First and second-order optimality conditions without differentiability in multivalued vector optimization

Abstract

By developing and using approximations as generalized derivatives of set-valued mappings, we establish both necessary conditions and sufficient conditions of order one and two for various kinds of solutions to a multivalued vector optimization problem with general inequality constraints and high level of nonsmoothness. An emphasis was paid on the important (but still not widely known) so-called envelope-like effect. Our results are more applicable than a number of recent existing ones in many situations as illustrated in detail by examples.