Finding Equilibria in the Traffic Assignment Problem with Primal-Dual Gradient Methods for Stable Dynamics Model and Beckmann Model

TL;DR

This paper explores gradient-based methods for finding traffic equilibria in the stable dynamics and Beckmann models, focusing on dual problem solutions and introducing a new flow reconstruction technique.

Contribution

It introduces primal-dual gradient methods for traffic assignment, providing complexity estimates and a novel flow reconstruction approach for the stable dynamics model.

Findings

Gradient methods effectively solve dual problems in traffic models.

A new admissible flow reconstruction method for the stable dynamics model.

Complexity bounds for the proposed gradient algorithms.

Abstract

In this paper we consider the application of several gradient methods to the traffic assignment problem: we search equilibria in the stable dynamics model (Nesterov and De Palma, 2003) and the Beckmann model. Unlike the celebrated Frank--Wolfe algorithm widely used for the Beckmann model, these gradients methods solve the dual problem and then reconstruct a solution to the primal one. We deal with the universal gradient method, the universal method of similar triangles, and the method of weighted dual averages, and estimate their complexity for the problem. Due to the primal-dual nature of these methods, we use a duality gap in a stopping criterion. In particular, we present a novel way to reconstruct admissible flows (i.e.,\ meeting the capacity constraints and induced by flows on the paths) in the stable dynamics model, which provides us with a computable duality gap.