Take $\mathcal{X}$ to be a compact and convex subset of $\mathbb{R}^{n\times n}$ such that all matrices $X\in\mathcal{X}$ are invertible.
I am trying to solve the following problem $$ \max_{X\in\mathcal{X}} a^TX^{-1}b $$ where $a$ and $b$ are column vectors.
I can of course compute the first-order conditions of this problem but I don't know if they are sufficient to characterize the optimum.
I am happy to impose more (non-trivial) constraints on $\mathcal{X}$ if this can lead to a characterization of the optimum.
Any help would be much appreciated!