Advertisement

On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory

  • C. Cromvik
  • M. Patriksson
Article

Abstract

We consider a stochastic mathematical program with equilibrium constraints (SMPEC) and show that, under certain assumptions, global optima and stationary solutions are robust with respect to changes in the underlying probability distribution. In particular, the discretization scheme sample average approximation (SAA), which is convergent for both global optima and stationary solutions, can be combined with the robustness results to motivate the use of SMPECs in practice. We then study two new and natural extensions of the SMPEC model. First, we establish the robustness of global optima and stationary solutions to an SMPEC model where the upper-level objective is the risk measure known as conditional value-at-risk (CVaR). Second, we analyze a multiobjective SMPEC model, establishing the robustness of weakly Pareto optimal and weakly Pareto stationary solutions. In the accompanying paper (Cromvik and Patriksson, Part 2, J. Optim. Theory Appl., 2010, to appear) we present applications of these results to robust traffic network design and robust intensity modulated radiation therapy.

Stochastic mathematical program with equilibrium constraints Solution stability and robustness Sample average approximation Weak Pareto optimality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kall, P., Wallace, S.W.: Stochastic Programming. Wiley, New York (1994) MATHGoogle Scholar
  2. 2.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer Series in Operations Research. Springer, New York (1997) MATHGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust optimization: methodology and applications. Math. Program. Ser. B 92(3), 453–480 (2002) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Zhang, Y.: General robust-optimization formulation for nonlinear programming. J. Optim. Theory Appl. 132(1), 111–124 (2007) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Patriksson, M., Wynter, L.: Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. 25(4), 159–167 (1999) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cromvik, C., Patriksson, M.: On the robustness of global optima and stationary solutions to stochastic mathematical programs with equilibrium constraints, part II: applications. J. Optim. Theory Appl. (2010, to appear). doi: 10.1007/s10957-009-9640-2
  7. 7.
    Römisch, W.: Stability of stochastic programming problems. In: Stochastic Programming. Handbooks Oper. Res. Management Sci., vol. 10, pp. 483–554. Elsevier, Amsterdam (2003) CrossRefGoogle Scholar
  8. 8.
    Römisch, W., Wets, R.J.B.: Stability of ε-approximate solutions to convex stochastic programs. SIAM J. Optim. 18(3), 961–979 (2007) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Heitsch, H., Römisch, W., Strugarek, C.: Stability of multistage stochastic programs. SIAM J. Optim. 17(2), 511–525 (2006) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Birbil, Ş.İ., Gürkan, G., Listeş, O.: Solving stochastic mathematical programs with complementarity constraints using simulation. Math. Oper. Res. 31(4), 739–760 (2006) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shapiro, A.: Stochastic programming with equilibrium constraints. J. Optim. Theory Appl. 128(1), 223–243 (2006) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Lin, G.H., Xu, H., Fukushima, M.: Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints. Math. Methods Oper. Res. 67(3), 423–441 (2008) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Evgrafov, A., Patriksson, M., Petersson, J.: Stochastic structural topology optimization: existence of solutions and sensitivity analyses. Z. Angew. Math. Mech. 83(7), 479–492 (2003) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Evgrafov, A., Patriksson, M.: Stochastic structural topology optimization: discretization and penalty function approach. Struct. Multidiscipl. Optim. 25(3), 17–188 (2003) Google Scholar
  15. 15.
    Evgrafov, A., Patriksson, M.: Stable relaxations of stochastic stress-constrained weight minimization problems. Struct. Multidiscipl. Optim. 25(3), 189–198 (2003) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17(1), 259–286 (2006) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Leyffer, S., López-Calva, G., Nocedal, J.: Interior methods for mathematical programs with complementarity constraints. SIAM J. Optim. 17(1), 52–77 (2006) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Bard, J.F.: Practical Bilevel Optimization. Nonconvex Optimization and Its Applications, vol. 30. Kluwer, Dordrecht (1998) MATHGoogle Scholar
  20. 20.
    Luo, Z., Pang, J., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) Google Scholar
  21. 21.
    Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and Its Applications, vol. 28. Kluwer Academic, Dordrecht (1998) MATHGoogle Scholar
  22. 22.
    Bendsøe, M.P., Sigmund, O.: Topology Optimization. Springer, Berlin (2003) Google Scholar
  23. 23.
    Stackelberg, H.V.: The Theory of Market Economy. Oxford University Press, London (1952) Google Scholar
  24. 24.
    Nash, J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951) CrossRefMathSciNetGoogle Scholar
  25. 25.
    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) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lin, G.H., Chen, X., Fukushima, M.: Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization. Math. Program. Ser. B 116(1–2), 343–368 (2009) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    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. B 42(10), 843–860 (2008) CrossRefGoogle Scholar
  28. 28.
    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) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5(1), 43–62 (1980) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1983) MATHGoogle Scholar
  31. 31.
    Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 3. Springer, New York (2000) MATHGoogle Scholar
  32. 32.
    Rockafellar, R., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000) Google Scholar
  33. 33.
    Shapiro, A.: Monte Carlo sampling methods. In: Stochastic Programming. Handbooks Oper. Res. Management Sci., vol. 10, pp. 353–425. Elsevier, Amsterdam (2003) CrossRefGoogle Scholar
  34. 34.
    Shapiro, A., Xu, H.: Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions. J. Math. Anal. Appl. 325(2), 1390–1399 (2007) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Ye, J.J., Zhu, Q.J.: Multiobjective optimization problem with variational inequality constraints. Math. Program. Ser. A 96(1), 139–160 (2003) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 331. Springer, Berlin (2006) Google Scholar
  37. 37.
    Mordukhovich, B.S.: Multiobjective optimization problems with equilibrium constraints. Math. Program. Ser. B 117(1–2), 331–354 (2009) MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005) MATHGoogle Scholar
  39. 39.
    Minami, M.: Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space. J. Optim. Theory Appl. 41(3), 451–461 (1983) MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Li, X.F.: Constraint qualifications in nonsmooth multiobjective optimization. J. Optim. Theory Appl. 106(2), 373–398 (2000) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Mathematical SciencesChalmers University of Technology and Mathematical Sciences, University of GothenburgGothenburgSweden

Personalised recommendations