Splitting methods for pseudomonotone mixed variational inequalities
type: Item
instance of: scholarly article
title: Splitting methods for pseudomonotone mixed variational inequalities
zbMATH Open document ID: 0966.49011
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-11-22T00:00:00Z
review text: The author analyzes some algorithms for solving a monotone mixed variational inequality \(\;\langle Tu,v-u\rangle + \varphi (v)-\varphi(u)\geq 0,\quad\) for all\(\;v\in H\) with a Hilbert space \(H\), a nonlinear operator \(T:H\to H\) and a lower semicontinuous function \(\varphi:H\to R\cup \{+\infty\}.\) The resolvent operator technique is used to study a number of splitting methods for solving the mixed variational inequalities. The methods are based on the fact that a solution \(u\in H\) of a variational inequality satisfies the relation \(u=J_\varphi [u-\rho Tu] \), where \(J_\varphi= (I+\rho \partial \varphi)^{-1}\) is the resolvent operator. The resolvent operator method requires the assumption that \(T\) must be strongly monotone for the convergence. In order to overcome this difficulty the author suggests the double resolvent formula \(\;u=J_\varphi \Big\lgroup u-\rho TJ_\varphi [u-\rho Tu]\Big\rgroup.\) Several splitting methods for solving the iteration method connected with fixed point algorithms are analyzed.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 1496350
zbMATH Keywords: variational inequalities
zbMATH Keywords: splitting methods
zbMATH Keywords: resolvent equations
zbMATH Keywords: fixed point
zbMATH Keywords: convergence
author: Muhammad Aslam Noor
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6776
OpenAlex ID: W1991326129
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DOI: 10.1006/JMAA.2000.6776
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description: scientific article; zbMATH DE number 1496350