Duality and saddle-points for convex-like vector optimization problems on real linear spaces

Abstract

Usually, finite dimensional linear spaces, locally convex topological linear spaces or normed spaces are the framework for vector and multiojective optimization problems. Likewise, several generalizations of convexity are used in order to obtain new results. In this paper we show several Lagrangian type duality theorems and saddle-points theorems. From these, we obtain some characterizations of several efficient solutions of vector optimization problems (VOP), such as weak and proper efficient solutions in Benson’s sense. These theorems are generalizations of preceding results in two ways. Firstly, because we consider real linear spaces without any particular topology, and secondly because we work with a recently appeared convexlike type of convexity. This new type, designated GVCL in this paper, is based on a new algebraic closure which we named vector closure.