On circumcentered direct methods for monotone variational inequality problems

TL;DR

This paper introduces two circumcentered direct methods for monotone variational inequalities that accelerate convergence by using approximate projections, outperforming classical algorithms like extragradient in complex scenarios.

Contribution

The paper proposes novel circumcentered direct methods for variational inequalities involving paramonotone and monotone operators, with convergence proofs and superior empirical performance.

Findings

Methods outperform classical algorithms in computational time.

Algorithms are effective for complex intersections of convex sets.

Convergence is established for both proposed schemes.

Abstract

Circumcentered techniques have been shown to significantly accelerate projection-based methods for convex feasibility problems. Motivated by this success, we propose two direct methods with circumcenter acceleration for solving variational inequality problems involving two classes of operators: paramonotone and monotone. Both schemes rely on approximate projections onto separating halfspaces, thereby avoiding computationally expensive exact projections. We establish convergence results for both methods and conduct numerical experiments, demonstrating that the proposed algorithms outperform classical methods, such as the extragradient algorithm, by orders of magnitude in terms of computational time, particularly when the feasible set is a complex intersection of convex sets.