Let $V$ be a finite-dimensional vector space with norm $\|\cdot\|$ and let $F : V \rightarrow \mathbb R$ be a bounded linear functional. It is only given as black-box.

I would like to estimate the norm of $F$ (from above and below). As $F$ is a black-box, the only way to do so is to test it with unit vectors from $V$ and, based on the result, find $v \in S^1 V$ that maximizes $|F(v)|$.

Do you know such an algorithm? In the application that I have in mind, $V$ is a finite-element space and $F$ is a complicated functional on that space.

EDIT:
My first idea is to choose $v \in S^1 V$ randomly, perturb it into several directions, say, $v\_1,\dots,v\_k$, and then repeat the procedure with the $v\_i$ that got the biggest $F(v\_i)$. I do not know where to find algorithms and analysis for this problem.