Solving stochastic mathematical programs with complementarity constraints using simulation

Mathematics of Operations Research

Volumn 31, Issue 4, 2006, Pages 739-760

  Birbil, S Ilker ^a   Gürkan, Gül ^b   Listeş, Ovidiu ^c  

^a SABANCI UNIVERSITY   (Turkey)

^b TILBURG UNIVERSITY   (Netherlands)

^c Paragon Decision Technology   (Netherlands)

Author keywords

Mathematical programs with equilibrium constraints; Simulation; Stability; Stochastic mathematical programs with complementarity constraints; Stochastic network equilibrium; Toll pricing problem

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; CONSTRAINT THEORY; HIGHWAY SYSTEMS; OPTIMIZATION; STOCHASTIC CONTROL SYSTEMS; THEOREM PROVING;

COMPLEMENTARITY CONSTRAINTS; EQUILIBRIUM CONSTRAINTS; STOCHASTIC NETWORK EQUILIBRIUM; TOLL PRICING PROBLEM;

STOCHASTIC PROGRAMMING;

EID: 33847039571     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1060.0215     Document Type: Article

References (45)

1. Bard, J. F. 1998. Practical Bilevel Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands.
2. Birbil, Ş. I., G. Gürkan, O. Listeş. 2004. Simulation-based solution of stochastic mathematical programs with complementarity constraints: Sample-path analysis. CentER discussion paper, ISSN: 0924-7815, Tilburg University, Tilburg, The Netherlands. Available at http://greywww.kub.nl:2080/greyfiles/center/2004/doc/25.pdf.
3. Birbil, Ş. I., G. Gürkan, O. Listeş. 2004. Solving stochastic mathematical programs with complementarity constraints using simulation. R. G. Ingalls, M. D. Rossetti, J. S. Smith, B. A. Peters, eds. Proc. 2004 Winter Simulation Conf., Washington, D.C., IEEE, Piscataway, NJ, 550-558.
4. Bureau of Public Roads. 1964. Traffic Assignment Manual. U.S. Department of Commerce, Urban Planning Division, Washington, D.C.
5. Dempe, S. 2002. Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht, The Netherlands.
6. Dirkse, S. P., M. C. Ferris. 1998. Traffic modeling and variational inequalities using GAMS. Operations Research and Decision Aid Methodologies in Traffic and Transportation Management. Ph. L. Toint, M. Labbe, K. Tanczos, G. Laporte, eds. NATO ASI Series F, Vol. 6. Springer-Verlag, 136-163.
7. Drud, A. 1985. CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems. Math. Programming 31 153-191.
8. Facchinei, F., J. S. Pang. 2003. Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, New York.
9. Ferris, M. C., S. P. Dirkse, A. Meeraus. 2005. Mathematical programs with equilibrium constraints: Automatic reformulation and solution via constrained optimization. T. J. Kehoe, T. N. Srinivasan, J. Whalley, eds. Frontiers in Applied General Equilibrium Modeling. Cambridge University Press, Cambridge, UK, 67-93. Available at http://web.comlab.ox.ac.uk/oucl/publications/natr/na- 02-11.html.
10. Ferris, M. C., J. S. Pang. 1997. Engineering and economic applications of complementarity problems. SIAM Rev. 39 669-713.
11. Florian, M., D. Hearn. 1995. Network equilibrium models and algorithms. M. O. Ball, T. Magnanti, C. Monma, G. Nemhauser, eds. Network Routing Handbooks in OR and MS, Vol. 8. Elsevier Science, Amsterdam, The Netherlands, 485-550.
12. Glasserman, P. 1991. Gradient Estimation via Perturbation Analysis. Kluwer, Norwell, MA.
13. Griewank, A. 2000. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia, PA.
14. Gürkan, G., A. Y. Özge, S. M. Robinson. 1996. Sample-path solution of stochastic variational inequalities, with applications to option pricing. J. M. Charnes, D. M. Morrice, D. M. Brunner, J. J. Swain, eds. Proc. 1996 Winter Simulation Conf., Coronado, CA. IEEE, Piscataway, NJ, 337-344.
15. Gürkan, G., A. Y. Özge, S. M. Robinson. 1999. Sample-path solution of stochastic variational inequalities. Math. Programming 84 313-333.
16. Gürkan, G., A. Y. Özge, S. M. Robinson. 1999. Solving stochastic optimization problems with stochastic constraints: An application in network design. D. T. Sturrock, P. A. Farrington, H. B. Nembhard, G. W. Evans, eds. Proc. 1999 Winter Simulation Conf., Phoenix, AZ. IEEE, Piscataway, NJ, 471-478.
17. Ho, Y. C., X. R. Cao. 1991. Perturbation Analysis of Discrete Event Dynamical Systems, Kluwer, Norwell, MA.
18. Kail, P. 1986. Approximation to optimization problems: An elementary review. Math. Oper. Res. 11 9-18.
19. Kleywegt, A. J., A. Shapiro, T. Homem-De-Mello. 2001. The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12 479-502.
20. Kojima, M. 1980. Strongly stable stationary solutions in nonlinear programming. S. M. Robinson, ed. Analysis and Computation of Fixed Points. Academic Press, New York, 93-138.
21. Lawphongpanich, S., D. W. Hearn. 2004. An MPEC approach to second-best toll pricing. Math. Programming 101 33-55.
22. Lin, G. H., X. Chen, M. Fukushima. 2003. Smoothing implicit programming approaches for stochastic mathematical programs with linear complementarity constraints. Technical report 2003-006, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, Japan.
23. Linderoth, J. T., A. Shapiro, S. J. Wright. 2002. The empirical behavior of sampling methods for stochastic programming. Optimization technical report 02-01, Computer Sciences Department, University of Wisconsin-Madison, Madison, WI. Available at http://www.cs.wisc.edu/~swright/papers/uwopt-0201.pdf.
24. Luo, Z. Q., J. S. Pang, D. Ralph. 1996. Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, UK.
25. Munkres, J. R. 1975. Topology: A First Course, Prentice Hall, Englewood Cliffs, NJ.
26. Outrata, J. V., M. Kocvara, J. Zowe. 1998. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results, Kluwer Academic Publishers, Boston, MA.
27. Patriksson, M., L. Wynter. 1999. Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. 25 159-167.
28. Plambeck, E. L., B. R. Fu, S. M. Robinson, R. Suri. 1993. Throughput optimization in tandem production lines via nonsmooth programming. J. Schoen, ed. Proc. 1993 Summer Computer Simulation Conf. Society for Computer Simulation, San Diego, CA, 70-75.
29. Plambeck, E. L., B. R. Fu, S. M. Robinson, R. Suri. 1996. Sample-path optimization of convex stochastic performance functions. Math. Programming 137-176.
30. Robinson, S. M. 1979. Generalized equations and their solutions, Part I: Basic theory. Math. Programming Study 10 128-141.
31. Robinson, S. M. 1980. Strongly regular generalized equations. Math. Oper. Res. 5 43-62.
32. Robinson, S. M. 1996. Analysis of sample-path optimization. Math. Oper. Res. 21 513-528.
33. Rockafellar, R. T., R. J.-B. Wets. 1998. Variational Analysis, Springer-Verlag, Berlin, Germany.
34. Rubinstein, R. Y., A. Shapiro. 1993. Discrete event systems: Sensitivity analysis and stochastic optimization by the score function method, Wiley, Chichester, UK.
35. Scheel, H. 1995. Ein Straffunktionsansatz für Optimierungsprobleme mit Gleichgewichtsrestriktionen, Diploma thesis, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Karlsruhe, Germany.
36. Scheel, H., S. Scholtes. 2000. Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity. Math. Oper. Res. 25 1-22.
37. Scholtes, S. 1994. Introduction to piecewise differentiable equations, Habilitation thesis, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Karlsruhe, Germany.
38. Shapiro, A. 2004. Stochastic programming with equilibrium constraints. Technical report, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA. Available at Optim. Online, http://www.optimization-online.org/DB_HTML/2004/08/930.html.
39. Shapiro, A., T. Homem-De-Mello. 1998. A simulation-based approach to two-stage stochastic programming with recourse. Math. Programming 81 301-325.
40. Still, G. 1999. Generalized semi-infinite programming: Theory and methods. Eur. J. Oper. Res. 119 301-303.
41. Transportation Research Board. 2000. Highway Capacity Manual. Washington, D.C.
42. Wardrop, J. G. 1952. Some theoretical aspects of road traffic research. Proc. Institution Civil Engineers, Vol. 2. 325-378.
43. Wynter, L. 2001. Stochastic bilevel programs. P. Pardalos, C. Floudas, eds. Encyclopedia of Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands.
44. Ying, J. Q., T. Miyagi. 2001. Sensitivity analysis for stochastic user equilibrium network flows - a dual approach. Transportation Sci. 35 124-133.
45. Zhang, H. M., Y. E. Ge. 1997. Modeling variable demand equilibrium under second-best road pricing. Transportation Res. B 38(4) 733-749.