Optimization or Global Oprimization toolbox for semidefinite programming

Hi all, just a quick question is it possible to use Optimization or Global Oprimization toolbox to solve semidefinite programming problem? I can't find the way to input semidefinite constraint for matrix. Thanks

Answer 1

[TED MOSBY]

Hi Frogy,

The Optimization Toolbox or Global Optimization Toolbox do not have a built-in solver that accepts a positive - semidefinite matix constraint. The conic solver that ships with MATLAB: "coneprog" used for second-order cones. Instead you can use a modelling layer that express variables as symmetric matrices, add the constraint X >= 0 (PSD), and call an external SDP solver.

For this you will have to download the following:

• YALMIP: using YALMIP you describe the variables, constrainsts, cost etc.
• SDP-capable solver like SDPT3 or SeDuMi: when you call "optimize", YALMIP packages your problem and hands it over to this solver that understands semi-definite problem.

Here is a little workflow showing how to get started:

1. Use sdpvar just like you’d write variables on paper:

X = sdpvar(n,n,'symmetric');   % matrix you want PSD

2. Tell YALMIP the constraints and cost in plain math:

F = [X >= 0, trace(X) == 1];   % “X is PSD and its trace is 1”
cost = trace(C*X);             % whatever you’re minimising

3. Pick a solver

ops = sdpsettings('solver','sdpt3');   % or 'sedumi', 'mosek', …

4. Then solve and record the answer

optimize(F, cost, ops);
Xopt = value(X);

Hope this helps!