All of the examples can be found in Jupyter notebook form here.

# Entropy Maximization

Here is a constrained entropy maximization problem:

$\begin{array}{ll} \mbox{maximize} & -\sum_{i=1}^n x_i \log x_i \\ \mbox{subject to} & \mathbf{1}' x = 1 \\ & Ax \leq b \end{array}$

where $x \in \mathbf{R}^n$ is our optimization variable and $A \in \mathbf{R}^{m \times n}, b \in \mathbf{R}^{m}$.

To solve this, we can simply use the entropy operation Convex.jl provides.

using Convex, SCS

n = 25;
m = 15;
A = randn(m, n);
b = rand(m, 1);

x = Variable(n);
problem = maximize(entropy(x), sum(x) == 1, A * x <= b)
solve!(problem, SCSSolver(verbose=0))
problem.optval
3.207225104404655
x.value
25×1 Array{Float64,2}:
0.04206886425093688
0.03457229652244405
0.035878056898433805
0.029020798307983226
0.052324629391367804
0.04247007662656345
0.03828009010340274
0.04047616962048803
0.039062862471573354
0.03640861151656186
⋮
0.05421299208385005
0.033986094211083215
0.04508071889024127
0.03871064663612938
0.04881740688341207
0.03073950762360894
0.04238196966674161
0.0377729388286942
0.03181532194215058