Strong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks

TL;DR

This paper introduces a new strongly convergent primal-dual algorithm for solving pseudomonotone variational inequalities, with applications to dynamic user equilibrium in traffic networks, improving convergence guarantees and practical performance.

Contribution

The authors develop a simple, computationally efficient augmentation of Tseng's forward-backward-forward scheme that guarantees strong convergence without requiring global monotonicity or knowledge of the Lipschitz constant.

Findings

The proposed algorithm guarantees strong convergence to a minimal norm solution.

It performs competitively and sometimes better than state-of-the-art solvers in traffic equilibrium problems.

The method is adaptable, removing the need for prior knowledge of the Lipschitz constant.

Abstract

In infinite-dimensional Hilbert spaces we device a class of strongly convergent primal-dual schemes for solving variational inequalities defined by a Lipschitz continuous and pseudomonote map. Our novel numerical scheme is based on Tseng's forward-backward-forward scheme, which is known to display weak convergence, unless very strong global monotonicity assumptions are made on the involved operators. We provide a simple augmentation of this algorithm which is computationally cheap and still guarantees strong convergence to a minimal norm solution of the underlying problem. We provide an adaptive extension of the algorithm, freeing us from requiring knowledge of the global Lipschitz constant. We test the performance of the algorithm in the computationally challenging task to find dynamic user equilibria in traffic networks and verify that our scheme is at least competitive to state-of-the-art solvers, and in some case even improve upon them.