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On circumcentered direct methods for monotone variational inequality problems

Roger Behling, Yunier Bello-Cruz, Alfredo Iusem, Di Liu, Luiz-Rafael Santos

TL;DR

This paper tackles solving variational inequality problems where the feasible set is an intersection of convex sets and the operator is monotone or paramonotone. It introduces two circumcenter-accelerated direct methods that use cheap approximate projections onto separating halfspaces, avoiding expensive exact projections. The authors prove convergence for the paramonotone case and establish ergodic convergence for the monotone case, with closed-form updates when the constraints are sublevel sets. Numerical experiments show dramatic speedups over classical methods (e.g., extragradient) and robust performance even in nonparamonotone settings, highlighting the practical impact of circumcenter acceleration for complex feasible regions.

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.

On circumcentered direct methods for monotone variational inequality problems

TL;DR

This paper tackles solving variational inequality problems where the feasible set is an intersection of convex sets and the operator is monotone or paramonotone. It introduces two circumcenter-accelerated direct methods that use cheap approximate projections onto separating halfspaces, avoiding expensive exact projections. The authors prove convergence for the paramonotone case and establish ergodic convergence for the monotone case, with closed-form updates when the constraints are sublevel sets. Numerical experiments show dramatic speedups over classical methods (e.g., extragradient) and robust performance even in nonparamonotone settings, highlighting the practical impact of circumcenter acceleration for complex feasible regions.

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.

Paper Structure

This paper contains 20 sections, 28 theorems, 95 equations, 3 figures, 12 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Variational inequalities
  3. Gradient-Type methods for variational inequalities
  4. The case of $C=\cap_{i=1}^mC_i$
  5. Circumcentered-Reflection methods
  6. Approximate projections
  7. Our contributions and assumptions
  8. Preliminaries
  9. A circumcenter method for the variational inequality problem with paramonotone operator
  10. The algorithm
  11. Convergence Analysis
  12. Algorithm \ref{['alg.ACRM_paramonotone']} when the $C_i$'s are sublevel sets of convex functions
  13. An explicit circumcenter method for the monotone variational inequality problem
  14. Analysis of the direct method
  15. Numerical experiments
  16. ...and 5 more sections

Key Result

Proposition 2.1

Assume that $K\subset \mathds{R}^n$ is a closed convex set. Then, for any $x\in \mathds{R}^n$ and any $y \in K$ we have the following:

Figures (3)

  • Figure 1: Performance profiles for Example \ref{['Example_1']} (gradient operator).
  • Figure 2: Performance profiles for Example \ref{['Example_2']} (paramonotone, non-gradient operator).
  • Figure 3: Performance profiles for Example \ref{['Example_3']} (monotone, non-paramonotone operator).

Theorems & Definitions (68)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 58 more