Some Goldstein's type methods for co-coercive variant variational inequalities
type: Item
instance of: scholarly article
title: Some Goldstein's type methods for co-coercive variant variational inequalities
zbMATH Open document ID: 1208.65093
author: Min Li
author: Li-Zhi Liao
author: Xiaoming Yuan
published in: Applied Numerical Mathematics
publication date: 2011-01-21T00:00:00Z
review text: This paper is devoted to the study of the so-called variant variational inequality VVI \((F, \Omega)\), (which is closely related to VI \((\Omega, F)\): find a vector \(u \in \Omega\) such that: \((v-u)^T F(u) \geq 0, \forall v \in \Omega\), where \(\Omega\) is a nonempty closed convex set in \(\mathbb R^n\) and \(F\) is a mapping from \(\mathbb R^n\) into itself) of the form. Find a vector \(u \in \mathbb R^n\) such that (1) \(F(u) \in \Omega, (v-F(u))^T u \geq 0, \forall v \in \Omega\). It is assumed that the solution set \(S^*\) of (1) is nonempty. Since, in general, it is not possible to identify analytic solutions of VI \((\Omega, F)\) and VVI \((\Omega, F)\) without specific structures of \(F\) and \(\Omega\), the researchers are particularly interested in designing numerical algorithms to approach to VI \((\Omega, F)\) and VVI \((\Omega, F)\), subject to certain criteria. The monotonicity of \(F\) plays a crucial role in designing numerical algorithms for solving VI \((\Omega, F)\) and VVI \((\Omega, F)\). With the restrictions that \(F\) is Lipschitz continuous and strongly monotone, the authors use the classical Goldstein method to get iteratively updates \(u^K\) (starting from any \(u^o \in \Omega\)) according to the formula (2) \(u^{K+1} = P_\Omega [u^K - \beta_K F(u^K)]\), where \(\beta_K > 0\) is the scaling parameter and \(P_\Omega (v)\) denotes the projection of \(v\) onto \(\Omega\) under the Euclidean norm, i.e. \(P_\Omega (v) = arg \min \{\|u-v\|: u \in \Omega \}\). The Goldstein method converges globally with linear speed under the condition (3) \(0 < \beta_L \leq \inf^\infty_{K=1} \{\beta_K\} \leq \sup^\infty_{K=1} \{\beta_K\} \leq \beta_U < \frac{2s}{L^2}\). To solve VVI \((\Omega, F)\) with Lipschitz continuous and strongly monotone \(F\), an extended Goldstein method is presented. The corresponding scheme is: (4) (He-Goldstein method) \(u^{k+1} = u^K-\frac{1}{\beta_K}\{F(u^K)- P_\Omega [F(u^K)-\beta_Ku^k]\}\). Similar to (3) the scaling parame…
reviewed by: Jan Lovíšek
Mathematics Subject Classification ID: 65K15
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49M37
zbMATH DE Number: 5840217
zbMATH Keywords: co-coercive
zbMATH Keywords: Goldstein method
zbMATH Keywords: optimal step-size
zbMATH Keywords: variant variational inequality
zbMATH Keywords: Lipschitz and strongly monotone operators
zbMATH Keywords: convergence
zbMATH Keywords: numerical results
MaRDI profile type: Publication
full work available at URL: j.apnum.2010.10.001
OpenAlex ID: W2102682107
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DOI: 10.1016/J.APNUM.2010.10.001
description: scientific article; zbMATH DE number 5840217