E-Benson proper efficiency in vector optimization

Abstract

Starting from the innovative ideas of Chicco et al. (Vector optimization problems via improvement sets. J. Optim. Theory Appl. 2011;150:516–529), in this paper, the concepts of improvement set and -efficiency are introduced in a real locally convex Hausdorff topological vector space. Furthermore, some properties of the improvement sets are given and a kind of proper efficiency, named as -Benson proper efficiency, which unifies some proper efficiency and approximate proper efficiency, is proposed via the improvement sets in vector optimization. Moreover, the concept of -subconvexlikeness of set-valued maps is introduced via the improvement sets and an alternative theorem is proved. In the end, some scalarization theorems and Lagrange multiplier theorems of -Benson proper efficiency are established for a vector optimization problem with set-valued maps.

Keywords:

• vector optimization
• E-Benson proper efficiency
• E-subconvexlikeness
• scalarization
• Lagrange multipliers

AMS Subject Classifications:

• 90C26
• 90C29
• 90C30

Acknowledgments

This work is partially supported by the National Science Foundation of China (Grants 11271391, 11126348, 11171363), the Natural Science Foundation Project of Chongqing (Grant 2011BA0030 and CSTS2012jjA00002) and the Education Committee Research Foundation of Chongqing (Grant KJ110625).