Approximation methods for a mixed integer convex optimization problem

Question

I have a convex objective function, e.g., minimizing the negative entropy function. My constraints are also linear. The only issue is that I also have binary variables.

I am currently aware of AIMMS's outer approximation (AOA) that is said to be a good options. My questions are:

1. Is such an outer approximation method using another solver to solve each relaxed problem? For example, if the variables were continuous, then minimizing the negative entropy function would be solved with MOSEK. Do you think solvers like AOA will still use MOSEK but will apply some sort of branch&bounding?
2. What other options than AOA do I have? I prefer using MATLAB and YALMIP.

Answer 1

Mosek 9.x can natively solve mixed-integer exponential cone problems.

Formulate the problem in YALMIP, specifying the binary variables as binvar, and Mosek as the solver. YALMIP will call Mosek to exploit its native mixed-integer exponential cone capability.

Here is a mixed-integer example (mixture of binary and continuous):

x = binvar(3,1); % binary
y = sdpvar(2,1); % continuous
optimize([A*[x;y] <= b,0 <= y <= 1],-entropy([x;y]),sdpsettings('solver','mosek'))