A Jacobian smoothing method for box constrained variational inequality problems
type: Item
instance of: scholarly article
title: A Jacobian smoothing method for box constrained variational inequality problems
zbMATH Open document ID: 1068.65089
author: Tong Kang
author: Changfeng Ma
published in: Applied Mathematics and Computation
publication date: 2005-02-23T00:00:00Z
review text: The authors propose a new Jacobian smoothing method such that it becomes well defined for general box constrained variational inequality problems: Find a vector \(x^*\in X\) such that: \(F(x^*)^T(x-x^*)\geq 0\) for all \(x\in X= \{x\in \mathbb R^n\mid \ell\leq x\leq u\}\), where \(\ell_i\in \mathbb R\cup\{-\infty\}\), \(u_i\in \mathbb R\cup \{+\infty\}\) and \(u_i >\ell_i\), \(i = 1,2\dots n\). Further, if \(X=\mathbb R^n_+\), this problem reduces to the nonlinear complementarity problem: Find \(x\in \mathbb R^n\) such that \(x\geq 0\), \(F(x)\geq 0\), \(x^TF(x)= 0\). By the Fischer-Burmeister function \(h: \mathbb R^2\to \mathbb R\), \(h(a, b) =\sqrt{a^2+ b^2} -a-b\), this problem is equivalent to the nonlinear system \[ H(z)=\left[\begin{aligned} & h(x_1-\ell_1,F_1(x)+y_1)\\ &\vdots \\ & h(x_n-\ell_n,\,F_n(x)+y,)\\ & h(u_1-x_1,y_1)\\ & \vdots \\ & h(u_n -x_n,y_n)\end{aligned}\right]=0\tag{1} \] where \(z=(x^Tmy^T)^T\in \mathbb R^{2n}\), the nonsmoothing mapping \(H: \mathbb R^{2n}\to \mathbb R^{2n}\). Main result: The globalization strategy for the proposed algorithm is based on the natural merit function \(\theta:\mathbb R^{2n}\to \mathbb R_+\) given by: \(\theta(z)=\frac12\| H(z)\|^2\), where \(z=(x^T,y^T)\in \mathbb R^{2n}\). The corresponding smooth operator \(H_\mu: \mathbb R^{2n}\to \mathbb R^{2n}\) is defined by (1) with \(h=h_\mu :\mathbb R^2\to \mathbb R\) (the smooth approximation). Here \[ h_\mu(a,b)=\begin{cases} \sqrt{a^2+b^2}-a-b, \sqrt{a^2+b^2}>\mu\\ \frac{a(a-2\mu)}{2\mu}+\frac{b(b-2\mu)}{2\mu}+\frac{\mu}{2},\;\sqrt{a^2+b^2}\leq \mu\end{cases} \] of the Fischer-Burmeister function. The Jacobian consistency property and the Jacobian smoothing idea are discussed. The algorithm is proposed in detail (precise proofs are devoted to proving global superlinear convergence of the algorithm). Finally, some numerical experiments for the proposed algorithm are presented.
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 2137762
zbMATH Keywords: variational inequality
zbMATH Keywords: obstacle problem
zbMATH Keywords: box constraints
zbMATH Keywords: smoothing method
zbMATH Keywords: global convergence
zbMATH Keywords: superlinear convergence
zbMATH Keywords: nonlinear complementarity problem
zbMATH Keywords: algorithm
zbMATH Keywords: numerical experiments
reviewed by: Jan Lovíšek
describes a project that uses: GQTPAR
MaRDI profile type: Publication
full work available at URL: j.amc.2004.03.018
OpenAlex ID: W1989239717
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DOI: 10.1016/J.AMC.2004.03.018
description: scientific article; zbMATH DE number 2137762