Continuity of Solution Mappings for Parametric Set Optimization Problems under Partial Order Relations

Yueming Sun

doi:10.4236/apm.2020.1011039

Abstract

This paper mainly investigates the semicontinuity of solution mappings for set optimization problems under a partial order set relation instead of upper and lower set less order relations. To this end, we propose two types of monotonicity definition for the set-valued mapping introduced by two nonlinear scalarization functions which are presented by these partial order relations. Then, we give some sufficient conditions for the semicontinuity and closedness of solution mappings for parametric set optimization problems. The results presented in this paper are new and extend the main results given by some authors in the literature.

Highlights

IntroductionFor set-valued optimization problems, there are two types of criteria which are vector optimization criterion and set optimization criterion
Set-valued optimization which is a generalization of vector optimization has been studied and applied in many fields, such as engineering, mathematical finance, medicine, robust and fuzzy optimization; see [1] [2] [3] [4] and the references therein.As we know, for set-valued optimization problems, there are two types of criteria which are vector optimization criterion and set optimization criterion
This paper mainly investigates the semicontinuity of solution mappings for set optimization problems under a partial order set relation instead of upper and lower set less order relations

Summary

Introduction

For set-valued optimization problems, there are two types of criteria which are vector optimization criterion and set optimization criterion. Based on the vector optimization criterion, discussing the continuity of solution set mapping for set-valued vector optimization problems is very similar to discuss vector variational inequalities or vector equilibrium problems [5]. This criterion is not always suitable for all types of set-valued optimization prob-. Many research results have been studied for parametric set optimization problems under different kinds of set order relations, such as the optimality conditions, convexity, well-posedness, existence, duality theory and algorithms; see [8]-[15] and the references therein

Objectives

Discussion

Conclusion