Error bounds for mixed set-valued vector inverse quasi-variational inequalities

Shih-Sen Chang,Z L Ma,Salahuddin Salahuddin,L Wang,G Wang

doi:10.1186/s13660-020-02424-7

Abstract

The purpose of this paper is to introduce and study the mixed set-valued vector inverse quasi-variational inequality problems (MSVIQVIPs) and to obtain error bounds for this kind of MSVIQVIP in terms of the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of mixed set-valued vector inverse quasi-variational inequality problem. The results presented in the paper improve and generalize some recent results.

Highlights

The inverse variational inequalities were developed by He et al [1, 2], which have many applications in various fields such as market equilibrium issues in economics, transportation networks and communication networks; see [3,4,5,6,7,8]
Motivated and inspired by the research going on in this direction, the purpose of this paper is to introduce and study the mixed set valued vector inverse quasi-variational inequality problems (MSVIQVIP)
Gap functions play a central role in deriving the so-called error bounds, which provide a measure of the distances between the solution set and an arbitrary feasible point

Summary

Introduction

The inverse variational inequalities were developed by He et al [1, 2], which have many applications in various fields such as market equilibrium issues in economics, transportation networks and communication networks; see [3,4,5,6,7,8]. We propose three gap functions, the residual gap function, the regularized gap function, the D-gap function By using these gap functions and the generalized f -projection operator, and under suitable conditions, we obtain error bounds for this kind of mixed set-valued vector inverse quasi-variational inequalities. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of MSVIQVIPs. The results presented in the paper improve and generalize the corresponding ones in [15, 19, 21]. More important is it to evaluate their convergence properties and to obtain useful stopping rules for iterative algorithms This motivates us to research and evaluate various gap functions for a mixed set-valued vector inverse quasi-variational inequality. Qi0 (u ), y – h(x) + f (y) – f h(x) ≤ 0, ∀y ∈ Ω(x), u ∈ F(x)

It ensures that

Rn and ρ