A smoothing Newton method for extended vertical linear complementarity problems

Hou Duo Qi*, Lizhi LIAO

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)


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, cursive Greek chi) 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,cursive Greek chik)} 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.

Original languageEnglish
Pages (from-to)45-66
Number of pages22
JournalSIAM Journal on Matrix Analysis and Applications
Issue number1
Publication statusPublished - 1999

Scopus Subject Areas

  • Analysis

User-Defined Keywords

  • Aggregation function
  • Global convergence
  • Semismoothness
  • Smoothing Newton method


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