_images/gplogo.png

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.

Table of contents

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 Signomials.

More precisely, GP objectives and inequalities are formed out of monomials and posynomials. In the context of GP, a monomial is defined as:

\[f(x) = c x_1^{a_1} x_2^{a_2} ... x_n^{a_n}\]

where \(c\) is a positive constant, \(x_{1..n}\) are decision variables, and \(a_{1..n}\) are real exponents. For example, taking \(x\), \(y\) and \(z\) to be positive variables, the expressions

\[7x \qquad 4xy^2z \qquad \frac{2x}{y^2z^{0.3}} \qquad \sqrt{2xy}\]

are all monomials. Building on this, a posynomial is defined as a sum of monomials:

\[g(x) = \sum_{k=1}^K c_k x_1^{a_1k} x_2^{a_2k} ... x_n^{a_nk}\]

For example, the expressions

\[x^2 + 2xy + 1 \qquad 7xy + 0.4(yz)^{-1/3} \qquad 0.56 + \frac{x^{0.7}}{yz}\]

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 written as:

\[\begin{split}\begin{array}{lll}\text{} \text{minimize} & g_0(x) & \\ \text{subject to} & f_i(x) = 1, & i = 1,....,m \\ & g_i(x) \leq 1, & i = 1,....,n \end{array}\end{split}\]

Boyd et. al. give the following example of a GP in standard form:

\[\begin{split}\begin{array}{llll}\text{} \text{minimize} & x^{-1}y^{-1/2}z^{-1} + 2.3xz + 4xyz \\ \text{subject to} & (1/3)x^{-2}y^{-2} + (4/3)y^{1/2}z^{-1} \leq 1 \\ & x + 2y + 3z \leq 1 \\ & (1/2)xy = 1 \end{array}\end{split}\]

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 Signomial Programming.

Where can I learn more?

To learn more about GPs, refer to the following resources:

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.

_images/inheritance.png

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.

Model objects

In GPkit, a Model object represents a symbolic problem declaration. That problem may be either GP-compatible 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.

_images/solvemethods.png

Installation Instructions

If you encounter bugs during installation, please email gpkit@mit.edu 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: sudo apt-get install python-pip
Windows
Install the Python 2.7 64-bit version of Anaconda.

Get python packages

Mac OS X
Run the following commands:
  • pip install pip --upgrade
  • pip install numpy
  • pip install scipy
  • pip install pint
Linux
Use your package manager to install numpy and scipy
Ubuntu: sudo apt-get 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.

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
Linux
  • Install ctypesgen with pip install ctypesgen --pre (for system python installs, use sudo pip)
  • Download MOSEK, then:
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 file and put it in C:\Users\(your_username)\mosek\
    • Make sure gcc is on your system path.
      • 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 gpkit@mit.edu 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
  • Run pip install --no-cache-dir --no-deps gpkit
  • Run python -c "import gpkit.tests; gpkit.tests.run()"
  • If any tests fail, please email gpkit@mit.edu 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.

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, install GPkit 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:

# 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

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

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:

print 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.

print "The optimal value is %s." % sol["cost"]
print "The x dimension is %s." % sol(x)
print "The y dimension is %s." % sol["variables"]["y"]

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 assumption-checking.

Using variable sensitivities

Fixed variable sensitivities can be accessed from a SolutionArray’s ["sensitivities"]["constants"] dict, as in this example:

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

These sensitivities are actually log derivatives (\(\frac{d \mathrm{log}(y)}{d \mathrm{log}(x)}\)); 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

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.

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:

"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
])

# 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
# 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
print

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
--------------
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         2        ]

Constants
---------
x_max : 1
x_min : 2

Sensitivities
-------------
x_min : +1
x_max : -0.99


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
        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

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().

_images/controlpanel.gif

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 Sweeps 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)
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:

_images/plot_sweep1d.png
_images/plot_autosweep1d.png

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.

"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")

    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 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")


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"
    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                                         [lbf] weight

         | Aircraft/Wing
       W : 190                                         [lbf] weight
       c : 2.653                                       [ft]  mean chord

         | Mission/FlightSegment/AircraftP
W_{burn} : [ 0.487     0.486     0.485     0.485    ]  [lbf] segment fuel burn
W_{fuel} : [ 1.94      1.46      0.97      0.485    ]  [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            surface area
\rho : +0.48            areal density
   A : -0.31            aspect ratio

Next Largest Sensitivities
--------------------------
     | Mission/FlightSegment/AircraftP/WingAero
   e : [ -0.093    -0.092    -0.092    -0.092   ] Oswald efficiency

     | Mission/FlightSegment/FlightState
   V : [ +0.1      +0.1      +0.1      +0.1     ] true airspeed
\rho : [ +0.034    +0.034    +0.035    +0.035   ] air density

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 Substitutions 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 \(\nu\) of a motor is not a GP-compatible variable, but \((1-\nu)\) is a valid GP variable, then \(\nu\) 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!

# 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 Sequential Geometric Programs.

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 1D Autosweeps 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 \(y\ \forall~y_i\in\left\{1,2\right\}\).

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 \((x,y)\in\left\{(1, 14),\ (1, 17),\ (3, 14),\ (3, 17)\right\}\). 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.

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 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.

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).

Substituting with replacement

Any of the substitutions above can be run with p.sub(*args, replace=True) to clobber any previously-substituted values.

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"]).

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.

Composite Objectives

Given \(n\) posynomial objectives \(g_i\), you can sweep out the problem’s Pareto frontier with the composite objective:

\(g_0 w_0 \prod_{i\not=0} v_i + g_1 w_1 \prod_{i\not=1} v_i + ... + g_n \prod_i v_i\)

where \(i \in 0 ... n-1\) and \(v_i = 1- w_i\) and \(w_i \in [0, 1]\)

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.

Signomial Programming

Signomial programming finds a local solution to a problem of the form:

\[\begin{split}\begin{array}{lll}\text{} \text{minimize} & g_0(x) & \\ \text{subject to} & f_i(x) = 1, & i = 1,....,m \\ & g_i(x) - h_i(x) \leq 1, & i = 1,....,n \end{array}\end{split}\]

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')

# 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:

\[\begin{split}\begin{array}{lll}\text{} \text{minimize} & y & \\ \text{subject to} & y \geq \sin(x) \\ & \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \end{array}\end{split}\]

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
--------------
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
    else:
        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):
    "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()


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

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 \(x \ge 1\).

"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))

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

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          upper limit, floor aspect ratio
    gamma : 2           lower limit, floor aspect ratio

Sensitivities
-------------
 A_{wall} : -1.5     upper limit, wall area
    alpha : +0.5     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                             [m**2] Surface Area of the Tank
V : 0.1                               [m**3] Volume of the Tank
d : [ 0.464     0.464     0.464    ]  [m]    Dimension Vector

Sensitivities
-------------
   M : +0.67 Mass of Water in the Tank
\rho : -0.67 Density of Water in the Tank

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
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              aspect ratio
C_D : 0.02059           Drag coefficient of wing
C_L : 0.4988            Lift coefficent of wing
C_f : 0.003599          skin friction coefficient
  D : 303.1      [N]    total drag force
 Re : 3.675e+06         Reynold's number
  S : 16.44      [m**2] total wing area
  V : 38.15      [m/s]  cruising speed
  W : 7341       [N]    total aircraft weight
W_w : 2401       [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
----
 [ 338       294       396       326       ] [N] 

Sweep Variables
---------------
      V : [ 45        45        55        55       ]  [m/s] cruising speed
V_{min} : [ 20        25        20        25       ]  [m/s] takeoff speed

Free Variables
--------------
  A : [ 6.2       8.84      4.77      7.16     ]         aspect ratio
C_D : [ 0.0146    0.0196    0.0123    0.0157   ]         Drag coefficient of wing
C_L : [ 0.296     0.463     0.198     0.31     ]         Lift coefficent of wing
C_f : [ 0.00333   0.00361   0.00314   0.00342  ]         skin friction coefficient
  D : [ 338       294       396       326      ]  [N]    total drag force
 Re : [ 5.38e+06  3.63e+06  7.24e+06  4.75e+06 ]         Reynold's number
  S : [ 18.6      12.1      17.3      11.2     ]  [m**2] total wing area
  W : [ 6.85e+03  6.97e+03  6.4e+03   6.44e+03 ]  [N]    total aircraft weight
W_w : [ 1.91e+03  2.03e+03  1.46e+03  1.5e+03  ]  [N]    wing weight

Most Sensitive
--------------
                W_0 : [ +0.92     +0.95     +0.85     +0.85    ] aircraft weight excluding wing
            V_{min} : [ -0.82     -0.41     -1        -0.71    ] takeoff speed
                  V : [ +0.59     +0.25     +0.97     +0.75    ] cruising speed
(\frac{S}{S_{wet}}) : [ +0.56     +0.45     +0.63     +0.54    ] wetted area ratio
                  k : [ +0.56     +0.45     +0.63     +0.54    ] form factor

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.

    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:
    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                                             [m]   Length of an element
     M : [ 1.98e+03  1.27e+03  713       317      ... ]  [N*m] Internal moment
     V : [ 660       528       396       264      ... ]  [N]   Internal shear
\theta : [  -        0.177     0.285     0.341    ... ]        Slope
     w : [  -        0.106     0.384     0.759    ... ]  [m]   Displacement

Most Sensitive
--------------
 L : +4                                             Overall beam length
EI : -1
 q : [ +0.0072   +0.042    +0.12     +0.23    ... ] Distributed load at each point

By plotting the deflection, we can see that the agreement between the analytical solution and the GP solution is good.

_images/beam.svg

Glossary

For an alphabetical listing of all commands, check out the Index

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.

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)

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.

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, constraints, substitutions=None, verbosity=1)

Bases: gpkit.constraints.costed.CostedConstraintSet, gpkit.nomials.data.NomialData

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
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, *args, **kwargs)

Bases: gpkit.constraints.costed.CostedConstraintSet

Symbolic representation of an optimization problem.

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.
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.

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, include_only=None, exclude=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
relaxvars : 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).
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

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.

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

class gpkit.constraints.signomial_program.SignomialProgram(cost, constraints, substitutions=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

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
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 #
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

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

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: gpkit.nomials.data.NomialData

Shared non-mathematical properties of all nomials

c = None
convert_to(arg)

Convert this signomial to new units

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

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, cs=1, require_positive=True, simplify=True, **descr)

Bases: gpkit.nomials.nomial_math.Posynomial

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.
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(wrt)

Derivative of this with respect to a Variable

wrt (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)

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)

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.
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)

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.

descr

a Variable’s descr is derived from its VarKey.

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)

class gpkit.nomials.variables.VectorizableVariable(*args, **descr)

Bases: gpkit.nomials.variables.Variable, gpkit.nomials.variables.ArrayVariable

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-name-in-python

gpkit.tests.from_paths.newtest_fn(name, solver, import_dict, path)

Doubly nested callbacks to run the test with getattr(self, name)()

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
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

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)
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: gpkit.constraints.model.Model

simple box for model testing

setup()
class gpkit.tests.t_model.BoxAreaBounds(cost=None, constraints=None, *args, **kwargs)

Bases: gpkit.constraints.model.Model

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
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

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: gpkit.constraints.model.Model

a thing, for model testing

setup(length)
gpkit.tests.t_model.test

alias of TestSP_cvxopt

gpkit.tests.t_model.testcase

alias of TestSP

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()
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 (!!)

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

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.TestGPSubs(methodName='runTest')

Bases: unittest.case.TestCase

Test substitution for Model and GP objects

test_calcconst()
test_getkey()
test_model_composition_units()
test_model_recursion()
test_persistence()
test_phantoms()
test_skipfailures()
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_bad_gp_sub()
test_bad_subinplace()
test_basic()

Basic substitution, symbolic

test_dimensionless_units()
test_numeric()

Basic substitution of numeric value

test_quantity_sub()
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_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)

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_hash()

Hashes should collide independent of units

test_init()

Test Variable initialization

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()
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
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

gpkit.tools.fmincon.generate_mfiles(model, algorithm='interior-point', guesstype='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

algorithm: [string] Algorithm used by fmincon
‘interior-point’: uses the interior point solver ‘SQP’: uses the sequential quadratic programming solver
guesstype: [string] The type of initial guess used

‘ones’: One for each variable ‘order-of-magnitude-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
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, guesstype='ones')

Returns initial guess

gpkit.tools.spdata module

Implements SPData class

class gpkit.tools.spdata.SPData(model)

Bases: gpkit.nomials.data.NomialData

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

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: gpkit.build.SolverBackend

CVXopt finder.

look()

Attempts to import cvxopt.

name = 'cvxopt'
class gpkit.build.Mosek

Bases: gpkit.build.SolverBackend

MOSEK finder and builder.

build()

Builds a dynamic library to GPKITBUILD or $HOME/.gpkit

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: gpkit.build.SolverBackend

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

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: gpkit.keydict.KeyDict

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

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.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 CootMatrix

class gpkit.small_classes.Count

Bases: object

Like python 2’s itertools.count, for Python 3 compatibility.

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

gpkit.small_scripts.try_str_without(item, excluded)

Try to call item.str_without(excluded); fall back to str(item)

gpkit.small_scripts.unitstr(units, into='%s', options='~', dimless='')

Returns the unitstr of a given object.

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
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, showvars=(), title='Sensitivities', **kwargs)

Returns sensitivity table lines

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

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.
**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(dimless='')

Returns string representation of 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.

>>> 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/v0.5.3/local/lib/python2.7/site-packages/gpkit/env/settings')

Load the settings file at SETTINGS_PATH; return settings dict

Citing GPkit

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}
     }

Acknowledgements

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.

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:
    • 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)

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

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)
  • 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)