Formulation of convex problem in CVX

Nonconvex
1

Hello,

I want to minimise the objective

f(X) = norm(X*A, 'fro') + trace(B - X*C*X')

with respect to the matrix X. A, B and C are matrices, where B, C and (B - XCX’) are positive semidefinite.

Is there a way of formulating this problem using the DCP ruleset so that CVX can solve it? I was thinking about adding the constraint that B - XCX’ is SDP via the Schur complement, such that CVX knows the problem is convex…

2

I don’t think this is convex, actually. It would be if you were minimizing norm(X*A,'fro')+trace(X*C*X'-B). But as written, I’m not seeing it.

3

Thanks for your fast reply. I agree that in general this as a non-convex problem. My thought was that with the additional constraint of the matrix (B - XCX’) being PSD the problem is convex on the restricted domain. Please correct me if I made a mistake here.

4

No, this is not the case.

5

Thanks for pointing that out. How would I implement
norm(XA,‘fro’)+trace(XC*X’-B)
in CVX?

6

norm(X*A,'fro')+sum_square(X*sqrtm(C))-trace(B)