IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i11p1217-d563377.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/11/1217/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/11/1217/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Meruza Kubentayeva & Demyan Yarmoshik & Mikhail Persiianov & Alexey Kroshnin & Ekaterina Kotliarova & Nazarii Tupitsa & Dmitry Pasechnyuk & Alexander Gasnikov & Vladimir Shvetsov & Leonid Baryshev & A, 2024. "Primal-dual gradient methods for searching network equilibria in combined models with nested choice structure and capacity constraints," Computational Management Science, Springer, vol. 21(1), pages 1-33, June.
    2. Chi Xie & Xing Wu & Stephen Boyles, 2019. "Traffic equilibrium with a continuously distributed bound on travel weights: the rise of range anxiety and mental account," Annals of Operations Research, Springer, vol. 273(1), pages 279-310, February.
    3. Maria Mitradjieva & Per Olov Lindberg, 2013. "The Stiff Is Moving---Conjugate Direction Frank-Wolfe Methods with Applications to Traffic Assignment ," Transportation Science, INFORMS, vol. 47(2), pages 280-293, May.
    4. Sang Nguyen & Stefano Pallottino & Federico Malucelli, 2001. "A Modeling Framework for Passenger Assignment on a Transport Network with Timetables," Transportation Science, INFORMS, vol. 35(3), pages 238-249, August.
    5. Benjamin Grimmer, 2023. "General Hölder Smooth Convergence Rates Follow from Specialized Rates Assuming Growth Bounds," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 51-70, April.
    6. Marta Rojo, 2020. "Evaluation of Traffic Assignment Models through Simulation," Sustainability, MDPI, vol. 12(14), pages 1-19, July.
    7. Olaf Jahn & Rolf H. Möhring & Andreas S. Schulz & Nicolás E. Stier-Moses, 2005. "System-Optimal Routing of Traffic Flows with User Constraints in Networks with Congestion," Operations Research, INFORMS, vol. 53(4), pages 600-616, August.
    8. A. Karakitsiou & A. Migdalas, 2016. "Convex optimization problems in supply chain planning and their solution by a column generation method based on the Frank Wolfe method," Operational Research, Springer, vol. 16(3), pages 401-421, October.
    9. Taesung Hwang, 2021. "Assignment of Freight Truck Shipment on the U.S. Highway Network," Sustainability, MDPI, vol. 13(11), pages 1-11, June.
    10. Tan Wang & L. Jeff Hong, 2023. "Large-Scale Inventory Optimization: A Recurrent Neural Networks–Inspired Simulation Approach," INFORMS Journal on Computing, INFORMS, vol. 35(1), pages 196-215, January.
    11. Léon Faure & Bastien Mollet & Wolfram Liebermeister & Jean-Loup Faulon, 2023. "A neural-mechanistic hybrid approach improving the predictive power of genome-scale metabolic models," Nature Communications, Nature, vol. 14(1), pages 1-14, December.
    12. Claudia Quinteros-Cartaya & Guillermo Solorio-Magaña & Francisco Javier Núñez-Cornú & Felipe de Jesús Escalona-Alcázar & Diana Núñez, 2023. "Microearthquakes in the Guadalajara Metropolitan Zone, Mexico: evidence from buried active faults in Tesistán Valley, Zapopan," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 116(3), pages 2797-2818, April.
    13. Furqan Dar & Samuel R. Cohen & Diana M. Mitrea & Aaron H. Phillips & Gergely Nagy & Wellington C. Leite & Christopher B. Stanley & Jeong-Mo Choi & Richard W. Kriwacki & Rohit V. Pappu, 2024. "Biomolecular condensates form spatially inhomogeneous network fluids," Nature Communications, Nature, vol. 15(1), pages 1-17, December.
    14. Guillaume Sagnol & Edouard Pauwels, 2019. "An unexpected connection between Bayes A-optimal designs and the group lasso," Statistical Papers, Springer, vol. 60(2), pages 565-584, April.
    15. Nina Tiel & Fabian Fopp & Philipp Brun & Johan Hoogen & Dirk Nikolaus Karger & Cecilia M. Casadei & Lisha Lyu & Devis Tuia & Niklaus E. Zimmermann & Thomas W. Crowther & Loïc Pellissier, 2024. "Regional uniqueness of tree species composition and response to forest loss and climate change," Nature Communications, Nature, vol. 15(1), pages 1-12, December.
    16. Abdelfettah Laouzai & Rachid Ouafi, 2022. "A prediction model for atmospheric pollution reduction from urban traffic," Environment and Planning B, , vol. 49(2), pages 566-584, February.
    17. López Pérez, Mario & Mansilla Corona, Ricardo, 2022. "Ordinal synchronization and typical states in high-frequency digital markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 598(C).
    18. Chou, Chang-Chi & Chiang, Wen-Chu & Chen, Albert Y., 2022. "Emergency medical response in mass casualty incidents considering the traffic congestions in proximity on-site and hospital delays," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 158(C).
    19. Jessica M. Vanslambrouck & Sean B. Wilson & Ker Sin Tan & Ella Groenewegen & Rajeev Rudraraju & Jessica Neil & Kynan T. Lawlor & Sophia Mah & Michelle Scurr & Sara E. Howden & Kanta Subbarao & Melissa, 2022. "Enhanced metanephric specification to functional proximal tubule enables toxicity screening and infectious disease modelling in kidney organoids," Nature Communications, Nature, vol. 13(1), pages 1-23, December.
    20. André de Palma & Claude Lefèvre, 2018. "Bottleneck models and departure time problems," Working Papers hal-01581519, HAL.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1217-:d:563377. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.