A Nesterov's Accelerated Projected Gradient Method for Monotone Variational Inequalities

TL;DR

This paper introduces a Nesterov's accelerated projected gradient method for variational inequalities, achieving faster convergence rates and demonstrating superior performance over existing methods through simulations.

Contribution

It is the first to prove convergence of Nesterov's accelerated method for variational inequalities, extending its application beyond convex optimization.

Findings

Achieves at least linear convergence rate under common assumptions.

Demonstrates significant reduction in iteration count compared to existing methods.

Shows superior performance in simulation results.

Abstract

In this technical note, we are concerned with the problem of solving variational inequalities with improved convergence rates. Motivated by Nesterov's accelerated gradient method for convex optimization, we propose a Nesterov's accelerated projected gradient algorithm for variational inequality problems. We prove convergence of the proposed algorithm with at least linear rate under the common assumption of Lipschitz continuity and strongly monotonicity. To the best of our knowledge, this is the first time that convergence of the Nesterov's accelerated protocol is proved for variational inequalities, other than the convex optimization or monotone inclusion problems. Simulation results are given to demonstrate the outperformance of the proposed algorithms over the well-known projected gradient approach, the reflected projected approach, and the golden ratio method. It is shown that the required number of iterations to reach the solution is greatly reduced in our proposed algorithm.