Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD but I am having some difficulty expressing the function as a semidefinite program, so I thought maybe casting a wider net would be beneficial).

UPDT: The specific function I have in mind goes like this. Take a fixed real matrix $A$ and let $f(X)=\min{diag(AX)}$ for all $X \in PSD$, excluding $X=0$. So actually, I'm looking at the cone without it's vertex, possibly complicating matters further.

UPDT2: Let's also assume we are taking a compact subset of the cone (in the PSD case, we can take all the diagonal entries to be $1$).