Have you ever seen any optimization problem over finite groups (or finitely generated groups)? That is, given a group $G$, we want to maximize or minimize a function $f$ over $G$.
An example that struck my mind, was $N$-queens problem. We can define a function $f$ as number of pairwise conflicts. In this problem, group $G$ is $S_N$.
You know, $(ℤ,+)$ is also a finitely generated group, and any integer programming problem will be the answer. But for this trivial groups, I mean a problem having a group theoretic approach.
I will also be grateful if you can mention the current approaches to solve that optimization problem.