A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints
type: Item
instance of: scholarly article
title: A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints
zbMATH Open document ID: 0987.90079
DOI: 10.1023/A:1026546230851
author: Bintong Chen
author: Xiaojun Chen
published in: Computational Optimization and Applications
publication date: 2001-02-18T00:00:00Z
review text: This paper deals with the Variational Inequality Problem (VIP): to find \(x \in [l,u]\): such that \( (v-x)^TF(x) \geq 0 \forall v \in [l,u]\), where \(F: \mathbb{R}^n \to \mathbb{R}^n\) is a continuously differentiable function, \(lu \in \mathbb{R}^n \bigcup\{ -\infty,+\infty\} \), \(l<u\) . Reformulating (VIP) as the non-smooth equations system: \((\phi_0)_i (x)= x_i-\text{mid}(l_i,u_i,x_i-F_i (x))=0\), \(i=1 \ldots n\), and approximating the componentwise median operator \(\text{mid}(.)\) by the Gabriel-More function: \[ (p_\mu)_i (s)= \int_{-\infty}^{+\infty} \text{mid}(l_i,u_i,s- \mu t) \rho(t) dt=0, \quad i=1, \dots,\rho, \] the authors formulate a continuation algorithm on the parametrized system \(H_{\mu}(x,y)=(y-F(x) x_i -(p_ \mu)_i (x_i-y_i)\), \( i=1,\dots)\). It is considered a neighborhood of the smoothing path and the slice \(N (\beta,\mu^0)=\{z \mid \|H_{\mu}(x,y)\|\leq \beta \mu\), \(0< \mu <\mu^0 \},\) \(\beta,\mu^0 > 0\). The properties of the Gabriel-More function, the \(\mu\)-reduction strategy, and some assumptions on \(F'\) and on \( \nabla H_{\mu}(x^ky^k)\) \(\forall k\) imply the global convergence of the proposed algorithm. For the local convergence analysis it is assumed that \({ x^k,y^k }\) converges to \((\bar x, \bar y)\), the Jacobian of \(H_0(\bar x, \bar y)\) is nonsingular and a non-degeneracy assumption (\(\min \{\bar x_i -l_i,u_i-x_i +{y_i}\} \in 0\), \(\forall i=1 \dots n.\)). In addition the authors present a hybrid continuation algorithm (A2) by adding an optional Newton step to the algorithm A1. A global convergence result for A2 is proved under the assumption that \(F\) is a \(P_0\) function and that the slice \({ N}(\beta,\mu^0)\) is bounded. Conditions for this last assumption to hold are also discussed in the paper, particularly in the context of nonlinear complementarity problem. The authors prove the local quadratic convergence of A2 under the additional assumption that all the Clarke generalized Jacobian of \(H_0(\bar x,…
reviewed by: S. M. Allende-Alonso
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 65K10
zbMATH DE Number: 1565498
zbMATH Keywords: variational inequality
zbMATH Keywords: continuation smoothing method
zbMATH Keywords: linear convergence
zbMATH Keywords: quadratic convergence
MaRDI profile type: Publication
Recommended article: Q4784421
Recommended article: A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation
Recommended article: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities
Recommended article: A Jacobian smoothing method for box constrained variational inequality problems
Recommended article: Q4901015
Recommended article: On Homotopy-Smoothing Methods for Box-Constrained Variational Inequalities
Recommended article: Q4353534
Recommended article: A Regularized Smoothing Newton Method for Box Constrained Variational Inequality Problems with P0-Functions
Recommended article: A nonmonotone smoothing Newton algorithm for solving general box constrained variational inequalities
Recommended article: A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a \(P_0\) function
description: scientific article; zbMATH DE number 1565498