A LMI Problem with nonsigular constraint

[mas tab]

I Have a LMI Problem(min t    S. T. AX+DY>tB) that Unkown Matrix X must be nonsingular (inerse of A exist). Pls tell me a way for formulate this Problem in MATLAB Code and YALMIP toolbox.

THANKS.

[Johan Löfberg]

X non-singular is a very hard non-convex constraint (unless X is symmetric)

Your notation is not clear. What does > mean?

[mas tab]

Thank John.

My Problem is:

We have 

1) Two Known Matrix : A , B & D

2) Two Unknown Matrix: X & Y

3) and a minimization Problem: min t

4) subject to: X is non-singular and AX+DY>tB   (> means Greater than so that  A>B means A-B is positive definite)

whats your Hint?

[Johan Löfberg]

What makes you know that A*X+D*Y is a symmetric matrix?. Generically. AX+DY will be unsymmetric, so it is not clear what you mean with that unsymmetric matrix being positive definite. Semidefinite programming is done over the cone of symmetric matrices

[mas tab]

Matrix A and B and D are Symetric(and Complex) ,

X and Y are unknown and complex.

[Johan Löfberg]

So effectively your constraints are

AX+DY> tB
AX+DY Hermitian
X non-singular

[mas tab]

Thanks John.

Yes The Problem is:

AX+DY> tB
AX+DY Hermitian
X non-singular

and Can It be solved with YALMIP?

And How If X be Hermitian?

[Johan Löfberg]

I strongly suspect you've misunderstood something and try to solve the wrong problems

As stated now, there are several practical problems. You have a strict inequality: That can not be used practice, you have to change it to a non-strict (using a margin of some sort to keep strict feasibility and avoid the trivial solution (X,Y,t)=0)

Non-singularity is extremely hard to do in practice. Generically, the solution from a numerical solver is non-singular, and if it is not, you can perturb it, and the problem will still will be feasible within any reasonable tolerance. Actually stating a non-singularity constraint is very hard (very non-convex set). The product t*X also introduces non-convexity, although it can be dealt with using bisection etc

You say AX+DY is Hermitian, and then you ask what happens if X is Hermitian. That question is strange. If AX+DY is Hermitian, then X has to be Hermitian, otherwise AX+DY >= t*X doesn't make sense (if this is to be interpreted in the semidefinite cone)

As I said, I think you are trying to solve the wrong problem. Where did this problem arise?

[mas tab]

Sorry Dr John.

I think the general form of problem could be solved. So, I formulate it simpler Now.

We have a minimization Problem with this formulation:

min t

s.t.

1)    [ t*I     Y*X

        (Y*X)~  I  ] >=0   ( ~:transposed conjugated, Y is  2x4 (Known) and X  is 4x2(Unknown) complex matrix. I: identity matrix )  

 2)  (X~)*X=I

Thank you for attention.

[Johan Löfberg]

X^TX = I is a very nasty nonlinear nonconvex constraint

[mas tab]

Thanks Dr.

Really X=[N;M] so  that M is non-singular. Do this condition help solving Problem.

And the last Question. In which condition My Problem can be solved? 

[Johan Löfberg]

Still hard. Quadratic equality is generally intractable. Of course, if the problem is small enough, a global solver might succeed, but you should not expect it to work.

[mas tab]

Plz introduce me a reference(sit, person, book,...) for my problem.

Thank u.

[Johan Löfberg]

A reference for what? Hard problems?

Google LMI with Stiefel-Grassman Manifold and you will find a lot of similiar stuff