Determinant maximization via SDP


#1

Hi, I have the following cvx code:

A is given, W is given D is given and I run the following:

     cvx_begin quiet
    variable Pi(n, n) semidefinite
    variable P(n, n) symmetric
    minimize(-det_rootn(Pi)) 
   A*P*A' - P + W <In> semidefinite(n)
    trace(P) <= D
    [P-Pi, P*A';
     A*P, A*P*A' + W] <In> semidefinite(2*n)
     cvx_end

So I would like to know if it is possible to include the constraint that matrix P commutes with APA’ + W. Does anyone know? Thanks a lot!


(Michael C. Grant) #2

You’re basically asking if you can include A*P*A'*P+W*P==P*A*P*A'+P*W, I think; and that is not convex, so no.


#3

Yes, I am aware of this. Is it possible to convexify it somehow? Thanks Michael.


#4

I do not mind as well having APA’P+WP semidefinite it is basically the same thing.


(Mark L. Stone) #5

That is a BMI (Bilinear Matrix Inequality), which is … drum roll, Smitty … non-convex, and requires a BMI solver, such as PENLAB or PENBMI.


#6

Hm…I was thinking to fix each parameter, namely Lambda=APA’+W and see what can I get. But I do not think is doable.