Convex Analysis in Groups and Semigroups and Applications to Separation and Optimization

The aim of this paper is to investigate some fundamental properties of convex sets, cones, and functions in groups and semigroups. In particular, we introduce the concept of quasi core of a convex set, which allows us to provide separation theorems between convex sets in groups where at least one of them has nonempty quasi core. We also analyze optimization problems defined on groups and semigroups and provide saddle point necessary or sufficient optimality conditions by considering suitable Lagrangian functions associated with the given problem.