Modified Projection-Type Methods for Monotone Variational Inequalities

Abstract

We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projection-type methods in which the projection direction is modified by a strongly monotone mapping of the form $I - \alpha F$ or, if F is affine with underlying matrix M, of the form $I + \alpha M^T $, with $\alpha \in (0,\infty )$. We show that these methods are globally convergent, and if in addition a certain error bound based on the natural residual holds locally, the convergence is linear. Computational experience with the new methods is also reported.