# How to solve LMIs with equality constraints using MATLAB?

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David Cho on 16 Jan 2018
Edited: Frank on 25 Nov 2018
Hi
I would like to find n by n matrices P and Q that minimize
*J = norm(P) + w*norm(Q), where w is a given weight,
subject to
P>=0, Q>=0, and f(P,Q)=0, where f(P,Q)=0 is a given function of P and Q.*
I tried to solve this problem using the lmi solver of MATLAB, but have no idea how to deal with the equality constraint: f(P,Q)=0.
f(P,Q)=0 is equivalent to f(P,Q)>=0 and f(P,Q)<=0, but MATLAB only solves strictly feasible constraints.
But the above problem contains not strictly feasible constraints.
Is there any good solution to handle this kind of problem using MATLAB?
Thank you.
##### 2 CommentsShow 1 older commentHide 1 older comment
David Cho on 17 Jan 2018
Torsten - Thank you for your comment.
How can we use 'fmincon' for multiple decision variables (P and Q)?
Also, in this problem the decision variables are matrices.

Frank on 25 Nov 2018
Edited: Frank on 25 Nov 2018
Section 2.5 of Byod's book Linear Matrix Inequalities in System and Control Theory specifically talks about this problem.

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