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The mathematical foundations of dynamic user equilibrium

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  • Friesz, Terry L.
  • Han, Ke

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.

Suggested Citation

  • Friesz, Terry L. & Han, Ke, 2019. "The mathematical foundations of dynamic user equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 126(C), pages 309-328.
    • Handle: RePEc:eee:transb:v:126:y:2019:i:c:p:309-328
    DOI: 10.1016/j.trb.2018.08.015
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    Cited by:

    1. Frascaria, Dario & Olver, Neil & Verhoef, Erik, 2020. "Emergent hypercongestion in Vickrey bottleneck networks," Transportation Research Part B: Methodological, Elsevier, vol. 139(C), pages 523-538.
    2. Yuncheng Zeng & Minhua Shao & Lijun Sun, 2023. "Network-Level Hierarchical Bottleneck Congestion Control Method for a Mixed Traffic Network," Sustainability, MDPI, vol. 15(23), pages 1-27, November.
    3. František Kolovský & Ivana Kolingerová, 2022. "The Piecewise Constant/Linear Solution for Dynamic User Equilibrium," Networks and Spatial Economics, Springer, vol. 22(4), pages 737-765, December.
    4. Friesz, Terry L. & Han, Ke & Bagherzadeh, Amir, 2021. "Convergence of fixed-point algorithms for elastic demand dynamic user equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 150(C), pages 336-352.
    5. Huang, Y.P. & Xiong, J.H. & Sumalee, A. & Zheng, N. & Lam, W.H.K. & He, Z.B. & Zhong, R.X., 2020. "A dynamic user equilibrium model for multi-region macroscopic fundamental diagram systems with time-varying delays," Transportation Research Part B: Methodological, Elsevier, vol. 131(C), pages 1-25.
    6. Mads Paulsen & Thomas Kjær Rasmussen & Otto Anker Nielsen, 2022. "Including Right-of-Way in a Joint Large-Scale Agent-Based Dynamic Traffic Assignment Model for Cars and Bicycles," Networks and Spatial Economics, Springer, vol. 22(4), pages 915-957, December.

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