Preprint C95/2009
Direct Methods for Monotone Variational Inequalities

JosÃ© Yunier Bello Cruz

**Keywords: **
Armijo-type search | Convex minimization problem | Korpelevich's me- thod | Nonsmooth optimization | Maximal monotone operators | monotone variational inequalities | Point-to-set operator | Projected gradient method | Projection method | quasi-Fejer convergence |

We analyze one-step direct methods for nonsmooth variational inequality problems, establishing convergence under paramonotonicity of the operator. Previous results on
the method required much more demanding assumptions, like strong or uniform monotonicity, which imply uniqueness of solution, which is not the case for our approach.
We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by
projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under the only assumptions of continuity
and monotonicity of the operator and existence of solutions.
We also introduce a two-step direct method, like Korpelevich's. The advantage of our method over that one is that ours converges strongly in Hilbert spaces, while only
weak convergence has been proved for Korpelevich's algorithm. Our method also has the following desirable property: the sequence converges to the solution of the problem
which lies closest to the initial iterate.