Given two positive semidefinite matrices, $X$ and $Y$, I want to minimize the product $\operatorname{Tr} (X) \operatorname{Tr} (Y)$.
Now as far as I understand the following holds:
$\operatorname{Tr} (X) \operatorname{Tr} (Y)$ is convex since $\operatorname{Tr} (X)$ and $\operatorname{Tr} (Y)$ are both convex (actually affine), positive and non-decreasing.
It cannot be done in CVX, since the DCP ruleset "...generally forbid products between nonconstant expressions, with the exception of scalar quadratic forms".
Is this two statements true? If so can I minimize $\operatorname{Tr} (X) \operatorname{Tr} (Y)$ with other convex optimiztion programs?
Consider the case when you have scalars. The product $xy$ is not convex (study the Hessian, or simply look at the definition of convexity using the two points $(1,0)$ and $(0,1)$)
More generally, it appears you think the products of two convex positive non-decreasing functions is convex. This is false. If memory serves me right, a necessary condition for the product $f(x)g(y)$ to be convex is that the two functions are log-convex.