Connectedness and Path Connectedness of Weak Efficient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods

Xin Xu,Yang Dong Xu

doi:10.3390/math7100947

Abstract

The connectedness and path connectedness of the solution sets to vector optimization problems is an important and interesting study in optimization theories and applications. Most papers involving the direction established the connectedness and connectedness for the solution sets of vector optimization problems or vector equilibrium problems by means of the linear scalarization method rather than the nonlinear scalarization method. The aim of the paper is to deal with the connectedness and the path connectedness for the weak efficient solution set to a vector optimization problem by using the nonlinear scalarization method. Firstly, the union relationship between the weak efficient solution set to the vector optimization problem and the solution sets to a series of parametric scalar minimization problems, is established. Then, some properties of the solution sets of scalar minimization problems are investigated. Finally, by using the union relationship, the connectedness and the path connectedness for the weak efficient solution set of the vector optimization problem are obtained.

Highlights

Whether the decision is made by a team or an individual, it usually involves several conflicting goals
It is well known that the scalarization method is one effective approach to deal with the connectedness of the solution sets to vector optimization problems, vector variational inequalities and vector equilibrium problems
By means of the linear scalarization method, the authors in [11,12,13,14,15] established the connectedness of the solution set to the class of vector optimization, weak vector variational inequalities and weak vector equilibrium problems

Summary

Introduction

Whether the decision is made by a team or an individual, it usually involves several conflicting goals. By means of the linear scalarization method, the authors in [11,12,13,14,15] established the connectedness of the solution set to the class of vector optimization, weak vector variational inequalities and weak vector equilibrium problems. Most papers mentioned above established the connectedness and the path connectedness for the weak efficient solution sets of vector optimization problems or vector equilibrium problems by means of the linear scalarization method rather than the nonlinear scalarization method. The aim of this paper is to establish the connectedness and path connectedness of the weak efficient solution set for a vector optimization problem via the nonlinear scalarization method.

Preliminaries

Scalarization for VOP

Connectedness and Path Connectedness of VOP

Conclusions