A new modified one-step smoothing Newton method for solving the general mixed complementarity problem
type: Item
instance of: scholarly article
title: A new modified one-step smoothing Newton method for solving the general mixed complementarity problem
zbMATH Open document ID: 1192.65079
author: Jia Tang
author: Sanyang Liu
author: Changfeng Ma
published in: Applied Mathematics and Computation
publication date: 2010-05-25T00:00:00Z
review text: The mixed complementarity problem to find a vector \(x \in X \subset \mathbb R^n\) such that \( F(x)^T (y-x) \geq0\), \(\forall y\in X\), where \(X=\prod^n_{i=1} [l_i, u_i]\), \(-\infty \leq l_i < u_i < + \infty\), \(i = 1,2, \dots, n\) is analyzed. Furthermore, if \(X=\mathbb R^n_+\), the above problem reduces to the nonlinear complementarity problem, which is to find \(x \in\mathbb R^n\), such that \(x\geq0\), \(F(x) \geq 0\), \(x^TF(x)=0\). This corresponds to the case that \(l_i=0\) and \(u_i= +\infty\) for all \(i=1,2,\dots, n\), and for \(l_i=-\infty\), \(u_i=+\infty\), \(i=1,2,\dots, n\). In this case, the mixed complementarity problem reduces to the nonlinear equation \(F(x)=0\) and can be reformulated as system of nonsmooth equations: \(\Phi(x)=0\) where \(\Phi:\mathbb R^n \rightarrow\mathbb R^n\) is a semismooth function and its square function, \(\Psi (x):=\frac{1}{2} \|\Phi(x)\|^2\), is continuously differentiable. Main result: A detailed description of the authors modified one-step smoothing Newton method (based on the well-known Chen-Harker-Kanzov-Smale smooth function) and the state of some of its elementary properties is given. It is shown that the presented algorithm is well-defined for an arbitrary mixed complementarity problem. The authors' algorithm consists of two parts: a Newton step, which is in a similar (but slightly different) way as this is done by some smoothing Newton method, and a gradient step. The two parts adopt different merit functions. The global convergence and local superlinear convergence of the algorithm are discussed.
reviewed by: Jan Lovíšek
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 5711778
zbMATH Keywords: mixed complementarity problem
zbMATH Keywords: modified one-step smoothing Newton method
zbMATH Keywords: Chen-Harker-Kanzov-Smale smooth function
zbMATH Keywords: global convergence
zbMATH Keywords: superlinear convergence
zbMATH Keywords: nonlinear equation
zbMATH Keywords: algorithm
zbMATH Keywords: iteration
zbMATH Keywords: updating rules
zbMATH Keywords: sequence strictly decreasing
MaRDI profile type: Publication
full work available at URL: j.amc.2010.02.006
OpenAlex ID: W2060759251
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DOI: 10.1016/J.AMC.2010.02.006
description: scientific article; zbMATH DE number 5711778