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
----
 0.7071

Free Variables
--------------
x : 0.7854
y : 0.7071

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 import ConstraintSet
from gpkit.exceptions import InvalidGPConstraint
from external_function import external_code


class ExternalConstraint(ConstraintSet):
    "Class for external calling"
    # Overloading the __init__ function here permits the constraint class to be
    # called more cleanly at the top level GP.
    def __init__(self, x, y, **kwargs):

        # Calls the ConstriantSet __init__ function
        super(ExternalConstraint, self).__init__([], **kwargs)

        # 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

    # Prevents the ExternalConstraint class from solving in a GP, thus forcing
    # iteration
    def as_posyslt1(self, substitutions=None):
        raise InvalidGPConstraint("ExternalConstraint cannot solve as a GP.")

    # Returns the ExternalConstraint class as a GP compatible constraint when
    # requested by the GPkit solver
    def as_gpconstr(self, x0, substitutions=None):

        # 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
----
 0.7071

Free Variables
--------------
x : 0.7854
y : 0.7071

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