We are given a set of unit vectors $U \subset \mathbb{C}^n$ which spans the space $\mathbb{C}^n$. Given another unit vector $x$, consider then the following optimization problem:

$$ \sup_H \left\{ x^* H x \;:\; H \text{ is Hermitian},\; 0 \leq u^* H u \leq 1 \;\forall u \in U \right\}. $$

This optimization comes up in a problem that I have been solving numerically, and to my surprise, the optimal solution $H$ seems to always be positive semidefinite (when the supremum is actually achieved). I am trying to understand whether this is a general property of the problem.

My question is thus: is the optimal $H$ necessarily positive semidefinite? If not, is there some property of the set $U$ that guarantees that the optimal solution is positive semidefinite?

One observation is that the problem is actually unbounded when the vectors in $U$ are mutually orthogonal, so we can assume this not to be the case. However, I do not see if this by itself implies the positive semidefiniteness of the optimal $H$ somehow. I would appreciate any help.