Self-adaptive implicit methods for monotone variant variational inequalities

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.