Hybrid Lagrange multiplier approaches for solving infinite dimensional equilibrium problems with cone constraints
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.90367
author: Mostafa Nasri
author: Javad Mashreghi
publication date: 2012-05-30T00:00:00Z
full work available at URL: p331.html
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 65K15
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 6039389
zbMATH Keywords: equilibrium problem
zbMATH Keywords: one constraint Bregman distance
zbMATH Keywords: Bregman projection
zbMATH Keywords: extragradient algorithm
zbMATH Keywords: projection algorithm
MaRDI profile type: Publication
title: Hybrid Lagrange multiplier approaches for solving infinite dimensional equilibrium problems with cone constraints
published in: Journal of Nonlinear and Convex Analysis
review text: Let \(K\subset B\) be a nonempty closed convex cone of a reflexive Banach space \(B\) and let \(f: K\times K\to\mathbb{R}\) be a smooth function. The authors provide hybrid Lagrange multiplier approaches for equilibrium problems according to NEWLINE\[NEWLINE\text{find an }x^*\in K\text{ such that }f(x^*, y)\geq 0\quad\forall y\in K.NEWLINE\]NEWLINE It is pointed out that many particular problems (e.g. optimization problems, fixed point problems, complementarity problems, variational inequality problems) can be formulated and solved as equilibrium problems.NEWLINENEWLINE The main goal of the paper is the generalization of two finite-dimensional Lagrange multiplier algorithms published in a former paper to the infinite-dimensional case. Both algorithms start with an approximate solution of an unconstrained equilibrium problem. Then, using the so-called Bregman distance with respect to a suitable regularizing function, the next iterations are computed by performing either an extragradient-type step or a Bregman projection onto a certain hyperplane separating the current iterate from the solution set. Under appropriate assumptions, it is shown that the generated sequences converge weakly and globally to a solution of the problem.
reviewed by: Jörg Thierfelder
description: scientific article; zbMATH DE number 6039389