Skip to main content
SpringerLink
Log in

On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 2: Applications

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In a companion paper (Cromvik and Patriksson, Part I, J. Optim. Theory Appl., 2010), the mathematical modeling framework SMPEC was studied; in particular, global optima and stationary solutions to SMPECs were shown to be robust with respect to the underlying probability distribution under certain assumptions. Further, the framework and theory were elaborated to cover extensions of the upper-level objective: minimization of the conditional value-at-risk (CVaR) and treatment of the multiobjective case. In this paper, we consider two applications of these results: a classic traffic network design problem, where travel costs are uncertain, and the optimization of a treatment plan in intensity modulated radiation therapy, where the machine parameters and the position of the organs are uncertain. Owing to the generality of SMPEC, we can model these two very different applications within the same framework. Our findings illustrate the large potential in utilizing the SMPEC formalism for modeling and analysis purposes; in particular, information from scenarios in the lower-level problem may provide very useful additional insights into a particular application.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Patriksson, M., Wynter, L.: Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. 25(4), 159–167 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Couchman, P., Kouvaritakis, B., Cannon, M., Prashad, F.: Gaming strategy for electric power with random demand. IEEE Trans. Power Syst. 20(3), 1283–1292 (2005)

    Article  Google Scholar 

  3. Xu, H.: An MPCC approach for stochastic Stackelberg–Nash–Cournot equilibrium. Optimization 54(1), 27–57 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Christiansen, S., Patriksson, M., Wynter, L.: Stochastic bilevel programming in structural optimization. Struct. Multidiscipl. Optim. 21(5), 361–371 (2001)

    Article  Google Scholar 

  5. Evgrafov, A., Patriksson, M., Petersson, J.: Stochastic structural topology optimization: existence of solutions and sensitivity analyses. ZAMM Z. Angew. Math. Mech. 83(7), 479–492 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Patriksson, M.: Robust bi-level optimization models in transportation science. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366(1872), 1989–2004 (2008)

    Article  MathSciNet  Google Scholar 

  7. Cromvik, C., Patriksson, M.: On the robustness of global optima and stationary solutions to stochastic mathematical programs with equilibrium constraints, part I: Theory. J. Optim. Theory Appl. (2010). doi:10.1007/s10957-009-9639-8

    Google Scholar 

  8. Patriksson, M.: On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs with equilibrium constraints. Transp. Res. 42B(10), 843–860 (2008)

    Article  Google Scholar 

  9. Evgrafov, A., Patriksson, M.: On the existence of solutions to stochastic mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 121(1), 65–76 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5(1), 43–62 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  11. Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II, pp. 325–378 (1952)

  12. Aashtiani, H.Z., Magnanti, T.L.: Equilibria on a congested transportation network. SIAM J. Algebr. Discrete Methods 2(3), 213–226 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Patriksson, M.: The Traffic Assignment Problem: Models and Methods. VSP, Zeist (1994)

    Google Scholar 

  14. Birbil, Ş.İ., Gürkan, G., Listeş, O.: Solving stochastic mathematical programs with complementarity constraints using simulation. Math. Oper. Res. 31(4), 739–760 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sheffi, Y.: Urban Transportation Networks. Englewood Cliffs, Prentice-Hall (1985)

    Google Scholar 

  16. Marcotte, P., Patriksson, M.: Traffic equilibrium. In: Barnhart, C., Laporte, G. (eds.) Transportation. Handbooks in Operations Research and Management Science, vol. 14, pp. 623–713. North-Holland, Amsterdam (2007)

    Chapter  Google Scholar 

  17. Sensitivity analysis of separable traffic equilibrium equilibria with application to bilevel optimization in network design. Transp. Res. B 41(1), 4–31 (2007)

  18. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gwinner, J., Raciti, F.: Random equilibrium problems on networks. Math. Comput. Model. 43(7–8), 880–891 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sumalee, A.: Optimal toll ring design with spatial equity impact constraint: An evolutionary approach. J. East. Asia Soc. Transp. Stud. 5, 1813–1828 (2003)

    Google Scholar 

  21. Connors, R., Sumalee, A., Watling, D.: Equitable network design. J. East. Asia Soc. Transp. Stud. 6, 1382–1397 (2005)

    Google Scholar 

  22. Maruyama, T., Sumalee, A.: Efficiency and equity comparison of cordon- and area-based road pricing schemes using a trip-chain equilibrium model. Transp. Res. A 41(7), 655–671 (2007)

    Google Scholar 

  23. Friesz, T.L., Anandalingam, G., Mehta, N.J., Nam, K., Shah, S.J., Tobin, R.L.: The multiobjective equilibrium network design problem revisited: A simulated annealing approach. Eur. J. Oper. Res. 65(1), 44–57 (1993)

    Article  MATH  Google Scholar 

  24. Migdalas, A.: Bilevel programming in traffic planning: Models methods and challenge. J. Global Optim. 7(4), 381–405 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Larsson, T., Patriksson, M.: Side constrained traffic equilibrium models—traffic management through link tolls. In: Marcotte, P., Nguyen, S. (eds.) Equilibrium and Advanced Transportation Modelling. pp. 125–151. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  26. Yin, Y.: Robust optimal traffic signal timing. Transp. Res. 42B(10), 911–924 (2008)

    Article  Google Scholar 

  27. Stavrev, P., Hristov, D., Warkentin, B., Sham, E., Stavreva, N., Fallone, B.G.: Inverse treatment planning by physically constrained minimization of a biological objective function. Med. Phys. 30(11), 2948–2958 (2003)

    Article  Google Scholar 

  28. Niemierko, A.: Reporting and analyzing dose distributions: A concept of equivalent uniform dose. Med. Phys. 24(1), 103–110 (1997)

    Article  Google Scholar 

  29. Olafsson, A., Wright, S.J.: Efficient schemes for robust IMRT treatment planning. Phys. Med. Biol. 51(21), 5621–5642 (2006)

    Article  Google Scholar 

  30. Chu, M., Zinchenko, Y., Henderson, S.G., Sharpe, M.B.: Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty. Phys. Med. Biol. 50(23), 5463–5477 (2005)

    Article  Google Scholar 

  31. Chan, T.C.Y., Bortfeld, T., Tsitsiklis, J.N.: A robust approach to IMRT optimization. Phys. Med. Biol. 51(10), 2567–2583 (2006)

    Article  Google Scholar 

  32. Baum, C., Alber, M., Birkner, M., Nüsslin, F.: Robust treatment planning for intensity modulated radiotherapy of prostate cancer based on coverage probabilities. Radiother. Oncol. 78(1), 27–35 (2006)

    Article  Google Scholar 

  33. Unkelbach, J., Oelfke, U.: Relating two techniques for handling uncertainties in IMRT optimization. Phys. Med. Biol. 51(23), 423–427 (2006)

    Article  Google Scholar 

  34. Nakamura, R.A., Monti, C.R., Castilho, L.N., Trevisan, F.A., Valim, A.C., Reinato, J.A.: Prognostic factors for late urinary toxicity grade 2–3 after conformal radiation therapy on patients with prostate cancer. Int. Braz J Urol 33(5), 652–659 (2007)

    Article  Google Scholar 

  35. Harsolia, A., Vargas, C., Yan, D., Brabbins, D., Lockman, D., Liang, J., Gustafson, G., Vicini, F., Martinez, A., Kestin, L.L.: Predictors for chronic urinary toxicity after the treatment of prostate cancer with adaptive three-dimensional conformal radiotherapy: dose-volume analysis of a phase II dose-escalation study. Int. J. Radiat. Oncol. Biol. Phys. 69(4), 1100–1109 (2007)

    Google Scholar 

  36. Kåver, G., Lind, B.K., Löf, J., Liander, A., Brahme, A.: Stochastic optimization of intensity modulated radiotherapy to account for uncertainties in patient sensitivity. Phys. Med. Biol. 44(12), 2955–2969 (1999)

    Article  Google Scholar 

  37. Lian, J., Xing, L.: Incorporating model parameter uncertainty into inverse treatment planning. Phys. Med. Biol. 31(9), 2711–2720 (2004)

    Google Scholar 

  38. Brahme, A.: Individualizing cancer treatment: biological optimization models in treatment planning and delivery. Int. J. Radiat. Oncol. Biol. Phys. 49(2), 327–337 (2001)

    Article  Google Scholar 

  39. Webb, S.: Intensity-Modulated Radiation Therapy. Institute of Physics Publishing, London (2001)

    Book  Google Scholar 

  40. Löf, J.: Development of a general framework for optimization of radiation therapy. Ph.D. thesis, Stockholm University, Stockholm (2000)

  41. Carlsson, J.: Utilizing problem structure in optimization of radiation therapy. Ph.D. thesis, Royal Institute of Technology, Stockholm (2008)

  42. Emami, B., Lyman, J., Brown, A., Coia, L., Goitein, M., Munzenrider, J.E., Solin, L.J., Wesson, M.: Tolerance of normal tissue to therapeutic irradiation. Int. J. Radiat. Oncol. Biol. Phys. 21(1), 109–122 (1991)

    Google Scholar 

  43. Deasy, J., Blanco, A.I., Clark, V.H.: CERR: A computational environment for radiotherapy research. Med. Phys. 30(5), 979–985 (2003)

    Article  Google Scholar 

  44. Gould, N.I.M., Orban, D., Toint, Ph.L.: GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization. ACM Trans. Math. Softw. 29(4), 353–372 (2004)

    Article  MathSciNet  Google Scholar 

  45. Convery, D.J., Rosebloom, M.E.: The generation of intensity-modulated fields for conformal radiotherapy by dynamic collimation. Phys. Med. Biol. 37(6), 1359–1374 (1992)

    Article  Google Scholar 

  46. Spirou, S.V., Chui, C.S.: Generation of arbitrary intensity profiles by dynamic jaws or collimators. Med. Phys. 21(7), 1031–1041 (1994)

    Article  Google Scholar 

  47. Chui, C.S., Spirou, S., LoSasso, T.: Testing of dynamic multileaf collimation. Med. Phys. 23(5), 635–641 (1996)

    Article  Google Scholar 

  48. Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, University of Gothenburg, 412 96, Gothenburg, Sweden

    C. Cromvik & M. Patriksson

Authors
  1. C. Cromvik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Patriksson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Patriksson.

Additional information

Communicated by F. Giannessi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cromvik, C., Patriksson, M. On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 2: Applications. J Optim Theory Appl 144, 479–500 (2010). https://doi.org/10.1007/s10957-009-9640-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-009-9640-2

Search

Navigation