Globally proper efficiency of set-valued optimization and vector variational inequality involving the generalized contingent epiderivative

Wang Chen,Zhiang Zhou

doi:10.1186/s13660-018-1891-8

Abstract

In this paper, firstly, a new property of the cone subpreinvex set-valued map involving the generalized contingent epiderivative is obtained. As an application of this property, a sufficient optimality condition for constrained set-valued optimization problem in the sense of globally proper efficiency is derived. Finally, we establish the relations between the globally proper efficiency of the set-valued optimization problem and the globally proper efficiency of the vector variational inequality.

Highlights

1 Introduction It is well known that convexity plays a crucial role in set-valued optimization
By considering the generalized cone preinvexity and the contingent epiderivative of set-valued maps, Yu [20] disclosed the relations between Henig global efficiency of (SVOP) and Henig global efficiency of a kind of vector variational inequality
We will use Lemma 3.1 to establish a sufficient optimality condition characterized by the generalized contingent epiderivative of set-valued maps in the sense of globally proper efficiency

Summary

Introduction

It is well known that convexity plays a crucial role in set-valued optimization. To generalize convexity of set-valued maps, some scholars introduced different kinds of generalized convex set-valued maps. By considering the generalized cone preinvexity and the contingent epiderivative of set-valued maps, Yu [20] disclosed the relations between Henig global efficiency of (SVOP) and Henig global efficiency of a kind of vector variational inequality. We introduce a new type of generalized vector variational inequality problem (shortly, (GVVIP)) by virtue of the generalized contingent epiderivative of set-valued maps, propose the notion of globally proper efficiency of (GVVIP) and obtain the optimality conditions of the globally proper efficiency of (GVVIP).

Results

Conclusion