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Optimization formulations and static equilibrium in congested transportation networks

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  • A. de Palma
  • Y. Nesterov

Abstract

In this paper we study the concepts of equilibrium and optimum in static transportation networks with elastic and non-elastic demands. The main mathematical tool of our paper is the theory of variational inequalities. We demonstrate that this theory is useful for proving the existence theorems. It also can justify Beckmann's formulation of the equilibrium problem.The main contribution of this paper is to propose a new definition of equilibrium, the normal equilibrium, which exists under very general assumptions. This concept can be used, in particular, when the travel costs are discontinuous and unbounded. As examples we consider the models of signalized intersections, traffic lights and unbounded travel-time relationships. For some of those cases, the standard concepts of user and Wardrop equilibria cannot be used.
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Suggested Citation

  • A. de Palma & Y. Nesterov, 1997. "Optimization formulations and static equilibrium in congested transportation networks," THEMA Working Papers 97-17, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    • Handle: RePEc:ema:worpap:97-17
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    References listed on IDEAS

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    1. Stella Dafermos, 1980. "Traffic Equilibrium and Variational Inequalities," Transportation Science, INFORMS, vol. 14(1), pages 42-54, February.
    2. Smith, M. J., 1979. "The existence, uniqueness and stability of traffic equilibria," Transportation Research Part B: Methodological, Elsevier, vol. 13(4), pages 295-304, December.
    3. David Bernstein & Tony E. Smith, 1994. "Equilibria for Networks with Lower Semicontinuous Costs: With an Application to Congestion Pricing," Transportation Science, INFORMS, vol. 28(3), pages 221-235, August.
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    Cited by:

    1. Erik T. Verhoef, 2000. "Second-Best Congestion Pricing in General Networks - Algorithms for Finding Second-Best Optimal Toll Levels and Toll Points," Tinbergen Institute Discussion Papers 00-084/3, Tinbergen Institute.
    2. José R. Correa & Andreas S. Schulz & Nicolás E. Stier-Moses, 2004. "Selfish Routing in Capacitated Networks," Mathematics of Operations Research, INFORMS, vol. 29(4), pages 961-976, November.
    3. Verhoef, Erik T., 2002. "Second-best congestion pricing in general static transportation networks with elastic demands," Regional Science and Urban Economics, Elsevier, vol. 32(3), pages 281-310, May.
    4. Chai, Huajun, 2019. "Dynamic Traffic Routing and Adaptive Signal Control in a Connected Vehicles Environment," Institute of Transportation Studies, Working Paper Series qt9ng3z8vn, Institute of Transportation Studies, UC Davis.
    5. Verhoef, Erik Teodoor, 2000. "The Generalized Second-Best Network Congestion Pricing Problem," ERSA conference papers ersa00p336, European Regional Science Association.
    6. repec:wsi:jeapmx:v:20:y:2018:i:04:n:s0219198918500068 is not listed on IDEAS
    7. A. de Palma & Y. Nesterova, 2000. "Stable Dynamics in Transportation Systems," THEMA Working Papers 2000-18, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    8. Correa, José R. & Schulz, Andreas S. & Stier-Moses, Nicolás E., 2008. "A geometric approach to the price of anarchy in nonatomic congestion games," Games and Economic Behavior, Elsevier, vol. 64(2), pages 457-469, November.
    9. Verhoef, Erik T., 2002. "Second-best congestion pricing in general networks. Heuristic algorithms for finding second-best optimal toll levels and toll points," Transportation Research Part B: Methodological, Elsevier, vol. 36(8), pages 707-729, September.
    10. Mahdi Takalloo & Changhyun Kwon, 2019. "On the Price of Satisficing in Network User Equilibria," Papers 1911.07914, arXiv.org.
    11. Fernando Ordóñez & Nicolás E. Stier-Moses, 2010. "Wardrop Equilibria with Risk-Averse Users," Transportation Science, INFORMS, vol. 44(1), pages 63-86, February.
    12. Bliemer, Michiel C.J. & Raadsen, Mark P.H. & Smits, Erik-Sander & Zhou, Bojian & Bell, Michael G.H., 2014. "Quasi-dynamic traffic assignment with residual point queues incorporating a first order node model," Transportation Research Part B: Methodological, Elsevier, vol. 68(C), pages 363-384.
    13. Correa, Jose R. & Schulz, Andreas S. & Stier Moses, Nicolas E., 2003. "Selfish Routing in Capacitated Networks," Working papers 4319-03, Massachusetts Institute of Technology (MIT), Sloan School of Management.

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