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Finding Equilibria in the Traffic Assignment Problem with Primal-Dual Gradient Methods for Stable Dynamics Model and Beckmann Model

Author

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  • Meruza Kubentayeva

    (Institute for Information Transmission Problems, RAS, Bolshoy Karetny Per. 19, Build.1, 127051 Moscow, Russia)

  • Alexander Gasnikov

    (Institute for Information Transmission Problems, RAS, Bolshoy Karetny Per. 19, Build.1, 127051 Moscow, Russia
    Moscow Institute of Physics and Technology, 9 Institutskiy Per., Dolgoprudny, 141701 Moscow Region, Russia
    Higher School of Economics, 20 Myasnitskaya Ulitsa, 101000 Moscow, Russia)

Abstract

In this paper, we consider the application of several gradient methods to the traffic assignment problem: we search equilibria in the stable dynamics model (Nesterov and De Palma, 2003) and the Beckmann model. Unlike the celebrated Frank–Wolfe algorithm widely used for the Beckmann model, these gradients methods solve the dual problem and then reconstruct a solution to the primal one. We deal with the universal gradient method, the universal method of similar triangles, and the method of weighted dual averages and estimate their complexity for the problem. Due to the primal-dual nature of these methods, we use a duality gap in a stopping criterion. In particular, we present a novel way to reconstruct admissible flows in the stable dynamics model, which provides us with a computable duality gap.

Suggested Citation

  • Meruza Kubentayeva & Alexander Gasnikov, 2021. "Finding Equilibria in the Traffic Assignment Problem with Primal-Dual Gradient Methods for Stable Dynamics Model and Beckmann Model," Mathematics, MDPI, vol. 9(11), pages 1-17, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1217-:d:563377
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    References listed on IDEAS

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    1. A. de Palma & Y. Nesterov, 2001. "Stationary Dynamic Solutions in Congested Transportation Networks: Summary and Perspectives," THEMA Working Papers 2001-19, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    2. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    3. NESTEROV, Yurii, 2015. "Universal gradient methods for convex optimization problems," LIDAM Reprints CORE 2701, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Charles R. Harris & K. Jarrod Millman & Stéfan J. Walt & Ralf Gommers & Pauli Virtanen & David Cournapeau & Eric Wieser & Julian Taylor & Sebastian Berg & Nathaniel J. Smith & Robert Kern & Matti Picu, 2020. "Array programming with NumPy," Nature, Nature, vol. 585(7825), pages 357-362, September.
    5. Larry J. LeBlanc & Richard V. Helgason & David E. Boyce, 1985. "Improved Efficiency of the Frank-Wolfe Algorithm for Convex Network Programs," Transportation Science, INFORMS, vol. 19(4), pages 445-462, November.
    6. Y. Arezki & D. Van Vliet, 1990. "A Full Analytical Implementation of the PARTAN/Frank–Wolfe Algorithm for Equilibrium Assignment," Transportation Science, INFORMS, vol. 24(1), pages 58-62, February.
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    Cited by:

    1. Evgenia Gasnikova & Alexander Gasnikov & Yaroslav Kholodov & Anastasiya Zukhba, 2023. "An Evolutionary View on Equilibrium Models of Transport Flows," Mathematics, MDPI, vol. 11(4), pages 1-8, February.

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