SDP Mixed Integer Problem

[Marcello]

Hi,

I'm trying to implement an SDP Mixed Integer (binary) problem using yalmip.

This is my example code

clear all
clc
c=rand(3,1);
x = binvar(3,1);

A = [-1 2 0;-3 -4 1;0 0 -2];
P = sdpvar(3,3);
a = sdpvar(3,1);
% Define LMI
F = [P >= 0, A'*P+P*A <= 0];
% Define MI elementwise-inequality
F = F + set(a<=9.*x);
F = F + set(a>=-9.*x);

F = [F, trace(P) == 1];
solvesdp(F,(c+a)'*(P)*(c+a));
Pfeasible = double(P);

The routine returns 'No suitable solvers'.

Also if I specify to use 'bnb' solver, I'm not able to obtain a result.

Am I doing something wrong?

Thanks

[Johan Löfberg]

You objective is cubic in P and a, so you don't have any solver to solve the integer relaxation

Easily fixed though, perform a congruence transformation of your LMI using Q=P^-1, and then optimize over Q and a instead. Perform an epigraph reformulation of objective as (c+a)inv(Q)(c+a) <= t, apply Schur complement and minimize t.

And don't use SET. It is obsolete.

[Johan Löfberg]

Did you solve your issues?

[Marcello]

Dear Johan,

sorry for my late reply.

Yes, I have solved the issue.

Now I'm able to solve SDP Mixed Integer programs.

Thank you once again.

Best Regards

[Qi Zhang]

How do you solve this?

[Johan Löfberg]

solve what?

Integer linear SDPs can be solved (well...) in YALMIP using the internal solvers bnb and cutsdp (relying on a fast SDP solver and LP solver respectively)

[Qi Zhang]

Thank you! I just find cutsdp as you mentioned! I would try this. Thanks again