Given function $f : P \to \mathbb R$, where $P \subset \mathbb{R}^3$ is a convex polytope, I want to find the minimum and maximum of $f$ provided that $x \in P$.
I can show that $f$ is monotonic over the set $P$, i.e., the elements of the gradient do not change their signs. For the case that $P$ is an interval, I can evaluate $f$ at the vertices of this interval to find the minimum and maximum. Is the same true if $P$ is a convex polytope?