The mathematical foundations of dynamic user equilibrium
File(s)
Accepted version
Author(s)
Friesz, TL
Han, K
Type
Journal Article
Abstract
This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples.
Date Issued
2019-08
Date Acceptance
2018-08-24
Citation
Transportation Research Part B: Methodological, 2019, 126, pp.309-328
ISSN
0191-2615
Publisher
Elsevier
Start Page
309
End Page
328
Journal / Book Title
Transportation Research Part B: Methodological
Volume
126
Copyright Statement
© 2018 Elsevier Ltd. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence http://creativecommons.org/licenses/by-nc-nd/4.0/
Identifier
https://www.sciencedirect.com/science/article/pii/S0191261517301960?via=ihub
Subjects
dynamic user equilibrium
differential variational inequality
optimal control
fixed point algorithm
Publication Status
Published
Article Number
TRB 2050
Date Publish Online
2018-09-07