English version:
Journal of Applied and Industrial Mathematics, 2016, 10:2, 243-256

Volume 23, No 2, 2016, P. 63-87

UDC 519.8
A. Yu. Krylatov
Network flow assignment as a fixed point problem

This paper deals with the user equilibrium problem (flow assignment with equal journey time by alternative routes) and system optimum (flow assignment with minimal average journey time) in a network consisting of parallel routes with a single origin-destination pair. The travel time is simulated by arbitrary smooth nondecreasing functions. We prove that the equilibrium and optimal assignment problems for such a network can be reduced to the fixed point problem expressed explicitly. A simple iterative method of finding equilibrium and optimal flow assignment is developed. The method is proved to converge geometrically; under some fairly natural conditions the method is proved to converge quadratically.
Bibliogr. 30.

Keywords: user-equilibrium, system optimum, fixed point, network routes.

DOI: 10.17377/daio.2016.23.503

Alexander Yu. Krylatov 1,2
1. Saint Petersburg State University,
7/9 Universitetskaya Nab., 199034 St. Petersburg, Russia
2. Solomenko Institute of Transport Problems of the RAS,
13 12th Line VO, 199178 St. Petersburg, Russia
e-mail: a.krylatov@spbu.ru, aykrylatov@yandex.ru

Received 31 July 2015
Revised 24 November 2015


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