TY - JOUR

T1 - Self-adaptive operator splitting methods for monotone variational inequalities

AU - He, Bingsheng

AU - LIAO, Lizhi

AU - Wang, Shengli

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2003/6

Y1 - 2003/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0037660844&partnerID=8YFLogxK

U2 - 10.1007/s00211-002-0408-y

DO - 10.1007/s00211-002-0408-y

M3 - Journal article

AN - SCOPUS:0037660844

SN - 0029-599X

VL - 94

SP - 715

EP - 737

JO - Numerische Mathematik

JF - Numerische Mathematik

IS - 4

ER -