Elastic demand dynamic network user equilibrium: Formulation, existence and computation

TL;DR

This paper formulates a dynamic user equilibrium model with elastic demand as an infinite-dimensional variational inequality, proves its existence, and develops three convergent algorithms with broad applicability.

Contribution

It introduces a novel VI formulation for E-DUE with elastic demand, proves its existence, and proposes three new algorithms with rigorous convergence analysis.

Findings

Algorithms demonstrate good convergence and solution quality.

Proposed methods outperform existing approaches in efficiency.

Convergence results apply to a broad class of infinite-dimensional VIs.

Abstract

This paper is concerned with dynamic user equilibrium with elastic travel demand (E-DUE) when the trip demand matrix is determined endogenously. We present an infinite-dimensional variational inequality (VI) formulation that is equivalent to the conditions defining a continuous-time E-DUE problem. An existence result for this VI is established by applying a fixed-point existence theorem (Browder, 1968) in an extended Hilbert space. We present three algorithms based on the aforementioned VI and its re-expression as a differential variational inequality (DVI): a projection method, a self-adaptive projection method, and a proximal point method. Rigorous convergence results are provided for these methods, which rely on increasingly relaxed notions of generalized monotonicity, namely mixed strongly-weakly monotonicity for the projection method; pseudomonotonicity for the self-adaptive projection method, and quasimonotonicity for the proximal point method. These three algorithms are tested and their solution quality, convergence, and computational efficiency compared. Our convergence results, which transcend the transportation applications studied here, apply to a broad family of infinite-dimensional VIs and DVIs, and are the weakest reported to date.