Abstract
In this paper, a method to approximate the directions of Clarke's generalized gradient of the upper level function for the demand adjustment problem on traffic networks is presented. Its consistency is analyzed in detail. The theoretical background on which this method relies is the known property of proximal subgradients of approximating subgradients of proximal bounded and lower semicountinuous functions using the Moreau envelopes. A double penalty approach is employed to approximate the proximal subgradients provided by these envelopes. An algorithm based on partial linearization is used to solve the resulting nonconvex problem that approximates the Moreau envelopes, and a method to verify the accuracy of the approximation to the steepest descent direction at points of differentiability is developed, so it may be used as a suitable stopping criterion. Finally, a set of experiments with test problems are presented, illustrating the approximation of the solutions to a steepest descent direction evaluated numerically.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anandalingam, G. and T.L. Friesz (eds.). (1992). “Hierarchical Optimization.” Annals of Operations Research 34(1–4).
Aubin, J-P. (1998). Optima and Equilibria. An Introduction to Nonlinear Analysis, Springer Verlag.
Barceló, J. (1997). “A Survey of Some Mathematical Programming Models in Transportation.” Top, 5(1), 1–40.
Bard, J.F. (1998). Practical Bilevel Optimization. Algorithms and Applications, Kluwer Academic Publishers.
Bazaraa, M. Sherali, H.D. and C. Shetty. (1993). Mathematical Programming. Theory and Algorithms, John Wiley & Sons.
Bell, M.G.H. and Y. Iida. (1997). Transportation Network Analysis, John Wiley & Sons.
Cascetta, E. (1984). “Estimation of Trip Matrices from Traffic Counts and Survey Data: A Generalised Least Square Estimator.” Transportation Research B, 18-B, 289–299.
Chen, Y. (1994). Bilevel Programming Problems: Analysis, Algorithms and Applications, Ph.D. dissertation, Départament d'Informatique et de Recherche Opérationelle, Université de Montréal.
Clarke, F.H. (1983). Optimization and Nonsmooth Analisys, John Wiley InterScience.
Codina, E. (2000a). “Lipschitz Continuity of O-D Travel Times for the Asymmetric Traffic Assignment Problem.” Working Paper, DR/2000-21, Departament d'Estadística i Investigació Operativa. Universitat Politècnica de Catalunya.
Codina, E. (2000b). “A Test for the Approximation of Gradients of the Upper Level Function in Demand Adjustment Problems.” Working Paper, DR/2000-23. Departament d'Estadística i Investigació Operativa. Universitat Politècnica de Catalunya.
Codina, E. and J. Barceló. (2000). “Adjustment of O-D Trip Matrices from Traffic Counts: An Algorithmic Approach Based on Conjugate Directions.” Proceedings of the 8th EWGT., Rome, Italy, July 2000, pp. 427–432. (Forthcoming in European Journal of Operational Research)
Codina, E. García, R. and A. Marín (2001). “New Algorithmic Alternatives for the O-D Matrix Adjustment Problem on Traffic Networks.” Proceedings of the 9th EWGT, Bari, Italy, June 2001, pp. 421–425.
Dafermos, S. and A. Nagurney. (1984). “Sensitivity Analysis for the Asymmetric Network Equilibrium Problem.” Mathematical Programming, 28, 174–184.
Dontchev, A.L. and R.T. Rockafellar (1996). “Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets.” SIAM Journal of Optimization, 12(1), 1087–1105.
Dontchev A.L. and R.T. Rockafellar. (2001). “Ample Parametrization of Variational Inclusions.” SIAM Journal of Optimization, 12(1), 170–187.
Fiacco., A.V. (1983a). “Sensibility Analysis for Nonlinear Programming Using Penalty Methods.” Mathematical Programming, 10, 287–311.
Fiacco., A.V. (1983b). Introduction to Sensibility and Stability Analysis in Nonlinear Programming, Academic Press.
Friesz., T.L. et al. (1990). “Sensitivity Analysis Based Heuristic Algorithms for Mathematical Programs with Variational Inequality Constraints.” Mathematical Programming, 48, 265–284.
Gartner, N.H. (1980). “Optimal Traffic Assignments with Elastic Demands: A Review. Part II: Algorithmic Approaches.” Transportation Science, 14, 192–208.
Hall, M.A. (1978). “Properties of the Equilibrium State in Transportation Networks.” Transportation Science, 12(3), 208–216.
Hearn, D.W., S. Lawphongpanich, and J.A. Ventura (1987). “Restricted Simplicial Decomposition: Computation and Extensions.” Mathematical Programming Study, 31, 99–118.
Jörnsten, K. and S. Nguyen. (1979). “On the Estimation of a Trip Matrix from Network Data.” Working Paper, Centre de Recherche sur les Transports de Montréal, Publication #153.
Kolstad, C.D. and L.S. Lasdon. (1986). “Derivative Evaluation and Computational Experience with Large Bi-Level Mathematical Programs.” Faculty Working Paper, No. 1266, BEBR College of Business and Administration, University of Illinois at Urbana-Champaign.
Liu, J. (1995). “Sensitivity Analysis in Nonlinear Programs and Variational Inequalities via Continuous Selection.” SIAM Journal of Control and Optimization, 33(4), 1040–1060.
Luo, Z.-Q., J. Pang, and D. Ralph. (1986). Mathematical Programming with Equilibrium Constraints, Cambridge University Press.
Patriksson, M. (1993). “Partial Linearization Methods in Nonlinear Programming.” Journal of Optimization Theory and Applications, 78, 227–246.
Patriksson, M. (1994). The Traffic Assignment Problem. Models and Methods, VSP, Topics in Transportation.
Patriksson, M. and R.T. Rockafellar (2000). “Sensitivity Analysis of Variational Inequalities over Aggregated Polyhedra, with Application to Traffic Equilibria.” Proceedings of the International School of Mathematics “Guido Stampacchia” 32th Workshop, Erice, Italy, June 2000.
Patriksson, M. (2003). “Sensitivity Analysis of Traffic Equilibria.” Transportation Science. (forthcoming)
Qiu, Y. and T.L. Magnanti. (1989). “Sensitivity Analysis for Variational Inequalities Defined on Polyhedral Sets.” Mathematics of Operations Research, 14(3), 410–432.
Rockafellar, R.T. R. Wets, and J.-B. Roger. (1998). Variational Analysis, Springer Verlag.
Spiess, H. (1980). A Gradient Approach for the O-D Matrix Adjustment Problem, Working Paper, Centre de Recherche sur les Transports de MontrÉal, Publication #693.
Spiess, H. (1987). “A Maximum Likelihood Model for Estimation of Origin-Destination Matrices.” Transportation Research B, 21B, 395–412.
Tobin, R.L. and T.L. Friesz. (1988). “Sensitivity Analysis for Network Equilibrium Flows.” Transportation Science, 22 242–250.
Van Vliet, D. and L.G. Willumsen. (1981). “Validation of the ME2 Model for Estimating Trip Matrices from Traffic Counts.” Proceedings of the 8th International Symposium on Transportation and Traffic Theory, June 1981.
Wolfe, P. (1975). “A Method of Conjugate Subgradients for Minimizing Nondifferentiable Optimization.” Mathematical Programming Study, 3 145–173.
Van Zuylen, H.J. and L.G. Willumsen. (1980). “The Most Likely Trip Matrix Estimated from Traffic Counts.” Transportation Research B, 14B, 281–293.
Yang, H., Iida, Y. and T. Sasaki. (1992). “Estimation of Origin-Destination Matrices from Traffic Counts on Congested Networks.” Transportation Research B, 26B(6), 417–434.
Yang, H. (1995). “Heuristic Algorithms for the Bilevel Origin-Destination Matrix Estimation Problem.” Transportation Research B, 29B(4), 231–242.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported under Spanish CICYT project TRA99-1156-C02-02.
Rights and permissions
About this article
Cite this article
Codina, E., Montero, L. Approximation of the steepest descent direction for the O-D matrix adjustment problem. Ann Oper Res 144, 329–362 (2006). https://doi.org/10.1007/s10479-006-0007-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-006-0007-x