTINGS_PATH; return settings dict
98 Chapter 11. Glossary
If you use GPkit, please cite it with the following bibtex:
@Misc
{gpkit, author = {Edward Burnell
and
Warren Hoburg}, title = {GPkit software
for
geometric programming}, howpublished = {\url{https: // github .
com / hoburg / gpkit}}, year = { 2017 },
} note = {Version 0.5
.
3 }
CHAPTER
12
Citing GPkit
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100 Chapter 12. Citing GPkit
We thank the following contributors for helping to improve GPkit:
• Marshall Galbraith for setting up continuous integration.
• Stephen Boyd for inspiration and suggestions.
• Kirsten Bray for designing the GPkit logo.
CHAPTER
13
Acknowledgements
101
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102 Chapter 13. Acknowledgements
CHAPTER
14
Release Notes
This page lists the changes made in each point version of gpkit.
Version 0.5.3
• faster SP solves (#1109)
• LinkedConstraintSet deprecated (#1110)
• Fixes to autosweep, ConstraintSet, interactive
• Solution time is now stored with soltutions (including sweeps/SPs)
• Model strings are divided with slashes (e.g. Airplane/Wing)
Version 0.5.2
• Added new sweep and autosweep methods to Model
– Added plot routines to the results of those routines to make it easy to plot a 1D sweep.
• Added new summary method to solution_array.
– It and table accept iterables of vars, will only print vars in that iterable (or, by default, all vars)
• Cleaned up and documented the interactive submodule
– removed contour and sensitivity plots
– added a 1D-sweep plotting function
– added that plotting function as an option within the control panel interface
• Overhauled and documented three types of variables whose value is determined by functions:
103
gpkit Documentation, Release 0.5.3
– calculated constants
– post-solve calculated variables
– between-GP-solves calculated variables (for Sequential Geometric Programs)
• Fix Bounded and implement debug() for SPs
• Apply subinplace to substitutions dictionary as well
• Require GP substitutions to be Numbers only
• Extend Bounded to one-sided bounds
• Print model’s numbers by default, unless "modelnums" in exclude
• Implement lazy keymapping, allowing GP/SP results to be KeyDicts
• Handle Signomial Inequalities that become Posynomial Inequalities after substitution
• Various documentation updates
• Various bug fixes
Version 0.5.1
• O(N) sums and monomial products
• Warn about invalid ConstraintSet elements
• allow setting Tight tolerance as a class attribute
• full backwards compatibility for __init__ methods
• scripts to test remote repositories
• minor fixes, tests, and refactors
• 3550 lines of code, 1800 lines of tests, 1700 lines of docstring. (not counting interactive)
Version 0.5.0
• No longer recommend the use of linked variables and subinplace (see below)
• Switched default solver to MOSEK
• Added Linked Variable diagram (PR #915)
• Changed how overloaded operators interact with pint (PR #938)
• Added and documented debugging tools (PR #933)
• Added and documented vectorization tools
• Documented modular model construction
• 3200 lines of code, 1800 lines of tests, 1700 lines of docstring. (not counting interactive)
104 Chapter 14. Release Notes
gpkit Documentation, Release 0.5.3
Changes to named models / Model inheritance
We are deprecating the creation of named submodels with custom __init__ methods. Previously, variables created during __init__ in any class inheriting from Model were replaced by a copy with
__class__.__name__
added as varkey metadata. This was slow, a bit irregular, and hacky.
We’re moving to an explicitly-irregular setup method, which (if declared for a class inheriting from Model) is automatically called during Model.__init__ inside a NamedVariables(self.
__class__.__name__) environment. This 1) handles the naming of variables more explicitly and efficiently, and 2) allows us to capture variables created within setup, so that constants that are not a part of any constraint can be used directly (several examples of such template models are in the new Building
Complex Models documentation).
Model.__init__
calls setup with the arguments given to the constructor, with the exception of the reserved keyword substitutions. This allows for the easy creation of a named model with custom parameter values (as in the documentation’s Beam example). setup methods should return an iterable
(list, tuple, ConstraintSet, ...) of constraints or nothing if the model contains no constraints. To declare a submodel cost, set self.cost during setup. However, we often find declaring a model’s cost explicitly just before solving to be a more legible practice.
In addition to permitting us to name variables at creation, and include unconstrained variables in a model, we hope that setup methods will clarify the side effects of named model creation.
Version 0.4.2
• prototype handling of SignomialEquality constraints
• fix an issue where solution tables printed incorrect units (despite the units being correct in the
SolutionArray data structure)
• fix controlpanel slider display for newer versions of ipywidgets
• fix an issue where identical unit-ed variables could have different hashes
• Make the text of several error messages more informative
• Allow monomial approximation of monomials
• bug fixes and improvements to TightConstraintSet
• Don’t print results table automatically (it was unwieldy for large models). To print it, print sol.table()
.
• Use cvxopt’s ldl kkt solver by default for more robustness to rank issues
• Improved ConstraintSet.__getitem__, only returns top-level Variable
• Move toward the varkeys of a ConstraintSet being an immutable set
• CPI update
• numerous pylint fixes
• BoundedConstraint sets added for dual feasibility debugging
• SP sweep compatibility
14.5. Version 0.4.2
105
gpkit Documentation, Release 0.5.3
Version 0.4.0
• New model for considering constraints: all constraints are considered as sets of constraints which may contain other constraints, and are asked for their substitutions / posynomial less than 1 representation as late as possible.
• Support for calling external code during an SP solve.
• New class KeyDict to allow referring to variables by name or with objects.
• Many many other bug fixes, speed ups, and refactors under the hood.
Version 0.3.4
• Modular / model composition fixes and improvements
• Working controlpanel() for Model
• ipynb and numpy dependency fixes
• printing fixes
• El Capitan fix
• slider widgets now have units
Version 0.3.2
• Assorted bug fixes
• Assorted internal improvements and simplifications
• Refactor signomial constraints, resulting in smarter SP heuristic
• Simplify and strengthen equality testing for nomials
• Not counting submodules, went from 2400 to 2500 lines of code and from 1050 to 1170 lines of docstrings and comments.
Version 0.3
• Integrated GP and SP creation under the Model class
• Improved and simplified under-the-hood internals of GPs and SPs
• New experimental SP heuristic
• Improved test coverage
• Handles vectors which are partially constants, partially free
• Simplified interaction with Model objects and made it more pythonic
• Added SP “step” method to allow single-stepping through an SP
• Isolated and corrected some solver-specific behavior
• Fully allowed substitutions of variables for 0 (commit 4631255)
106 Chapter 14. Release Notes
gpkit Documentation, Release 0.5.3
• Use “with” to create a signomials environment (commit cd8d581)
• Continuous integration improvements, thanks @galbramc !
• Not counting subpackages, went from 2200 to 2400 lines of code (additions were mostly longer error messages) and from 650 to 1050 lines of docstrings and comments.
• Add automatic feasibility-analysis methods to Model and GP
• Simplified solver logging and printing, making it easier to access solver output.
Version 0.2
• Various bug fixes
• Python 3 compatibility
• Added signomial programming support (alpha quality, may be wrong)
• Added composite objectives
• Parallelized sweeping
• Better table printing
• Linked sweep variables
• Better error messages
• Closest feasible point capability
• Improved install process (no longer requires ctypesgen; auto-detects MOSEK version)
• Added examples: wind turbine, modular GP, examples from 1967 book, maintenance (part replacement)
• Documentation grew by ~70%
• Added Advanced Commands section to documentation
• Many additional unit tests (more than doubled testing lines of code)
14.10. Version 0.2
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108 Chapter 14. Release Notes
Python Module Index g
gpkit
,
gpkit.build
,
gpkit.constraints
,
gpkit.constraints.array
,
gpkit.constraints.bounded
,
gpkit.constraints.costed
,
gpkit.constraints.geometric_program
,
gpkit.constraints.model
,
gpkit.constraints.prog_factories
,
gpkit.constraints.relax
,
gpkit.constraints.set
,
gpkit.constraints.sigeq
,
gpkit.constraints.signomial_program
,
gpkit.constraints.single_equation
,
gpkit.constraints.tight
,
gpkit.exceptions
,
gpkit.interactive
,
gpkit.interactive.linking_diagram
,
gpkit.interactive.plot_sweep
,
gpkit.interactive.plotting
,
gpkit.interactive.ractor
,
gpkit.interactive.sensitivity_map
,
gpkit.keydict
,
gpkit.nomials
,
gpkit.nomials.array
,
gpkit.nomials.data
,
gpkit.nomials.nomial_core
,
gpkit.nomials.nomial_math
,
gpkit.nomials.substitution
,
gpkit.nomials.variables
,
gpkit.repr_conventions
,
gpkit.small_classes
,
gpkit.small_scripts
,
gpkit.solution_array
,
gpkit.tests
,
gpkit.tests.diff_output
,
gpkit.tests.from_paths
,
gpkit.tests.helpers
,
gpkit.tests.run_tests
,
gpkit.tests.t_constraints
,
gpkit.tests.t_examples
,
gpkit.tests.t_keydict
,
gpkit.tests.t_model
,
gpkit.tests.t_nomial_array
,
gpkit.tests.t_nomials
,
gpkit.tests.t_small
,
gpkit.tests.t_solution_array
,
gpkit.tests.t_sub
,
gpkit.tests.t_tools
,
gpkit.tests.t_vars
,
gpkit.tests.test_repo
,
gpkit.tools
,
gpkit.tools.autosweep
,
gpkit.tools.fmincon
,
gpkit.tools.spdata
,
gpkit.tools.tools
,
gpkit.varkey
,
109
gpkit Documentation, Release 0.5.3
110 Python Module Index
Index
A
add() (gpkit.keydict.KeySet method),
add_filetest() (in module gpkit.tests.from_paths),
add_split() (gpkit.tools.autosweep.BinarySweepTree
method),
add_splitcost() (gpkit.tools.autosweep.BinarySweepTree
assert_logtol() (in module gpkit.tests.t_tools),
assign_axes() (in module gpkit.interactive.plot_sweep),
atindex() (gpkit.small_classes.DictOfLists method),
autosweep() (gpkit.constraints.model.Model method),
autosweep_1d() (in module gpkit.tools.autosweep),
method),
append() (gpkit.constraints.set.ConstraintSet method),
B
append() (gpkit.small_classes.CootMatrix method),
begin_variable_naming() (in module gpkit),
append() (gpkit.small_classes.DictOfLists method),
BinarySweepTree (class in gpkit.tools.autosweep), append_sub() (in module gpkit.nomials.substitution),
Bounded (class in gpkit.constraints.bounded),
array_constraint() (in module gpkit.nomials.array),
Box (class in gpkit.tests.t_model),
ArrayConstraint (class in gpkit.constraints.array),
BoxAreaBounds (class in gpkit.tests.t_model),
ArrayVariable (class in gpkit.nomials.variables),
build (gpkit.build.SolverBackend attribute), as_approxsgt() (gpkit.nomials.nomial_math.SignomialInequality
method),
build_gpkit() (in module gpkit.build),
as_approxsgt() (gpkit.nomials.nomial_math.SingleSignomialEquality
method),
C
as_approxslt() (gpkit.nomials.nomial_math.SignomialInequality
method),
c (gpkit.nomials.nomial_core.Nomial attribute),
as_approxslt() (gpkit.nomials.nomial_math.SingleSignomialEquality
method),
call_and_retry() (in module gpkit.tests.test_repo),
as_gpconstr() (gpkit.constraints.set.ConstraintSet
check_solution() (gpkit.constraints.geometric_program.GeometricProgram
method),
method),
as_gpconstr() (gpkit.nomials.nomial_math.PosynomialInequality
method),
clean() (in module gpkit.tests.from_paths),
as_gpconstr() (gpkit.nomials.nomial_math.SignomialInequality
method),
close() (gpkit.tests.helpers.NullFile method),
as_gpconstr() (gpkit.nomials.nomial_math.SingleSignomialEquality
method),
collapse_arrays (gpkit.keydict.KeySet attribute),
as_posyslt1() (gpkit.constraints.set.ConstraintSet
colorfn_gen() (in module method),
kit.interactive.sensitivity_map),
as_posyslt1() (gpkit.nomials.nomial_math.PosynomialInequality
gpmethod),
composite_objective() (in module gpkit.tools.tools),
as_posyslt1() (gpkit.nomials.nomial_math.SignomialInequality
method),
constraint_latex_list() as_posyslt1() (gpkit.nomials.nomial_math.SingleSignomialEquality
(gpkit.interactive.sensitivity_map.SensitivityMap
method),
method),
assert_logtol() (in module gpkit.tests.t_examples),
ConstraintSet (class in gpkit.constraints.set),
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gpkit Documentation, Release 0.5.3
ConstraintsRelaxed (class in gpkit.constraints.relax),
ConstraintsRelaxedEqually (class in gpget() (gpkit.keydict.KeyDict method),
get_settings() (in module gpkit.tests.test_repo),
kit.constraints.relax),
get_tol() (in module gpkit.tools.autosweep),
controlpanel() (gpkit.constraints.costed.CostedConstraintSet
git_clone() (in module gpkit.tests.test_repo),
method),
convert_to() (gpkit.nomials.nomial_core.Nomial
method),
CootMatrix (class in gpkit.small_classes),
CootMatrixTuple (in module gpkit.small_classes),
cost_at() (gpkit.tools.autosweep.BinarySweepTree
method),
cost_lb() (gpkit.tools.autosweep.SolutionOracle method),
gp() (gpkit.constraints.model.Model method), gpkit (module),
gp() (gpkit.constraints.signomial_program.SignomialProgram
method),
gpkit.build (module),
gpkit.constraints (module),
gpkit.constraints.array (module),
gpkit.constraints.bounded (module),
cost_ub()
D
(gpkit.tools.autosweep.SolutionOracle
method),
CostedConstraintSet (class in gpkit.constraints.costed),
Count (class in gpkit.small_classes),
CVXopt (class in gpkit.build),
gpkit.constraints.costed (module),
gpkit.constraints.geometric_program (module),
gpkit.constraints.model (module),
gpkit.constraints.prog_factories (module),
gpkit.constraints.relax (module),
gpkit.constraints.set (module),
gpkit.constraints.sigeq (module),
gpkit.constraints.signomial_program (module),
gpkit.constraints.single_equation (module),
debug() (gpkit.constraints.model.Model method),
gpkit.constraints.tight (module),
DictOfLists (class in gpkit.small_classes),
gpkit.exceptions (module),
diff() (gpkit.nomials.data.NomialData method),
gpkit.interactive (module),
diff() (gpkit.nomials.nomial_math.Signomial method),
gpkit.interactive.linking_diagram (module),
diff() (in module gpkit.build),
gpkit.interactive.plot_sweep (module),
diff() (in module gpkit.tests.diff_output),
gpkit.interactive.plotting (module),
disable_units() (in module gpkit),
gpkit.interactive.ractor (module),
dot() (gpkit.small_classes.CootMatrix method),
gpkit.interactive.sensitivity_map (module),
E
gpkit.keydict (module),
gpkit.nomials (module),
enable_units() (in module gpkit),
gpkit.nomials.array (module),
end_variable_naming() (in module gpkit),
gpkit.nomials.data (module),
eq_ignores (gpkit.varkey.VarKey attribute),
gpkit.nomials.nomial_core (module),
F
gpkit.nomials.nomial_math (module),
gpkit.nomials.substitution (module),
firstgp() (gpkit.constraints.signomial_program.SignomialProgram
method),
gpkit.repr_conventions (module),
flat() (gpkit.constraints.set.ConstraintSet method),
gpkit.small_classes (module),
gpkit.small_scripts (module),
format_and_label_axes() (in module gpkit.interactive.plot_sweep),
attribute),
gpkit.solution_array (module), func_opers (gpkit.constraints.single_equation.SingleEquationConstraint
gpkit.tests.diff_output (module),
gpkit.tests.from_paths (module),
G
gpkit.tests.helpers (module),
gen() (gpkit.constraints.geometric_program.GeometricProgram
method),
gpkit.tests.t_constraints (module),
genA() (in module gpkit.constraints.geometric_program), gpkit.tests.t_examples (module),
gpkit.tests.t_keydict (module),
generate_example_tests() (in module gpkit.tests.helpers), gpkit.tests.t_model (module),
gpkit.tests.t_nomial_array (module),
generate_mfiles() (in module gpkit.tools.fmincon),
gpkit.tests.t_nomials (module),
GeometricProgram (class in gpgpkit.tests.t_small (module),
kit.constraints.geometric_program),
gpkit.tests.t_solution_array (module),
112 Index
gpkit Documentation, Release 0.5.3
I
gpkit.tests.t_sub (module),
gpkit.tests.t_tools (module),
gpkit.tests.t_vars (module),
gpkit.tests.test_repo (module),
gpkit.tools (module),
gpkit.tools.autosweep (module),
gpkit.tools.fmincon (module),
gpkit.tools.spdata (module),
gpkit.tools.tools (module),
gpkit.varkey (module),
GPkitUnits (class in gpkit),
H
HashVector (class in gpkit.small_classes),
import_tests() (in module gpkit.tests.run_tests),
init_from_nomials() (gpkit.nomials.data.NomialData
method),
insenss_table() (in module gpkit.solution_array),
installed (gpkit.build.SolverBackend attribute),
interact() (gpkit.constraints.costed.CostedConstraintSet
method),
InvalidGPConstraint,
is_sweepvar() (in module gpkit.small_scripts),
isfile() (in module gpkit.build),
K
log() (in module gpkit.build),
logged_example_testcase() (in module kit.tests.helpers),
look (gpkit.build.SolverBackend attribute),
look() (gpkit.build.CVXopt method),
look() (gpkit.build.Mosek method),
look() (gpkit.build.MosekCLI method),
M
gpmag() (in module gpkit.small_scripts),
make_initial_guess() (in module gpkit.tools.fmincon),
matrix_converter() (in module gpkit.small_classes),
mdmake() (in module gpkit.tools.tools),
mdparse() (in module gpkit.tools.tools),
min_bst() (gpkit.tools.autosweep.BinarySweepTree
method),
Model (class in gpkit.constraints.model),
mono_approximation() (gpkit.nomials.nomial_math.Monomial
method),
mono_approximation() (gpkit.nomials.nomial_math.Signomial
method),
mono_lower_bound() (gpkit.nomials.nomial_math.Posynomial method),
Monomial (class in gpkit.nomials.nomial_math),
MonomialEquality (class in gpkit.nomials.nomial_math),
Mosek (class in gpkit.build),
MosekCLI (class in gpkit.build),
key (gpkit.nomials.variables.Variable attribute),
KeyDict (class in gpkit.keydict),
keymapping (gpkit.keydict.KeyDict attribute),
KeySet (class in gpkit.keydict),
N
L
name (gpkit.build.CVXopt attribute),
latex (gpkit.interactive.sensitivity_map.SensitivityMap
name (gpkit.build.Mosek attribute),
attribute),
name (gpkit.build.MosekCLI attribute),
latex() (gpkit.constraints.set.ConstraintSet method),
name (gpkit.build.SolverBackend attribute),
latex() (gpkit.constraints.single_equation.SingleEquationConstraint
method),
name (gpkit.tests.t_model.TestGP attribute),
latex() (gpkit.nomials.array.NomialArray method),
name (gpkit.tests.t_model.TestSP attribute),
NamedVariables (class in gpkit),
latex() (gpkit.nomials.nomial_core.Nomial method),
latex() (gpkit.varkey.VarKey method),
latex_unitstr() (gpkit.varkey.VarKey method),
naming (gpkit.constraints.model.Model attribute), latex_num() (in module gpkit.small_scripts),
naming (gpkit.varkey.VarKey attribute),
latex_opers (gpkit.constraints.single_equation.SingleEquationConstraint
attribute),
ndig (gpkit.tests.t_model.TestSP attribute),
new_test() (in module gpkit.tests.helpers),
left (gpkit.nomials.array.NomialArray attribute),
new_unnamed_id() (gpkit.varkey.VarKey method),
NewDefaultSolver (class in gpkit.tests.helpers),
linking_diagram() (in module gpkit.interactive.linking_diagram),
newtest_fn() (in module gpkit.tests.from_paths),
load_settings() (in module gpkit),
next() (gpkit.small_classes.Count method),
localsolve() (gpkit.constraints.model.Model method),
Nomial (class in gpkit.nomials.nomial_core),
localsolve() (gpkit.constraints.signomial_program.SignomialProgram
method),
NomialArray (class in gpkit.nomials.array),
NomialData (class in gpkit.nomials.data),
Index 113
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nomials (gpkit.nomials.nomial_math.ScalarSingleEquationConstraint
attribute),
non_dimensionalize() (in kit.nomials.nomial_math),
module
NullFile (class in gpkit.tests.helpers),
num (gpkit.constraints.model.Model attribute),
gpreltol (gpkit.constraints.tight.Tight attribute), replacedir() (in module gpkit.build),
reset_varkeys() (gpkit.constraints.costed.CostedConstraintSet
method),
reset_varkeys() (gpkit.constraints.set.ConstraintSet
O
outer() (gpkit.nomials.array.NomialArray method),
method),
results_table() (in module gpkit.solution_array),
right (gpkit.nomials.array.NomialArray attribute),
rootconstr_latex() (gpkit.constraints.costed.CostedConstraintSet
P
method),
rootconstr_latex() (gpkit.constraints.set.ConstraintSet
method),
rootconstr_str() (gpkit.constraints.costed.CostedConstraintSet
padleft() (gpkit.nomials.array.NomialArray method),
padright() (gpkit.nomials.array.NomialArray method),
parse_and_index() (gpkit.keydict.KeyDict method),
parse_subs() (in module gpkit.nomials.substitution),
patches (gpkit.build.Mosek attribute),
pathjoin() (in module gpkit.build),
pip_install() (in module gpkit.tests.test_repo),
plot() (gpkit.solution_array.SolutionArray method),
plot() (gpkit.tools.autosweep.SolutionOracle method),
plot_1dsweepgrid() (in module gpmethod),
rootconstr_str() (gpkit.constraints.set.ConstraintSet
method),
run() (in module gpkit.tests.from_paths),
run() (in module gpkit.tests.run_tests),
run_sweep() (in module gpkit.constraints.prog_factories),
run_tests() (in module gpkit.tests.helpers),
kit.interactive.plot_sweep),
plot_convergence() (in module gpkit.interactive.plotting),
posy_at() (gpkit.tools.autosweep.BinarySweepTree
method),
posy_negy() (gpkit.nomials.nomial_math.Signomial
method),
S
sample_at() (gpkit.tools.autosweep.BinarySweepTree
method),
save() (gpkit.tools.spdata.SPData method),
ScalarSingleEquationConstraint (class in gp-
Posynomial (class in gpkit.nomials.nomial_math),
PosynomialInequality (class in gpkit.nomials.nomial_math),
process_result() (gpkit.constraints.bounded.Bounded
method),
process_result() (gpkit.constraints.relax.ConstantsRelaxed
method),
process_result() (gpkit.constraints.set.ConstraintSet
method),
kit.nomials.nomial_math),
sens_from_dual() (gpkit.constraints.bounded.Bounded
method),
sens_from_dual() (gpkit.constraints.set.ConstraintSet
method),
sens_from_dual() (gpkit.nomials.nomial_math.MonomialEquality
method),
sens_from_dual() (gpkit.nomials.nomial_math.PosynomialInequality
method),
method),
process_result() (gpkit.constraints.tight.Tight method),
prod() (gpkit.nomials.array.NomialArray method),
prod() (gpkit.nomials.nomial_core.Nomial method),
program (gpkit.constraints.model.Model attribute),
program (gpkit.solution_array.SolutionArray attribute),
(class in kit.interactive.sensitivity_map),
senss_table() (in module gpkit.solution_array),
setup() (gpkit.tests.t_model.Box method),
gpsetup() (gpkit.tests.t_model.BoxAreaBounds method),
setup() (gpkit.tests.t_model.Thing method),
showcadtoon() (in module gpkit.interactive.ractor),
Signomial (class in gpkit.nomials.nomial_math),
signomial_print() (in module kit.interactive.sensitivity_map),
gp-
R
ractorjs() (in module gpkit.interactive.ractor),
ractorpy() (in module gpkit.interactive.ractor),
raise_badelement() (in module gpkit.constraints.set),
raise_elementhasnumpybools() (in module gpkit.constraints.set),
rebuild() (in module gpkit.build),
SignomialEquality (class in gpkit.constraints.sigeq),
SignomialInequality (class in gpkit.nomials.nomial_math),
SignomialProgram (class in kit.constraints.signomial_program),
SignomialsEnabled (class in gpkit),
gp-
114 Index
gpkit Documentation, Release 0.5.3
simplify_exps_and_cs() (in module gpkit.nomials.data),
SingleEquationConstraint (class kit.constraints.single_equation),
in gp-
SingleSignomialEquality (class kit.nomials.nomial_math),
in gpsolarray (gpkit.tools.autosweep.BinarySweepTree
attribute),
solarray (gpkit.tools.autosweep.SolutionOracle attribute),
sollist (gpkit.tools.autosweep.BinarySweepTree
attribute),
solution (gpkit.constraints.model.Model attribute),
solution (gpkit.interactive.sensitivity_map.SensitivityMap
attribute),
SolutionArray (class in gpkit.solution_array),
subinplace() (gpkit.constraints.array.ArrayConstraint
method),
subinplace() (gpkit.constraints.costed.CostedConstraintSet
method),
subinplace() (gpkit.constraints.set.ConstraintSet method),
subinplace() (gpkit.nomials.nomial_math.ScalarSingleEquationConstraint
method),
subinplace() (gpkit.nomials.nomial_math.Signomial
method),
subinto() (gpkit.solution_array.SolutionArray method),
subscripts (gpkit.varkey.VarKey attribute),
substitution() (in module gpkit.nomials.substitution),
sum() (gpkit.nomials.array.NomialArray method),
sum() (gpkit.nomials.nomial_core.Nomial method),
SolutionOracle (class in gpkit.tools.autosweep),
summary() (gpkit.solution_array.SolutionArray method), solve() (gpkit.constraints.geometric_program.GeometricProgram
sweep() (gpkit.constraints.model.Model method),
method),
solve() (gpkit.constraints.model.Model method),
solver (gpkit.tests.t_model.TestGP attribute),
T
solver (gpkit.tests.t_model.TestSP attribute),
SolverBackend (class in gpkit.build),
table() (gpkit.solution_array.SolutionArray method),
table_titles (gpkit.solution_array.SolutionArray attribute),
SolverLog (class in gpkit.small_classes),
sp() (gpkit.constraints.model.Model method),
te_exp_minus1() (in module gpkit.tools.tools),
SPData (class in gpkit.tools.spdata),
StdoutCaptured (class in gpkit.tests.helpers),
te_secant() (in module gpkit.tools.tools),
te_tangent() (in module gpkit.tools.tools),
str_without() method),
(gpkit.constraints.set.ConstraintSet
test (in module gpkit.tests.t_model),
test_601() (gpkit.tests.t_model.TestGP method),
method),
str_without() (gpkit.nomials.array.NomialArray method),
str_without() (gpkit.nomials.nomial_core.Nomial
method),
str_without() (gpkit.varkey.VarKey method),
sub (gpkit.nomials.nomial_core.Nomial attribute),
(gpkit.tests.t_model.TestGP
method),
test_additive_scalar() (gpkit.tests.t_constraints.TestConstraint method),
test_additive_scalar_gt1() (gpkit.tests.t_constraints.TestConstraint method),
method),
method),
sub() (gpkit.nomials.array.NomialArray method),
test_autosweep() (gpkit.tests.t_examples.TestExamples
sub() (gpkit.nomials.nomial_math.Signomial method),
method),
sub() (gpkit.nomials.variables.Variable method),
test_bad_elements() (gpsubconstr_latex() (gpkit.constraints.model.Model
kit.tests.t_constraints.TestConstraint method), method),
subconstr_latex() (gpkit.constraints.set.ConstraintSet
test_bad_gp_sub() (gpkit.tests.t_sub.TestModelSubs
method),
method),
subconstr_latex() (gpkit.constraints.single_equation.SingleEquationConstraint
(gpkit.tests.t_sub.TestModelSubs
method),
method),
subconstr_str() (gpkit.constraints.model.Model method), test_basic() (gpkit.tests.t_sub.TestNomialSubs method),
subconstr_str() (gpkit.constraints.set.ConstraintSet
test_beam() (gpkit.tests.t_examples.TestExamples
method),
method),
subconstr_str() (gpkit.constraints.single_equation.SingleEquationConstraint
(gpkit.tests.t_tools.TestTools
method),
method),
Index 115
gpkit Documentation, Release 0.5.3
test_calcconst() method),
(gpkit.tests.t_sub.TestModelSubs
test_call() (gpkit.tests.t_solution_array.TestSolutionArray
method),
method),
test_eq_ne() (gpkit.tests.t_vars.TestVariable method),
test_eq_neq() (gpkit.tests.t_vars.TestVarKey method),
test_eq_units() (gpkit.tests.t_nomials.TestPosynomial
test_call_units() (gpkit.tests.t_solution_array.TestSolutionArray
method),
test_call_vector() (gpkit.tests.t_solution_array.TestSolutionArray
method),
method),
test_equality_relaxation() (gpkit.tests.t_constraints.TestConstraint method), test_composite_objective() (gpkit.tests.t_tools.TestTools
method),
test_constants_in_objective_1() kit.tests.t_model.TestGP method),
(gptest_evalfn() (gpkit.tests.t_constraints.TestConstraint
method),
test_exps_is_tuple() method),
(gpkit.tests.t_model.TestGP
test_constants_in_objective_2() kit.tests.t_model.TestGP method),
test_constraint_creation_units() kit.tests.t_vars.TestVectorVariable
test_constraint_gen() kit.tests.t_nomial_array.TestNomialArray
method),
test_constraint_gen() (gpkit.tests.t_nomials.TestPosynomial
method),
test_constraintget() (gpkit.tests.t_constraints.TestConstraint method),
test_cost_freeing() (gpkit.tests.t_model.TestGP method),
test_cvxopt_kwargs() (gpkit.tests.t_model.TestModelSolverSpecific
method),
test_debug() (gpkit.tests.t_examples.TestExamples
method),
(gp-
(gpmethod),
(gptest_dict_key() (gpkit.tests.t_vars.TestVarKey method),
test_dictlike() test_diff() method),
(gpkit.tests.t_nomials.TestPosynomial
method),
test_dimensionless_units() kit.tests.t_sub.TestNomialSubs
(gpmethod), test_div() (gpkit.tests.t_nomials.TestMonomial method),
test_elementwise_mult() kit.tests.t_nomial_array.TestNomialArray
(gpmethod), method),
test_eq_ne() method), test_eq_ne()
(gpkit.tests.t_keydict.TestKeyDict
test_empty() (gpkit.tests.t_nomial_array.TestNomialArray
test_eq() (gpkit.tests.t_nomials.TestPosynomial method),
(gpkit.tests.t_nomials.TestMonomial
(gpkit.tests.t_nomials.TestSignomial
test_external_sp() (gpkit.tests.t_examples.TestExamples
method),
test_external_sp2() (gpkit.tests.t_examples.TestExamples
method),
test_failed_getattr() (gpkit.tests.t_keydict.TestKeyDict
method),
test_fmincon_generator() (gpkit.tests.t_tools.TestTools
method),
test_fmincon_generator_logspace() (gpkit.tests.t_tools.TestTools method), test_getattr() (gpkit.tests.t_keydict.TestKeyDict method),
test_getitem() (gpkit.tests.t_nomial_array.TestNomialArray
method), test_getkey() (gpkit.tests.t_sub.TestModelSubs method),
test_hash() (gpkit.tests.t_vars.TestVariable method),
test_inheritance() (gpkit.tests.t_constraints.TestMonomialEquality
method),
test_init() (gpkit.tests.t_constraints.TestConstraint
method),
test_init() (gpkit.tests.t_constraints.TestMonomialEquality
method),
test_init() (gpkit.tests.t_constraints.TestSignomialInequality
test_init() (gpkit.tests.t_nomials.TestMonomial method), test_init() method),
(gpkit.tests.t_nomials.TestPosynomial
method),
test_init() (gpkit.tests.t_nomials.TestSignomial method),
test_init() (gpkit.tests.t_small.TestHashVector method),
test_init() (gpkit.tests.t_vars.TestVariable method),
test_init() (gpkit.tests.t_vars.TestVarKey method),
test_init() (gpkit.tests.t_vars.TestVectorVariable method),
test_initially_infeasible() method),
(gpkit.tests.t_model.TestSP
test_integer_division() (gpkit.tests.t_nomials.TestPosynomial
method),
116 Index
gpkit Documentation, Release 0.5.3
test_is_vector_variable() kit.tests.t_vars.TestArrayVariable
(gpmethod),
test_issue180() (gpkit.tests.t_model.TestSP method),
test_key_options() kit.tests.t_solution_array.TestSolutionArray
(gpmethod),
test_performance_modeling() (gpkit.tests.t_examples.TestExamples
method),
test_persistence() method),
(gpkit.tests.t_sub.TestModelSubs
method),
test_latex() (gpkit.tests.t_nomials.TestMonomial
test_phantoms() method),
(gpkit.tests.t_sub.TestModelSubs
method),
test_pint_366() test_left_right() (gpkit.tests.t_nomial_array.TestNomialArray
method),
(gpkit.tests.t_small.TestSmallScripts
method),
test_mdd_example() (gpkit.tests.t_model.TestGP
method),
test_model_composition_units() (gpkit.tests.t_sub.TestModelSubs method),
test_model_recursion() (gpkit.tests.t_sub.TestModelSubs
method),
test_model_var_access() (gpkit.tests.t_examples.TestExamples
method), test_posy_simplification() (gpkit.tests.t_model.TestGP
method),
test_posyconstr_in_gp() kit.tests.t_constraints.TestTight
test_posyconstr_in_sp() kit.tests.t_constraints.TestTight
(gpmethod),
(gpmethod), test_posyposy_mult() (gpkit.tests.t_nomials.TestPosynomial
method),
test_modelname_added() (gpkit.tests.t_model.TestModelNoSolve method),
test_mono_lower_bound() (gpkit.tests.t_nomials.TestPosynomial
method),
test_mul() (gpkit.tests.t_nomials.TestMonomial method),
test_mul_add() method),
(gpkit.tests.t_small.TestHashVector
test_nan_printing() kit.tests.t_solution_array.TestResultsTable
(gptest_posyslt1() (gpkit.tests.t_constraints.TestSignomialInequality
method),
test_pow() (gpkit.tests.t_nomials.TestMonomial method),
test_pow() (gpkit.tests.t_small.TestHashVector method),
test_primal_infeasible_ex1() kit.tests.t_examples.TestExamples
(gpmethod), method), method),
test_ndim() (gpkit.tests.t_nomial_array.TestNomialArray
test_neg() (gpkit.tests.t_small.TestHashVector method), test_no_naming_on_var_access() (gpkit.tests.t_model.TestModelNoSolve method),
test_non_monomial() (gpkit.tests.t_constraints.TestMonomialEquality
test_primal_infeasible_ex2() (gpkit.tests.t_examples.TestExamples
method),
test_prod() (gpkit.tests.t_nomial_array.TestNomialArray
method),
test_quantity_sub() (gpkit.tests.t_sub.TestModelSubs
method),
test_relaxation() (gpkit.tests.t_examples.TestExamples
method),
test_relaxation() (gpkit.tests.t_model.TestSP method),
method),
test_numeric() kit.tests.t_nomials.TestMonomial
test_oper_overload()
(gpkit.tests.t_sub.TestNomialSubs
method),
test_numerical_precision() (gpmethod),
(gpkit.tests.t_constraints.TestConstraint method),
test_repo() (in module gpkit.tests.test_repo),
test_repos() (in module gpkit.tests.test_repo),
test_repr() (gpkit.tests.t_nomials.TestMonomial method),
test_repr() (gpkit.tests.t_vars.TestVarKey method),
test_result_access() (gpkit.tests.t_solution_array.TestResultsTable
method),
test_scalar_units() (gpkit.tests.t_sub.TestNomialSubs
method),
test_sensitivities() (gpkit.tests.t_model.TestGP method), test_outer() (gpkit.tests.t_nomial_array.TestNomialArray
method),
test_partial_sub_signomial() (gpkit.tests.t_model.TestSP
test_setattr() (gpkit.tests.t_keydict.TestKeyDict method),
Index 117
gpkit Documentation, Release 0.5.3
test_shape() (gpkit.tests.t_nomial_array.TestNomialArray
method),
test_shape() (gpkit.tests.t_small.TestCootMatrix method),
test_shapes() (gpkit.tests.t_vars.TestVectorize method),
test_sigconstr_in_sp() (gpkit.tests.t_constraints.TestTight
method),
test_sigeq() (gpkit.tests.t_model.TestGP method),
test_signomial() (gpkit.tests.t_sub.TestNomialSubs
method),
test_sigs_not_allowed_in_cost() (gpkit.tests.t_model.TestSP method),
test_simple_box() (gpkit.tests.t_examples.TestExamples
method),
test_simple_sp() (gpkit.tests.t_examples.TestExamples
method),
test_simple_united_gp() (gpkit.tests.t_model.TestGP
method),
test_simpleflight() (gpkit.tests.t_examples.TestExamples
method),
test_simplification() (gpkit.tests.t_nomials.TestPosynomial
method),
test_sin_approx_example() (gpkit.tests.t_examples.TestExamples
method),
test_singular() (gpkit.tests.t_model.TestGP method),
test_skipfailures() method),
(gpkit.tests.t_sub.TestModelSubs
test_small_named_signomial() kit.tests.t_model.TestSP method),
(gptest_sp_bounded() (gpkit.tests.t_model.TestSP method),
test_sp_initial_guess_sub() (gpkit.tests.t_model.TestSP
method),
test_sp_substitutions() method),
(gpkit.tests.t_model.TestSP
test_str() (gpkit.tests.t_constraints.TestMonomialEquality
method),
test_str() (gpkit.tests.t_vars.TestArrayVariable method),
test_str_with_units() kit.tests.t_nomials.TestMonomial
test_string_mutation() (gpkit.tests.t_sub.TestNomialSubs
method),
(gpmethod), method),
method),
test_substitution_issue905() (gpkit.tests.t_constraints.TestBounded
method),
test_sum() (gpkit.tests.t_nomial_array.TestNomialArray
method),
test_table() (gpkit.tests.t_solution_array.TestSolutionArray
method),
test_te_exp_minus1() method),
(gpkit.tests.t_tools.TestTools
test_te_secant() (gpkit.tests.t_tools.TestTools method),
test_te_tangent() (gpkit.tests.t_tools.TestTools method),
test_terminating_constant_() (gpkit.tests.t_model.TestGP
method),
test_to() (gpkit.tests.t_vars.TestVariable method),
test_trivial_gp() (gpkit.tests.t_model.TestGP method),
test_trivial_sp() (gpkit.tests.t_model.TestSP method),
test_trivial_sp2() (gpkit.tests.t_model.TestSP method),
test_trivial_vector_gp() (gpkit.tests.t_model.TestGP
method),
test_unbounded() (gpkit.tests.t_examples.TestExamples
method),
test_unbounded_debugging() (gpkit.tests.t_model.TestSP
method),
test_unit_parsing() method),
(gpkit.tests.t_vars.TestVariable
test_united_sub_sweep() kit.tests.t_sub.TestModelSubs method),
(gptest_unitless_monomial_sub() kit.tests.t_sub.TestNomialSubs
(gpmethod),
test_units() (gpkit.tests.t_nomial_array.TestNomialArray
method),
test_units() (gpkit.tests.t_nomials.TestMonomial
method),
test_units_attr() (gpkit.tests.t_vars.TestVarKey method),
test_units_sub() (gpkit.tests.t_solution_array.TestSolutionArray
method),
test_unitstr() (gpkit.tests.t_small.TestSmallScripts
method),
test_value() (gpkit.tests.t_vars.TestVariable method),
test_values_vs_subs() (gpkit.tests.t_model.TestSP
method),
test_variable() (gpkit.tests.t_sub.TestNomialSubs
method),
test_vector() (gpkit.tests.t_keydict.TestKeyDict method),
test_vector() (gpkit.tests.t_sub.TestNomialSubs method),
(gpkit.tests.t_sub.TestModelSubs
method),
(gpkit.tests.t_sub.TestModelSubs
method),
test_vector_sweep() (gpkit.tests.t_sub.TestModelSubs
method),
test_vectorize() (gpkit.tests.t_examples.TestExamples
118 Index
gpkit Documentation, Release 0.5.3
method),
test_water_tank() (gpkit.tests.t_examples.TestExamples
method),
test_x_greaterthan_1() test_zero_lower_unbounded() kit.tests.t_model.TestGP method),
(gpkit.tests.t_examples.TestExamples
method),
(gptest_zeroing() (gpkit.tests.t_model.TestGP method),
TestArrayVariable (class in gpkit.tests.t_vars),
TestBounded (class in gpkit.tests.t_constraints),
testcase (in module gpkit.tests.t_model),
TestConstraint (class in gpkit.tests.t_constraints),
TestCootMatrix (class in gpkit.tests.t_small),
TestExamples (class in gpkit.tests.t_examples),
TestFiles (class in gpkit.tests.from_paths),
TestGP (class in gpkit.tests.t_model),
TestHashVector (class in gpkit.tests.t_small),
TestKeyDict (class in gpkit.tests.t_keydict),
TestModelNoSolve (class in gpkit.tests.t_model),
TestModelSolverSpecific (class in gpkit.tests.t_model),
TestModelSubs (class in gpkit.tests.t_sub),
TestMonomial (class in gpkit.tests.t_nomials),
TestMonomialEquality (class in gpkit.tests.t_constraints),
TestNomialArray (class in gpkit.tests.t_nomial_array),
TestNomialSubs (class in gpkit.tests.t_sub),
TestPosynomial (class in gpkit.tests.t_nomials),
TestResultsTable (class in gpkit.tests.t_solution_array),
TestSignomial (class in gpkit.tests.t_nomials),
TestSignomialInequality (class in gpkit.tests.t_constraints),
TestSmallScripts (class in gpkit.tests.t_small),
TestSolutionArray (class in gpkit.tests.t_solution_array),
TestSP (class in gpkit.tests.t_model),
TestTight (class in gpkit.tests.t_constraints),
TestTools (class in gpkit.tests.t_tools),
TestVariable (class in gpkit.tests.t_vars),
TestVarKey (class in gpkit.tests.t_vars),
TestVectorize (class in gpkit.tests.t_vars),
TestVectorVariable (class in gpkit.tests.t_vars),
Thing (class in gpkit.tests.t_model),
Tight (class in gpkit.constraints.tight),
to() (gpkit.nomials.nomial_core.Nomial method),
to() (gpkit.nomials.variables.Variable method),
to_united_array() (gpkit.small_classes.DictOfLists
method),
tocoo() (gpkit.small_classes.CootMatrix method),
tocsc() (gpkit.small_classes.CootMatrix method),
tocsr() (gpkit.small_classes.CootMatrix method),
todense() (gpkit.small_classes.CootMatrix method),
todia() (gpkit.small_classes.CootMatrix method),
todok() (gpkit.small_classes.CootMatrix method),
topsenss_filter() (in module gpkit.solution_array),
topsenss_table() (in module gpkit.solution_array),
topvar() (gpkit.constraints.set.ConstraintSet method),
try_str_without() (in module gpkit.small_scripts),
trycall() (in module gpkit.constraints.single_equation),
U
unique_varkeys (gpkit.constraints.set.ConstraintSet attribute),
units (gpkit.nomials.array.NomialArray attribute),
unitstr() (gpkit.nomials.nomial_core.Nomial method),
unitstr() (gpkit.varkey.VarKey method),
unitstr() (in module gpkit.repr_conventions),
update() (gpkit.keydict.KeyDict method),
update() (gpkit.keydict.KeySet method),
update_keymap() (gpkit.keydict.KeyDict method),
V
value (gpkit.nomials.nomial_core.Nomial attribute),
values (gpkit.nomials.data.NomialData attribute),
Variable (class in gpkit.nomials.variables),
variables_byname() (gpkit.constraints.set.ConstraintSet
method),
VarKey (class in gpkit.varkey),
varkey_bounds() (in module gpkit.constraints.bounded),
varkeys (gpkit.constraints.set.ConstraintSet attribute),
varkeys (gpkit.nomials.data.NomialData attribute),
veckeyed() (in module gpkit.small_scripts),
VectorizableVariable (class in gpkit.nomials.variables),
Vectorize (class in gpkit),
vectorize() (gpkit.nomials.array.NomialArray method),
W
write() (gpkit.small_classes.SolverLog method),
write() (gpkit.tests.helpers.NullFile method),
Z
zero_lower_unbounded_variables() kit.constraints.model.Model method),
(gp-
Index 119
* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project