Self-adaptive operator splitting methods for monotone variational inequalities

Bingsheng He*, Lizhi LIAO, Shengli Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

36 Citations (Scopus)

Abstract

Solving a variational inequality problem VI(Ω, F) is equivalent to finding a solution of a system of nonsmooth equations (a hard problem). The Peaceman-Rachford and /or Douglas-Rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems, because they solve the original problem via solving a series of systems of nonlinear smooth equations (a series of easy problems). Although the solution of VI(Ω, F) is invariant under multiplying F by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is a constant in the original operator splitting methods. In general, it is difficult to choose a proper parameter β for individual problems. In this paper, we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates. Exact and inexact forms of the modified method with self-adaptive variable parameter are suggested and proved to be convergent under mild assumptions. Finally, preliminary numerical tests show that the self-adaptive adjustment rule is proper and necessary in practice.

Original languageEnglish
Pages (from-to)715-737
Number of pages23
JournalNumerische Mathematik
Volume94
Issue number4
DOIs
Publication statusPublished - Jun 2003

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

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