Generalized properly efficient solutions of vector optimization problems

Generalized properly efficient solutions of a vector optimization problem (VP) are defined in terms of various tangent cones and a generalized directional derivative. We study their basic properties and relationships and show that under certain conditions, a generalized properly efficient solution of (VP), defined by the adjacent cone, is a generalized Karush–Kuhn–Tucker properly efficient solution of (VP). Furthermore, using subgradients defined by closed convex tangent cones, we give necessary optimality condition for a generalized properly efficient solution of (VP) defined by the adjacent cone.