Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China
Academic Editor: Ram U. Verma
Copyright © 2009 Zhili Ge and Deren Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The efficiency of the implicit
method proposed by He (1999) depends on the parameter heavily; while it varies for individual problem, that is, different problem has
different “suitable” parameter, which is difficult to find. In this
paper, we present a modified implicit method, which adjusts the
parameter automatically per iteration, based on the
message from former iterates. To improve the performance of the algorithm,
an inexact version is proposed, where the subproblem is just solved approximately.
Under mild conditions as those for variational inequalities, we prove the
global convergence of both exact and inexact versions of the new method. We
also present several preliminary numerical results, which demonstrate that the self-adaptive
implicit method, especially the inexact version, is efficient and robust.