The Forward-Backward-Forward Method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces

TL;DR

This paper extends Tseng's forward-backward-forward method to pseudo-monotone variational inequalities in Hilbert spaces, providing convergence proofs and demonstrating linear convergence under strong pseudo-monotonicity, supported by numerical experiments.

Contribution

It introduces a new convergence analysis for the forward-backward-forward algorithm applied to pseudo-monotone variational inequalities, including a dynamical systems perspective and linear convergence results.

Findings

Algorithm converges for pseudo-monotone variational inequalities.

Linear convergence under strong pseudo-monotonicity.

Numerical experiments confirm theoretical results.

Abstract

Tseng's forward-backward-forward algorithm is a valuable alternative for Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed by monotone and Lipschitz continuous operators, as it requires in every step only one projection operation. However, it is well-known that Korpelevich's method converges and can therefore be used also for solving variational inequalities governed by pseudo-monotone and Lipschitz continuous operators. In this paper, we first associate to a pseudo-monotone variational inequality a forward-backward-forward dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into Tseng's forward-backward-forward algorithm with relaxation parameters, which we prove to converge also when it is applied to pseudo-monotone variational inequalities. In addition, we show that linear convergence is guaranteed under strong pseudo-monotonicity. Numerical experiments are carried out for pseudo-monotone variational inequalities over polyhedral sets and fractional programming problems.