Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions
type: Item
instance of: scholarly article
title: Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions
zbMATH Open document ID: 1205.90279
published in: Journal of Optimization Theory and Applications
publication date: 2010-08-13T00:00:00Z
review text: This paper applies a scalarization approach of Konnov for studying strong and weak vector variational inequalities with set valued mappings. More precisely, let \(X\) be a real Banach space, \((Y,P)\) an ordered Banach space induced by a closed convex and pointed cone \(P\). Let \(L(X,Y)\) be the set of continuous linear mappings from \(X\) to \(Y\), and let \(K\subset X\) be nonempty, closed and convex. Given \(T:X\to 2^{L(X,Y)}\) a set-valued mapping, the authors consider two kinds of vector variational inequalities as follows. The \textit{Strong Vector Variational Inequality}: Find \(x^*\in K\) such that \[ \exists\,\,t\in T(x^*) : \langle t^*,y-x^* \rangle \not< 0,\;\;\forall\;y\in K. \] The \textit{Weak Vector Variational Inequality}: Find \(x^*\in K\) such that \[ \forall\;y\in K,\,\exists\,\,t\in T(x^*) : \langle t^*,y-x^* \rangle \not< 0. \] The authors consider several kinds of strong and weak variational inequalities and their gap functions. Under certain conditions, they prove the equivalence between the scalar and vector version of these problems and analyze the relations among their corresponding gap functions. They apply their results to derive error bounds for gap functions. Finally, they obtain some existence results of global error bounds for gap functions and derive characterizations of global and local error bounds fo these gap functions.
reviewed by: R. S. Burachik
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 5770313
zbMATH Keywords: vector variational inequalities
zbMATH Keywords: set-valued mappings
zbMATH Keywords: scalarization
zbMATH Keywords: gap functions
zbMATH Keywords: error bounds
author: Jun Li
author: Giandomenico Mastroeni
MaRDI profile type: Publication
full work available at URL: s10957-009-9625-1
OpenAlex ID: W2132008822
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DOI: 10.1007/S10957-009-9625-1
description: scientific article; zbMATH DE number 5770313