Inexact implicit methods for monotone general variational inequalities

Abstract.

Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear smooth (whenever the operator is smooth) equations. 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 $Q\big(u^*\big) \in \Omega, \qquad \bigl( v - Q\big(u^*\big) \bigr)^T F\big(u^*\big) \ge 0, \qquad \forall v\in \Omega.$ Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to preserve the same convergence properties as the original implicit method.