A class of gap functions for variational inequalities
type: Item
instance of: scholarly article
title: A class of gap functions for variational inequalities
zbMATH Open document ID: 0819.65101
DOI: 10.1007/BF01582565
author: Torbjörn Larsson
author: Michael Patriksson
published in: Mathematical Programming. Series A. Series B
publication date: 1994-08-10T00:00:00Z
review text: Let \(K\) be a closed convex set in \(R^ n\) and \(T : K \to R^ n\) be a continuous operator. The problem of finding \(u \in K\) such that \(\langle Tu,v - u \rangle \geq 0\) for all \(v \in K\), is known as the variational inequality problem (VIP). A function \(g : K \to R \cup \{-\infty, \infty\}\) is a gap (merit) function for (VIP) if (i) \(g\) is restricted in sign on \(K\) and (ii) \(g(u) = 0\) if and only if \(u \in \Omega\), where \(\Omega\) is the solution set of (VIP). In this interesting paper, the authors consider and study a new class of gap functions. They also discuss some algorithmic equivalence results relating their work to that of \textit{S. Dafermos} [Math. Program. 26, 40- 47 (1983; Zbl 0506.65026)], \textit{G. Cohen} [J. Optimization Theory Appl. 59, No. 2, 325-334 (1988; Zbl 0653.90062)] and \textit{M. A. Noor} [ibid. 73, No. 2, 409-413 (1992; Zbl 0794.49009)]. Using the gap functions, a general descent framework has been developed for finding the approximate solution of variational inequalities. The results presented in this paper represent an improvement of the previously known results in this area. Remark: In a recent paper, the reviewer [Some nonlinear variational inequalities, Tamkang J. Math. Vol. 26, No. 2 (1995)] has proved that the gap functions discussed in this paper can be derived by using the auxiliary principle technique.
reviewed by: Muhammad Aslam Noor
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49M30
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 617331
zbMATH Keywords: iterative algorithms
zbMATH Keywords: convergence
zbMATH Keywords: variational inequality
zbMATH Keywords: gap functions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01582565
OpenAlex ID: W2073584813
description: scientific article; zbMATH DE number 617331