The Popov's Algorithm with Optimal Bounded Stepsize for Generalized Monotone Variational Inequalities

TL;DR

This paper improves the understanding of stepsize bounds for Popov's algorithm in solving generalized monotone variational inequalities, establishing tight bounds and introducing a new Lyapunov function for convergence analysis.

Contribution

It proves tight bounds for stepsizes in Popov's algorithm for constrained and unconstrained cases and introduces a novel Lyapunov function for convergence analysis.

Findings

Upper bound of 1/(2L) is tight for constrained problems.

Upper bound of 1/(rac{{3}L) is tight for unconstrained problems.

New Lyapunov-type function facilitates convergence analysis.

Abstract

For solving constrained (pseudo)-monotone variational inequality, we prove that the upper bound of stepsize $\frac{1}{2L}$ established for the Popov's algorithm and the forward-reflected-backward algorithm is tight. For unconstrained case, we can enlarge the upper bound to $\frac{1}{\sqrt{3}L}$ and show that this upper bound is also tight. The convergence analysis is carried out by using a new Lyapunov-type function.