On the maximization of (not necessarily) convex functions on convex sets

The global solutions of the problem of maximizing a convex function on a convex set were characterized by several authors using the Fenchel (approximate) subdifferential. When the objective function is quasiconvex it was considered the differentiable case or used the Clarke subdifferential. The aim of the present paper is to give necessary and sufficient optimality conditions using several subdifferentials adequate for quasiconvex functions. In this way we recover almost all the previous results related to such global maximization problems with simple proofs.