Is $|| AXB-C ||_F$ convex?

Question

Let $A \in \mathbb R^{n\times n}$, $B \in \mathbb R^{n\times n}$, $C \in \mathbb R^{n\times n}$ be constant matrices.

Is the following convex?

minimize $|| AXB-C ||_F$ for $X>0$,

where $|| \dots ||_F$ denots the frobenius norm.

Answer 1

Yes, both the objective function and the domain are convex. For the objective function, recall that the composition of a convex function with an affine map is also convex. (This is immediate from the definition of convexity). Here $X\mapsto AXB-C$ is an affine map, and $Y\mapsto \|Y\|_F $ is a convex function.