Inexact implicit methods for monotone general variational inequalities

Solving a variational inequality problem is equivalent to finding a solutio n of a system of nonsmooth equations. Recently, we proposed an implicit met hod, which solves monotone:variational inequality problem via solving a ser ies of systems of nonlinear smooth (whenever the operator is smooth) equati ons. It can exploit the facilities of the classical Newton-like methods for smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems Q(u*) is an element of Omega, (upsilon - Q(u*))(T) F(u*) greater than or eq ual to 0, For All upsilon is an element of Omega. Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to pr eserve the same convergence properties as the original implicit method.