Entropy maximization

A derivative work by Judson Wilson, 6/2/2014. Adapted from the CVX example of the same name, by Joëlle Skaf, 4/24/2008.

Introduction

Consider the linear inequality constrained entropy maximization problem:

\[\begin{split}\begin{array}{ll} \mbox{maximize} & -\sum_{i=1}^n x_i \log(x_i) \\ \mbox{subject to} & \sum_{i=1}^n x_i = 1 \\ & Fx \succeq g, \end{array}\end{split}\]

where the variable is \(x \in \mathbf{{\mbox{R}}}^{n}\).

This problem can be formulated in CVXPY using the entr atom.

Generate problem data

import cvxpy as cp
import numpy as np

# Make random input repeatable.
np.random.seed(0)

# Matrix size parameters.
n = 20
m = 10
p = 5

# Generate random problem data.
tmp = np.random.rand(n)
A = np.random.randn(m, n)
b = A.dot(tmp)
F = np.random.randn(p, n)
g = F.dot(tmp) + np.random.rand(p)

Formulate and solve problem

# Entropy maximization.
x = cp.Variable(shape=n)
obj = cp.Maximize(cp.sum(cp.entr(x)))
constraints = [A*x == b,
               F*x <= g ]
prob = cp.Problem(obj, constraints)
prob.solve(solver=cp.CVXOPT, verbose=True)

# Print result.
print("\nThe optimal value is:", prob.value)
print('\nThe optimal solution is:')
print(x.value)

     pcost       dcost       gap    pres   dres
 0:  0.0000e+00 -2.8736e+00  2e+01  1e+00  1e+00
 1: -6.0720e+00 -5.9687e+00  2e+00  8e-02  2e-01
 2: -5.4688e+00 -5.5883e+00  4e-01  8e-03  4e-02
 3: -5.4595e+00 -5.4889e+00  5e-02  6e-04  1e-02
 4: -5.4763e+00 -5.4816e+00  1e-02  1e-04  5e-03
 5: -5.4804e+00 -5.4809e+00  1e-03  1e-05  2e-03
 6: -5.4809e+00 -5.4809e+00  3e-05  5e-07  3e-04
 7: -5.4809e+00 -5.4809e+00  4e-07  6e-09  1e-05
 8: -5.4809e+00 -5.4809e+00  4e-09  6e-11  3e-07
 9: -5.4809e+00 -5.4809e+00  4e-11  6e-13  4e-09
Optimal solution found.

The optimal value is: 5.480901486350394

The optimal solution is:
[0.43483319 0.66111715 0.49201039 0.36030618 0.38416629 0.30283658
 0.41730232 0.79107794 0.76667302 0.38292365 1.2479328  0.50416987
 0.68053832 0.67163958 0.13877259 0.5248668  0.08418897 0.56927148
 0.50000248 0.78291311]