 Robust linear optimization over the probability simplex

Worst-case linear revenue over a polyhedral uncertainty set is optimized by the linear programHere ${\Delta }_n$ is the probability simplex. Finite decisions are exactly its intersection with $\mathrm{range}{P}^T$. If this intersection is empty, all decisions have worst-case value $-\infty$ and the displayed linear program is infeasible; its supremum over the empty feasible set is also $-\infty$. Full column rank of $P$ suffices to make all simplex decisions finite.