A smoothing Newton method for general nonlinear complementarity problems
type: Item
instance of: scholarly article
title: A smoothing Newton method for general nonlinear complementarity problems
zbMATH Open document ID: 0987.90078
DOI: 10.1023/A:1026554432668
author: Li-Zhi Liao
author: Houduo Qi
published in: Computational Optimization and Applications
publication date: 2001-02-18T00:00:00Z
review text: The paper deals with the nonlinear complementarity problem: to find \(x \in \mathbb{R}^n_+\) such that \(F(x) \geq 0\), \(x \geq 0\), \(x^TF(x)=0\), where \(F: \mathbb{R}^n\to \mathbb{R}^n\) is a continuously differentiable function. The authors consider the system: \(H= ( e^{\mu}-1 \Phi_{\mu} (x))^T\), where \(\Phi_{\mu}\) is defined by Kanzow's smoothing approximation of the Fisher-Burmeister function (\(\sqrt{\mu^2 + x_i^2 +F_i (x)^2} -x_i- F_i(x)\)). Some properties of \(H\) , \(\Phi_\mu\) and \(\chi\) which play an important role in the analysis of the methods are discussed in the paper. For solving the above system the authors present an algorithm, which combines Newton steps and gradient steps. Two different merit functions are used: \(f(\mu,x)= \frac{ \|H(\mu,x) \|^2}{2}\) for Newton step and \(\chi (x)= \frac{\|\Phi_ \mu(x) \|^2}{2}\) for gradient step. In the Newton step \(\mu^k\) is a variable, in the gradient step \(\mu_k\) is viewed as a smoothing parameter which has to be updated. The rules for updating \(\mu\), the restriction of the step length along Newton direction assure the global convergence of the algorithm without boundedness assumption. The nonsingularity of any \(H(0,x^*)\) (\(x^*\) accumulation point of the sequence generated by the algorithm) is assumed for proving the local convergence of the algorithm. The paper includes a brief description of the implementation of the algorithm and reports numerical results for the test problems in MCPLIB and GAMSLIB libraries.
reviewed by: S. M. Allende-Alonso
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 1565502
zbMATH Keywords: nonlinear complementarity problem
zbMATH Keywords: smoothing Newton method
zbMATH Keywords: global convergence
zbMATH Keywords: linear convergence
zbMATH Keywords: superlinear convergence
describes a project that uses: PATH Solver
MaRDI profile type: Publication
full work available at URL: a:1026554432668
OpenAlex ID: W1569858081
Recommended article: A smoothing Newton method for nonlinear complementarity problems
Recommended article: A smoothing Newton-type method for generalized nonlinear complementarity problem
Recommended article: Q4783278
Recommended article: A smoothing inexact Newton method for generalized nonlinear complementarity problem
Recommended article: A smoothing inexact Newton method for nonlinear complementarity problems
Recommended article: Q5282602
Recommended article: A nonmonotone smoothing Newton method for complementarity problems
Recommended article: Smoothing Newton method for generalized complementarity problems based on a new smoothing function
description: scientific article; zbMATH DE number 1565502