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, Volume 4, Issue 2, pp 225–256

A simplicial decomposition algorithm for solving the variational inequality formulation of the general traffic assignment problem for large scale networks

  • L. Montero
  • J. Barceló
Article

Summary

The class of simplicial decomposition methods has been shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. This paper presents a particular implementation of such an algorithm, with emphasis on its ability to solve large scale problems efficiently.

The convergence of the algorithm is monitored by the primal gap function, which arises naturally in simplicial decomposition schemes. The gap function also serves as an instrument for maintaining a reasonable subproblem size, through its use in column dropping criteria. The small dimension and special structure of the subproblems also allows for the use of very efficient algorithms; several algorithms in the class of linearization methods are presented.

When restricting the number of retained extremal flows in a simplicial decomposition scheme, the number of major iterations tends to increase. For large networks the shortest path calculations, leading to new extremal flow generation, require a large amount of the total computation time. A special study is therefore made in order to choose the most efficient extremal flow generation technique.

Computational results on symmetric problems are presented for networks of some large cities, and on asymmetric problems for some of the networks used in the literature. Computational results for bimodal models of some large cities leading to asymmetric problems are also discussed.

Keywords

Traffic Equilibria Variational Inequalities Simplicial Decomposition Methods Projection Methods Quadratic Programming 

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References

  1. Aashtiani, H.Z., and T.L. Magnanti (1981).Equilibria on a congested transportation network. SIAM J. Algebraic and Discrete Methods2, 213–226.Google Scholar
  2. Beckmann, M., C.B. McGuire and C.B. Winsten (1956).Studies in the Economics of Transportation. Yale University Press, New Haven CT.Google Scholar
  3. Bertsekas, D.P. and E.M. Gafni (1982).Projection methods for variational inequalities with application to the traffic assignment problem. Mathematical Programming Study17, 139–159.Google Scholar
  4. Bureau of Public Roads (1964).Traffic Assignment Manual. U.S. Department of Commerce, Urban Planning Division, Washington, DC.Google Scholar
  5. Dafermos, S.C. (1980).Traffic assignment and variational inequalities. Transportation Science14 (1), 42–54.Google Scholar
  6. Dafermos, S.C. (1982).Relaxation algorithms for the general asymmetric traffic equilibrium problem. Transportation Science16 (2), 231–240.Google Scholar
  7. Dafermos, S.C. (1983).An iterative scheme for variational inequalities. Mathematical Programming26 (1), 40–47.Google Scholar
  8. Deo, N. and C. Pang (1984).Shortest-path algorithms: taxonomy and annotation. Networks Vol14, 275–323.Google Scholar
  9. Fisk, C. and S. Nguyen (1982).Solution algorithms for network equilibrium models with asymmetric user costs. Transp. Sci.16 (3), 361–381.Google Scholar
  10. Florian, M. and H. Spiess (1982).The convergence of diagonalization algorithms for asymmetric network equilibrium problems. Trans. Res.16B (6), 447–483.Google Scholar
  11. Florian, M. (1986).Nonlinear Cost Network Models in Transportation Analysis. Mathematical Programming Study26, 167–196.Google Scholar
  12. Frank, M. and P. Wolfe (1956).An algorithm for quadratic programming. Naval Research Logistics Quarterly3, 95–110.Google Scholar
  13. Gallo, G. and S. Pallotino (1984).Shortest Path Methods in Transportation Models. In Transportation Planning Models (Edited by M. Florian), 227–256. Elsevier, New York.Google Scholar
  14. Gill, P.E., W. Murray and M. Wright (1981). Practical Optimization. Academic Press.Google Scholar
  15. Harker, P. and J. Pang (1990).Finite-dimensional variational inequality and linear complementary problems: a survey of theory, algorithms and applications. Mathematical Programming48, 161–220.CrossRefGoogle Scholar
  16. Hearn, D.W. (1982).The gap function of a convex program. Oper. Res. Lett.1, 67–71.CrossRefGoogle Scholar
  17. Hohenbalken, B. (1977).Simplicial Decomposition in Nonlinear Programming algorithms. Mathematical Programming13, 49–68.CrossRefGoogle Scholar
  18. Holloway, C.A. (1974).An extension of the Frank and Wolfe method of feasible directions. Mathematical Programming,6, 14–27.CrossRefGoogle Scholar
  19. Kinderlehrer, D. and G. Stampacchia (1980).An Introduction to Variational Inequalities and Their Applications. Academic, New York.Google Scholar
  20. Knuth, D.E. (1973).The Art of Computer Programming. Addison-Wesley.Google Scholar
  21. Larsson, T. and M. Patriksson (1992).A dual scheme for traffic assignment problems. Transportation Science26, 4–17.CrossRefGoogle Scholar
  22. Lawphongpanich, S. and D.W. Hearn (1984).Simplicial decomposition of the asymmetric traffic assignment problem. Transp. Res18B, 123–133.CrossRefGoogle Scholar
  23. Luenberger, D.G. (1974).Introduction to Linear and Nonlinear Programming. Addison-Wesley, Reading, MA.Google Scholar
  24. Magnanti, T.L. (1984).Models and Algorithms for Predicting Urban Traffic Equilibria. In Transportation Planning Models (Edited by M. Florian), 153–186, Elsevier, New York.Google Scholar
  25. Marcotte, P. and J.P. Dussault (1987).A note on a globally convergent Newton method for solving monotone variational inequalities. Operations Research Letters6, No 1, 35–42.CrossRefGoogle Scholar
  26. Marcotte P. and J. Guélat (1988).Adaptation of a modified Newton method for solving the asymmetric traffic equilibrium problem. Transportation Science,22, 112–124.Google Scholar
  27. Montero L. (1992).A Simplicial Decomposition Approach for Solving the Variational Inequality Formulation of the General Traffic Assignment Problem for Large Scale Networks. Ph.D. Thesis supervised by Professor Jaume Barceló, Politechnical University of Catalunya in Barcelona (Spain).Google Scholar
  28. Nguyen, S. and C. Depuis (1984).An efficient method for computing traffic equilibria in network with asymmetric transportation costs. Transp.Google Scholar
  29. Pang, J.S. and D. Chan (1982).Iterative Methods for variational and complementary problems. Mathematical Programming24 (3), 284–313.CrossRefGoogle Scholar
  30. Pang, J.S. and C.S. Yu (1984).Linearized simplicial decomposition methods for computing traffic equilibria on networks. Networks14 (3), 427–438.Google Scholar
  31. Patriksson, M. (1990).The traffic assignment problem. Theory and Algorithms Report LiTH-MAT-R-90-29, Department of Mathematics, Institute of Technology, Linköping University, Sweden.Google Scholar
  32. Sheffi, Y. (1985). Urban transportation networks. Equilibrium analysis with mathematical methods. Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  33. Smith, M.J. (1979b).Existence, uniqueness and stability of traffic equilibria. Transp. Res.13B (4), 295–304.CrossRefGoogle Scholar
  34. Smith, M.J. (1981a).The existence of an equilibrium solution, to the traffic assignment problem when there are junction interactions. Transportation Research B,15B No 6, 443–451.CrossRefGoogle Scholar
  35. Smith, M.J. (1981b).Properties of a traffic control policy which ensure the existence of a traffic equilibrium consistent with the policy. Transportation Research B,15B No 6, 453–462.CrossRefGoogle Scholar
  36. Smith, M.J. (1983a).The existence and calculation of traffic equilibria. Transp. Res.17B (4), 291–303.CrossRefGoogle Scholar
  37. Smith, M.J. (1983b).Art algorithm for solving asymmetric equilibrium problems with continuous cost-flow function. Transp. Res.17B (5), 365–372.CrossRefGoogle Scholar
  38. Smith, M.J. (1985).Traffic Signals in assignment. Transportation Research B,19B, No 2, 155–160.CrossRefGoogle Scholar
  39. Smith, M.J., and M. Ghali (1989).The interaction between traffic flow and traffic control in congested urban networks. Paper for the Italian/USA Traffic Conference, Naples.Google Scholar
  40. Wardrop, J.G. (1952).Some theoretical aspects of road traffic research. Proc. Inst. Civ. Eng. Part II,1 (2), 325–378.Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 1996

Authors and Affiliations

  • L. Montero
    • 1
  • J. Barceló
    • 1
  1. 1.Statistics and Operations Research Dept. Facultat d'InformàticaUniversitat Politècnica de CatalunyaBarcelonaSpain

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