Projected Reflected Gradient Methods for Monotone Variational Inequalities

TL;DR

This paper introduces new projection algorithms for monotone variational inequalities that require fewer projections per iteration, achieve linear convergence under strong monotonicity, and are supported by promising numerical results.

Contribution

The paper proposes the projected reflected gradient method with constant stepsize requiring only one projection, and a modified version needing at most two, both with proven convergence and efficiency.

Findings

Methods are globally convergent to solutions.

Algorithms require fewer projections per iteration.

Preliminary numerical results are promising.

Abstract

This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only one value of the mapping per iteration. This distinguishes our method from most other projection-type methods for variational inequalities with monotone mapping. Also we prove that it has R-linear rate of convergence under the strong monotonicity assumption. The usual drawback of algorithms with constant stepsize is the requirement to know the Lipschitz constant of the mapping. To avoid this, we modify our first algorithm so that the algorithm needs at most two projections per iteration. In fact, our computational experience shows that such cases with two projections are very rare. This scheme, at least theoretically, seems to be very effective. All methods are shown to be globally convergent to a solution of the variational inequality. Preliminary results from numerical experiments are quite promising.