LP Well-Posedness for Bilevel Vector Equilibrium and Optimization Problems with Equilibrium Constraints

Phan Quoc Khanh,Somyot Plubtieng,Kamonrat Sombut

doi:10.1155/2014/792984

Phan Quoc Khanh, Somyot Plubtieng + Show 1 more

Open Access

https://doi.org/10.1155/2014/792984

Copy DOI

Abstract

The purpose of this paper is introduce several types of Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Base on criterion and characterizations for these types of Levitin-Polyak well-posedness we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescus measures of noncompactness of approximate solution sets under suitable conditions, and we prove the Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Obtain a gap function for bilevel vector equilibrium problems with equilibrium constraints using the nonlinear scalarization function and consider relations between these types of LP well-posedness for bilevel vector optimization problems with equilibrium constraints and these types of Levitin-Polyak well-posedness for bilevel vector equilibrium problems with equilibrium constraints under suitable conditions; we prove the Levitin-Polyak well-posedness for bilevel equilibrium and optimization problems with equilibrium constraints.

Highlights

Well-posedness is one of most important topics for optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution
We focus on vector equilibria with equilibrium constraints and optimization with equilibrium constraints, as well as an abstract set constraint, and investigate criteria and characterizations for these types of LP well-posedness with a gap function for bilevel vector equilibrium problems with equilibrium constraints and optimization problems with equilibrium constraints
Problems (BVEPEC) is called type I LP well-posed if and only if (i) the solution set of (BVEPEC) is nonempty; (ii) for any type I, LP approximating sequence of (BVEPEC) has a subsequence converging to a solution

Summary

Introduction

Well-posedness is one of most important topics for optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution. The constraints appear in this paper are solution sets of the following (parametric) vector quasi-equilibrium problem, for each z ∈ Z: (VQEPz) find x ∈ K1 (x, z) s.t. f (x, y, z) ∉ − int C (x) ,

Objectives

Results

Conclusion