On the Existence of Super Efficient Solutions and Optimality Conditions for Set-Valued Vector Optimization Problems

Lu-Chuan Ceng,Ching-Feng Wen,Yeong-Cheng Liou

doi:10.3390/math10030316

Abstract

In this paper, by using the normal subdifferential and equilibrium-like function we first obtain some properties for K-preinvex set-valued maps. Secondly, in terms of this equilibrium-like function, we establish some sufficient conditions for the existence of super minimal points of a K-preinvex set-valued map, that is, super efficient solutions of a set-valued vector optimization problem, and also attain necessity optimality terms for a general type of super efficiency.

Highlights

During the past more than 20 years, extending and characterizing definitions/properties of generalized convexity from the real-valued to the multi-valued mappings had been investigated by many scholars; the readers are referred to Benoist and Popovici [1,2], Jabarootian and Zafarani [3], Oveisiha and Zafarani [4], Sach and Yen [5], Yang [6] and the references cited therein
Motivated by Oveisiha and Zafarani [31], Ceng and Latif [32] considered Stampacchia equilibrium-like problems by virtue of a normal subdifferential for multifunctions, established their relations with set-valued vector optimization issues, and attained certain characterizations of a solution set of a set-valued generalized K-pseudoinvexity program
In this paper, inspired by the above research works, we first deduce some properties for K-preinvex set-valued maps using their marginal functions, equilibrium-like function and normal subdifferential concept. In terms of this equilibrium-like function we establish some sufficient conditions for the existence of super minimal points of a

Summary

Introduction

During the past more than 20 years, extending and characterizing definitions/properties of generalized convexity from the real-valued to the multi-valued mappings had been investigated by many scholars; the readers are referred to Benoist and Popovici [1,2], Jabarootian and Zafarani [3], Oveisiha and Zafarani [4], Sach and Yen [5], Yang [6] and the references cited therein. By virtue of the scalarizing technique, Oveisiha and Zafarani investigated Stampacchia variational-like inequalities by using a normal subdifferential for multifunctions and built their relations with set-valued vector optimization issues. They attained certain characterizations of the solution sets of pseudoinvexity extremum issues. Motivated by Oveisiha and Zafarani [31], Ceng and Latif [32] considered Stampacchia equilibrium-like problems by virtue of a normal subdifferential for multifunctions, established their relations with set-valued vector optimization issues, and attained certain characterizations of a solution set of a set-valued generalized K-pseudoinvexity program.

Concepts and Basic Tools

K-Preinvexity Mappings

Super Efficient Solutions

Methods