A simplicial decomposition algorithm for solving the variational inequality formulation of the general traffic assignment problem for large scale networks
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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.
- Aashtiani, H.Z., and T.L. Magnanti (1981).Equilibria on a congested transportation network. SIAM J. Algebraic and Discrete Methods2, 213–226.
- Beckmann, M., C.B. McGuire and C.B. Winsten (1956).Studies in the Economics of Transportation. Yale University Press, New Haven CT.
- 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.
- Bureau of Public Roads (1964).Traffic Assignment Manual. U.S. Department of Commerce, Urban Planning Division, Washington, DC.
- Dafermos, S.C. (1980).Traffic assignment and variational inequalities. Transportation Science14 (1), 42–54.
- Dafermos, S.C. (1982).Relaxation algorithms for the general asymmetric traffic equilibrium problem. Transportation Science16 (2), 231–240.
- Dafermos, S.C. (1983).An iterative scheme for variational inequalities. Mathematical Programming26 (1), 40–47.
- Deo, N. and C. Pang (1984).Shortest-path algorithms: taxonomy and annotation. Networks Vol14, 275–323.
- Fisk, C. and S. Nguyen (1982).Solution algorithms for network equilibrium models with asymmetric user costs. Transp. Sci.16 (3), 361–381.
- Florian, M. and H. Spiess (1982).The convergence of diagonalization algorithms for asymmetric network equilibrium problems. Trans. Res.16B (6), 447–483.
- Florian, M. (1986).Nonlinear Cost Network Models in Transportation Analysis. Mathematical Programming Study26, 167–196.
- Frank, M. and P. Wolfe (1956).An algorithm for quadratic programming. Naval Research Logistics Quarterly3, 95–110.
- 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.
- Gill, P.E., W. Murray and M. Wright (1981). Practical Optimization. Academic Press.
- 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. CrossRef
- Hearn, D.W. (1982).The gap function of a convex program. Oper. Res. Lett.1, 67–71. CrossRef
- Hohenbalken, B. (1977).Simplicial Decomposition in Nonlinear Programming algorithms. Mathematical Programming13, 49–68. CrossRef
- Holloway, C.A. (1974).An extension of the Frank and Wolfe method of feasible directions. Mathematical Programming,6, 14–27. CrossRef
- Kinderlehrer, D. and G. Stampacchia (1980).An Introduction to Variational Inequalities and Their Applications. Academic, New York.
- Knuth, D.E. (1973).The Art of Computer Programming. Addison-Wesley.
- Larsson, T. and M. Patriksson (1992).A dual scheme for traffic assignment problems. Transportation Science26, 4–17. CrossRef
- Lawphongpanich, S. and D.W. Hearn (1984).Simplicial decomposition of the asymmetric traffic assignment problem. Transp. Res18B, 123–133. CrossRef
- Luenberger, D.G. (1974).Introduction to Linear and Nonlinear Programming. Addison-Wesley, Reading, MA.
- 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.
- 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. CrossRef
- Marcotte P. and J. Guélat (1988).Adaptation of a modified Newton method for solving the asymmetric traffic equilibrium problem. Transportation Science,22, 112–124.
- 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).
- Nguyen, S. and C. Depuis (1984).An efficient method for computing traffic equilibria in network with asymmetric transportation costs. Transp.
- Pang, J.S. and D. Chan (1982).Iterative Methods for variational and complementary problems. Mathematical Programming24 (3), 284–313. CrossRef
- Pang, J.S. and C.S. Yu (1984).Linearized simplicial decomposition methods for computing traffic equilibria on networks. Networks14 (3), 427–438.
- 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.
- Sheffi, Y. (1985). Urban transportation networks. Equilibrium analysis with mathematical methods. Prentice Hall, Englewood Cliffs, New Jersey.
- Smith, M.J. (1979b).Existence, uniqueness and stability of traffic equilibria. Transp. Res.13B (4), 295–304. CrossRef
- 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. CrossRef
- 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. CrossRef
- Smith, M.J. (1983a).The existence and calculation of traffic equilibria. Transp. Res.17B (4), 291–303. CrossRef
- Smith, M.J. (1983b).Art algorithm for solving asymmetric equilibrium problems with continuous cost-flow function. Transp. Res.17B (5), 365–372. CrossRef
- Smith, M.J. (1985).Traffic Signals in assignment. Transportation Research B,19B, No 2, 155–160. CrossRef
- 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.
- Wardrop, J.G. (1952).Some theoretical aspects of road traffic research. Proc. Inst. Civ. Eng. Part II,1 (2), 325–378.
- A simplicial decomposition algorithm for solving the variational inequality formulation of the general traffic assignment problem for large scale networks
Volume 4, Issue 2 , pp 225-256
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- Traffic Equilibria
- Variational Inequalities
- Simplicial Decomposition Methods
- Projection Methods
- Quadratic Programming