Spectral analysis of the logit mapping and implications for stochastic user equilibrium algorithms

TL;DR

This paper analyzes the Jacobian of the logit mapping in stochastic user equilibrium and introduces two improved algorithms that leverage spectral properties for faster convergence, including an adaptive step-size method and a Newton-based approach.

Contribution

It develops novel algorithms for path-based SUE that exploit Jacobian structure, achieving linear and superlinear convergence with practical efficiency on large networks.

Findings

MSA with small step-size converges linearly at rate 1-s

Adaptive step-size rule retains global convergence and achieves asymptotic linear convergence

Newton-based method exploits Jacobian structure, showing superlinear convergence and outperforming existing methods

Abstract

We analyze the Jacobian of the logit mapping for stochastic user equilibrium (SUE) and use it to develop two improved algorithms for path-based SUE. We show that the Jacobian decomposes into two matrices: one that annihilates differences of feasible path flow vectors, and another whose eigenvalues are all non-positive reals, provided link costs are monotone non-decreasing and separable. Using these properties, we first show that the method of successive averages (MSA) with a small constant step-size $s$ converges linearly at a rate $1-s$, with the largest admissible step-size depending on the eigenvalues of the Jacobian of the logit mapping. Building on this result, we develop an adaptive constant step-size rule that retains the global convergence of MSA while achieving asymptotic linear convergence. Our second algorithm is a Newton-based method using a reformulation of SUE as a root-finding problem. Unlike gradient-projection approaches that operate on the Hessian of the SUE objective function (a dense matrix), our method exploits the structure of the Jacobian of the logit mapping, making computations tractable and removing the need for manifold optimization. Numerical experiments show superlinear convergence on most tested networks, with our methods outperforming existing approaches on large networks or when demand is high. To our knowledge, this article is the first to report runtimes for logit-based SUE on networks as large as Chicago Regional and Philadelphia, providing a benchmark for future algorithmic development.