TY - JOUR

T1 - A Smoothing Newton Method for Extended Vertical Linear Complementarity Problems

AU - Qi, Hou Duo

AU - Liao, Li Zhi

N1 - Funding information:
t Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, P.O. Box 2719, Beijing, China (qhd@lsec.cc.ac.cn). The work of this author was supported in part by Hong Kong Baptist University grant FRG/96-97/II-105.
*Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong (liliao@ hkbu.edu.hk). The work of this author was supported in part by Hong Kong Baptist University grant FRG/96-97/II-105.
Publisher copyright:
Copyright © 1999 Society for Industrial and Applied Mathematics

PY - 1999/8/3

Y1 - 1999/8/3

N2 - In this paper, we reformulate the extended vertical linear complementarity problem (EVLCP(m, q)) as a nonsmooth equation H(t, x) = 0, where H : ℝn+l → ℝn+1, t ∈ ℝ is a parameter variable, and cursive Greek chi ∈ ℝ is the original variable. H is continuously differeritiable except at such points (t, x) with t = 0. Furthermore H is strongly semismooth. The reformulation of EVLCP(m, q) as a nonsmooth equation is based on the so-called aggregation (smoothing) function. As a result, a Newton-type method is proposed which generates a sequence {wk = (tk, xk)} with all tk > 0. We prove that every accumulation point of this sequence is a solution of EVLCP(M, q) under the assumption of row W0-property. If row W-property holds at the solution point, then the convergence rate is quadratic. Promising numerical results are also presented.

AB - In this paper, we reformulate the extended vertical linear complementarity problem (EVLCP(m, q)) as a nonsmooth equation H(t, x) = 0, where H : ℝn+l → ℝn+1, t ∈ ℝ is a parameter variable, and cursive Greek chi ∈ ℝ is the original variable. H is continuously differeritiable except at such points (t, x) with t = 0. Furthermore H is strongly semismooth. The reformulation of EVLCP(m, q) as a nonsmooth equation is based on the so-called aggregation (smoothing) function. As a result, a Newton-type method is proposed which generates a sequence {wk = (tk, xk)} with all tk > 0. We prove that every accumulation point of this sequence is a solution of EVLCP(M, q) under the assumption of row W0-property. If row W-property holds at the solution point, then the convergence rate is quadratic. Promising numerical results are also presented.

KW - Aggregation function

KW - Global convergence

KW - Semismoothness

KW - Smoothing Newton method

UR - http://www.scopus.com/inward/record.url?scp=0033630888&partnerID=8YFLogxK

U2 - 10.1137/S0895479897329837

DO - 10.1137/S0895479897329837

M3 - Article

AN - SCOPUS:0033630888

VL - 21

SP - 45

EP - 66

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 1

ER -