# Determinant maximization via SDP

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!

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.

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

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

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

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.