Can the minima in this trace optimization be characterised?

Question

Let $\mathcal{A}$ be a finite set of vectors in $\mathbb{R}^d$. Assume $span\{\mathcal{A}\}=d$. Let $V_{\pi}=\sum_{a \in \mathcal{A}}\pi(a)aa^{t}$, be a PD matrix, where $\pi$ is a probability measure on $\mathcal{A}$. Let $U$ be any fixed PD matrix. What is the optimizer in $\underset{\pi}{\min}Tr(V_{\pi}^{-1}U)$? Is there a "nice" characterisation of its support? I know this is a convex optimization in $\pi$ but am unable to write up a dual or make any other progress.