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 directi onal derivative. We study their basic properties and relationships and show that under certain conditions, a generalized properly efficient solution o f (VP), defined by the adjacent cone, is a generalized Kuhn-Tucker properly efficient solution of (VP). Furthermore, using subgradients defined by clo sed convex tangent cones, we give a necessary optimality condition for a ge neralized properly efficient solution of (VP) defined by the adjacent cone.