Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 458134, 20 pages
Research Article

Self-Adaptive Implicit Methods for Monotone Variant Variational Inequalities

Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China

Received 26 January 2009; Accepted 24 February 2009

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 β 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 β 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.