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The mathematical foundations of dynamic user equilibrium

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Title: The mathematical foundations of dynamic user equilibrium
Authors: Friesz, TL
Han, K
Item 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.
Issue Date: Aug-2019
Date of Acceptance: 24-Aug-2018
URI: http://hdl.handle.net/10044/1/63887
DOI: https://doi.org/10.1016/j.trb.2018.08.015
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/
Keywords: Social Sciences
Science & Technology
Technology
Economics
Engineering, Civil
Operations Research & Management Science
Transportation
Transportation Science & Technology
Business & Economics
Engineering
Dynamic user equilibrium
Differential variational inequality
Optimal control
Fixed point algorithm
VARIATIONAL INEQUALITY FORMULATION
DIFFERENTIAL-EQUATION FORMULATION
VICKREYS BOTTLENECK MODEL
SIMULTANEOUS ROUTE
EXISTENCE
COMPLEMENTARITY
ALLOCATION
SYSTEMS
DEMAND
dynamic user equilibrium
differential variational inequality
optimal control
fixed point algorithm
Logistics & Transportation
1507 Transportation and Freight Services
0102 Applied Mathematics
Publication Status: Published
Article Number: TRB 2050
Online Publication Date: 2018-09-07
Appears in Collections:Faculty of Engineering
Civil and Environmental Engineering