Entropy maximization with nonlinear constraint

Specific Domains Optimization (Mathematical)
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4

NLopt doesn’t support the entropic cone, so you’ll have to write that as a nonlinear constraint.

Just write the JuMP model exactly as you formulated above. Perhaps:

using JuMP, Ipopt
N = 3
model = Model(Ipopt.Optimizer)
@variable(model, 0 <= p[1:N, 1:N] <= 1, Symmetric, start = 1 / N)
@NLobjective(model, Max, -sum(p[i, j] * log(p[i, j]) for i in 1:N, j in 1:N))
@constraint(model, sum(p) == 1)
nl_expr = @NLexpression(model, sum(sum(p[i,j] for j in 1:N)^2 for i in 1:N))
@NLconstraint(model, sum(p[i, i] for i in 1:N) - nl_expr == r * (1 - nl_expr))
optimize!(model)
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