Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives

This paper deals with maximization and minimization of quasiconvex functions in a finite dimensional setting. Firstly, some existence results on closed convex sets, possibly containing lines, are presented. This is given via a careful study of reduction to the boundary and/or extremality of the feasible set. Necessary or sufficient optimality conditions are derived in terms of radial epiderivatives. Then, the problem of minimizing quasiconvex functions are analyzed via asymptotic analysis. Finally, some attempts to define asymptotic functions under quasiconvexity are also outlined. Several examples illustrating the applicability of our results are shown.