Structure of Pareto Solutions of Generalized Polyhedral-Valued Vector Optimization Problems in Banach Spaces

Qinghai He,Weili Kong

doi:10.1155/2013/619206

Abstract

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

Highlights

Let X and Y be Banach spaces, Γ a closed convex subset of X, C a closed convex cone of Y, b ∈ Y, and T : X → Y a linear function
Zheng and Yang [11] proved that the weak Pareto set Sw and the weak Pareto optimal value set Vw of (LMOP) are pathwise connected, respectively, when the ordering cone C has a nonempty interior and the mapping T(⋅) + b is C-convex
Zheng [13] proved that the Pareto set S and the Pareto optimal value set V of (SVOP) are pathwise connected, respectively, when the ordering cone C is a pointed, closed, convex cone with a weakly compact base and F is a C-convex multifunction whose graph is the union of finitely many convex polyhedra

Summary

Introduction

Let X and Y be Banach spaces, Γ a closed convex subset of X, C a closed convex cone of Y, b ∈ Y, and T : X → Y a linear function. One of our main aims in this work is to investigate the structure of the (weak) Pareto solution set and the (weak) Pareto optimal set of (SVOP) whose graph Gr(F) is the union of finitely many G-polyhedra. Zheng and Yang [11] proved that the weak Pareto set Sw and the weak Pareto optimal value set Vw of (LMOP) are pathwise connected, respectively, when the ordering cone C has a nonempty interior and the mapping T(⋅) + b is C-convex. Zheng [13] proved that the Pareto set S and the Pareto optimal value set V of (SVOP) are pathwise connected, respectively, when the ordering cone C is a pointed, closed, convex cone with a weakly compact base and F is a C-convex multifunction whose graph is the union of finitely many convex polyhedra. The other of our main aims is to study the connectedness of the Pareto set S and the Pareto optimal value set V of (SVOP) without the assumption of the ordering cone C having a weakly compact base but with that of the cone C being polyhedral

Preliminaries

The Structure of Solution Sets and Optimal Value Sets

Connectedness of Pareto Solution Sets and Pareto Optimal Value Sets