#!/usr/bin/env python
# coding: utf-8

# # Entropy maximization
# 
# A derivative work by Judson Wilson, 6/2/2014.<br>
# Adapted from the CVX example of the same name, by Joëlle Skaf, 4/24/2008.
# 
# ## Introduction
# 
# Consider the linear inequality constrained entropy maximization problem:
#     $$\begin{array}{ll}
#     \mbox{maximize}   & -\sum_{i=1}^n x_i \log(x_i) \\
#     \mbox{subject to} & \sum_{i=1}^n x_i = 1 \\
#                       & Fx \succeq g,
#     \end{array}$$
# where the variable is $x \in \mathbf{{\mbox{R}}}^{n}$.   
# 
# This problem can be formulated in CVXPY using the `entr` atom.
# 
# ## Generate problem data

# In[1]:


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

# In[2]:


# 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.ECOS, verbose=True)

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