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

Entropy Maximization

Here is a constrained entropy maximization problem:

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

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, SCS.Optimizer; silent_solver = true)
problem.optval

3.1480332350684095

evaluate(x)

25-element Vector{Float64}:
 0.02090112150925664
 0.026662221322873127
 0.03088782116314491
 0.05861220536048732
 0.05449215264938552
 0.02645247054167646
 0.03366533267317568
 0.06379294998140604
 0.0732298484881859
 0.02313412240924475
 ⋮
 0.027717200127803588
 0.049568617076695645
 0.04157917069184767
 0.04260298097492467
 0.032397618155268273
 0.05250042455119695
 0.05960820012403955
 0.04838519200843289
 0.06074430288075965

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