Using ACCPM in a simplicial decomposition algorithm for the traffic assignment problem

Article

Abstract

The purpose of the traffic assignment problem is to obtain a traffic flow pattern given a set of origin-destination travel demands and flow dependent link performance functions of a road network. In the general case, the traffic assignment problem can be formulated as a variational inequality, and several algorithms have been devised for its efficient solution. In this work we propose a new approach that combines two existing procedures: the master problem of a simplicial decomposition algorithm is solved through the analytic center cutting plane method. Four variants are considered for solving the master problem. The third and fourth ones, which heuristically compute an appropriate initial point, provided the best results. The computational experience reported in the solution of real large-scale diagonal and difficult asymmetric problems—including a subset of the transportation networks of Madrid and Barcelona—show the effectiveness of the approach.

Keywords

Traffic assignment problem Variational inequalities Simplicial decomposition Analytic center cutting plane method 

References

  1. 1.
    Aashtiani, H.Z., Magnanti, T.L.: A linearization and decomposition algorithm for computing urban traffic equilibria. In: Proceedings of the IEEE Large Scale Systems Symposium, pp. 8–19, 1982 Google Scholar
  2. 2.
    Auslender, A.: Optimization. Métodes Numériques. Masson, Paris (1976) Google Scholar
  3. 3.
    Bar-Gera, H.: Origin-based algorithm for the traffic assignment problem. Transp. Sci. 36(4), 398–417 (2002) MATHCrossRefGoogle Scholar
  4. 4.
    Beckmann, M.J., McGuire, C., Wisten, C.: Studies in the economics of transportation (1956) Google Scholar
  5. 5.
    Bertsekas, D.P., Gafni, E.M.: Projection methods for variational inequalities with application to the traffic assignment problem. Math. Program. Study 17, 139–159 (1982) MATHMathSciNetGoogle Scholar
  6. 6.
    Dafermos, S.: Traffic equilibrium and variational inequalities. Transp. Sci. 14, 42–54 (1980) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dafermos, S.: Relaxation algorithms for the general asymmetric traffic equilibrium problem. Transp. Sci. 16, 231–240 (1982) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Denault, M., Goffin, J.L.: On a primal-dual analytic center cutting plan method for variational inequalities. Comput. Optim. Appl. 12, 127–156 (1999) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I–II. Springer, New York (2003) Google Scholar
  10. 10.
    Florian, M.: Nonlinear cost network models in transportation analysis. Math. Program. Study 26, 167–196 (1986) MATHMathSciNetGoogle Scholar
  11. 11.
    Florian, M., Spiess, H.: The convergence of diagonalization algorithms for asymmetric network equilibrium problems. Transp. Res. 16B, 477–483 (1982) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Res. Logist. Q. 3, 95–110 (1956) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fukushima, M., Itoh, T.: A dual approach to asymmetric traffic equilibrium problems. Math. Jpn. 32(5), 701–721 (1987) MATHMathSciNetGoogle Scholar
  15. 15.
    Goffin, J.-L., Haurie, A., Vial, J.-P.: Decomposition and nondifferentiable optimization with the projective algorithm. Manag. Sci. 38-2, 284–302 (1992) CrossRefGoogle Scholar
  16. 16.
    Goffin, J.-L., Gondzio, J., Sarkissian, R., Vial, J.-P.: Solving nonlinear multicommodity problems by the analytic center cutting plane method. Math. Program. 76, 131–154 (1997) MathSciNetGoogle Scholar
  17. 17.
    Goffin, J.-L., Marcotte, P., Zhu, D.: An analytic cutting plane method for pseudo-monotone variational inequalities. Oper. Res. Lett. 20, 1–6 (1997) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Goffin, J.-L., Sarkissian, R., Vial, J.-P.: Using an interior point method for the master problem in a decomposition approach. Eur. J. Oper. Res. 101, 577–587 (1997) CrossRefGoogle Scholar
  19. 19.
    Hearn, D.W., Lawphongpanich, S., Ventura, J.A.: Restricted simplicial decomposition: computation and extensions. Math. Program. Study 31, 99–118 (1987) MATHMathSciNetGoogle Scholar
  20. 20.
    Katsura, R., Fukushima, M., Ibaraki, T.: Interior methods for nonlinear minimum cost network flow problems. J. Oper. Res. Soc. Jpn. 32(2), 174–199 (1989) MATHMathSciNetGoogle Scholar
  21. 21.
    Larsson, T., Patriksson, M.: Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transp. Sci. 26, 4–17 (1992) MATHCrossRefGoogle Scholar
  22. 22.
    Lawphongpanich, S., Hearn, D.W.: Simplicial decomposition of the asymmetric traffic assignment problem. Transp. Res. 18B, 123–133 (1984) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Lawphongpanich, S., Hearn, D.W.: Restricted simplicial decomposition with application to the traffic assignment problem. Ric. Oper. 38, 97–120 (1986) Google Scholar
  24. 24.
    LeBlanc, L.J., Morlok, E.K., Pierskalla, W.P.: An efficient approach to solving the road network equilibrium traffic assignment problem. Transp. Res. 9, 309–318 (1975) CrossRefGoogle Scholar
  25. 25.
    Marcotte, P.: A new algorithm for solving the variational inequalities with application to the traffic assignment problem. Math. Program. Study 33, 339–351 (1985) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Marcotte, P., Dussault, J.-P.: A modified newton method for solving variational inequalities. In: Proceedings of the 24th IEEE Conference on Decision and Control, vol. 33, pp. 1433–1436, 1985 Google Scholar
  27. 27.
    Marcotte, P., Guélat, J.: Adaptation of a modified newton method for solving the asymmetric traffic equilibrium problem. Transp. Sci. 22(2), 112–124 (1988) MATHCrossRefGoogle Scholar
  28. 28.
    Minty, G.J.: On the generalization of a direct method of the calculus of variations. Bull. Am. Math. Soc. 73, 315–321 (1967) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Montero, L.: A simplicial decomposition approach for solving the variational inequality formulation of the general traffic assignment problem for large scale networks. PhD thesis, Universitat Politècnica de Catalunya, Barcelona, Spain (1992) Google Scholar
  30. 30.
    Montero, L., Barceló, J.: A simplicial decomposition algorithm for solving the variational inequality formulation of the general traffic assignment problem for large scale networks. TOP 4(2), 225–256 (1996) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994) MATHGoogle Scholar
  32. 32.
    Nesterov, Y., Vial, J.-P.: Homogeneous analytic center cutting plane methods for convex problems and variational inequalities. Tech. Rep. 4 Logilab (1997) Google Scholar
  33. 33.
    Newell, G.: Traffic Flow on Transportation Networks. MIT Press, Cambridge (1980) Google Scholar
  34. 34.
    Nguyen, S., Dupuis, C.: An efficient method for computing traffic equilibria in networks with asymmetric transportation costs. Transp. Sci. 18, 185–202 (1984) CrossRefGoogle Scholar
  35. 35.
    Patriksson, M.: The Traffic Assignment Problem: Models and Methods. VSP B.V., Zeist (1994) Google Scholar
  36. 36.
    Patriksson, M.: Nonlinear Programming and Variational Inequality Problems—A Unified Approach. Kluwer Academic, Dordrecht (1998) Google Scholar
  37. 37.
    Rosas, D., Castro, J., Montero, L.: Solving the traffic assignment problem using ACCPM. Tech. Rep. DR 2002-18, Statistics and Operation Research Department: Universitat Politècnica de Catalunya, Barcelona, Spain (2002) Google Scholar
  38. 38.
    Smith, M.J.: Existence, uniqueness and stability of traffic equilibria. Transp. Res. 13B, 295–304 (1979) CrossRefGoogle Scholar
  39. 39.
    Smith, M.J.: The existence and calculation of traffic equilibria. Transp. Res. 17B, 291–303 (1983) CrossRefGoogle Scholar
  40. 40.
    Smith, M.J.: An algorithm for solving asymmetric equilibrium problems with a continuous cost-flow function. Transp. Res. 17B, 365–371 (1983) CrossRefGoogle Scholar
  41. 41.
    Solodov, M., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 34(5), 1814–1830 (1996) MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Sonnevend, G.: New algorithms in convex programming based on a notion of “centre” (for systems of analytic inequalities) and on rational extrapolation, Trends Math. Optim. 311–326 (1988) Google Scholar
  43. 43.
    Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II, vol. 1, pp. 325–378, 1952 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.CENITUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Dept. of Statistics and Operations ResearchUniversitat Politècnica de CatalunyaBarcelonaSpain

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