Skip to main content
SpringerLink
Log in

Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints

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

Abstract

We study the quantitative stability of the solution sets, optimal value and M-stationary points of one stage stochastic mathematical programs with complementarity constraints when the underlying probability measure varies in some metric probability space. We show under moderate conditions that the optimal solution set mapping is upper semi-continuous and the optimal value function is Lipschitz continuous with respect to probability measure. We also show that the set of M-stationary points as a mapping is upper semi-continuous with respect to the variation of the probability measure. A particular focus is given to empirical probability measure approximation which is also known as sample average approximation (SAA). It is shown that optimal value and M-stationary points of SAA programs converge to their true counterparts with probability one (w.p.1.) at exponential rate as the sample size increases.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Theory, Applications, and Numerical. Kluwer Academic, Dordrecht (1998)

    MATH  Google Scholar 

  2. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Birbil, S.I., Gürkan, G., Listes, O.: Solving stochastic mathematical programs with complementarity constraints using simulation. Math. Oper. Res. 31, 739–760 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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, 423–441 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Liu, Y., Lin, G.-H.: Convergence analysis of a regularized sample average approximation method for stochastic mathematical programs with complementarity constraints. Asia-Pac. J. Oper. Res. (2011, to appear)

  7. Meng, F., Xu, H.: Exponential convergence of sample average approximation methods for a class of stochastic mathematical programs with complementarity constraints. J. Comput. Math. 24, 733–748 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Shapiro, A.: Stochastic mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 128, 223–243 (2006)

    Article  MathSciNet  Google Scholar 

  9. Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation. Optimization 57, 395–418 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Henrion, R., Römisch, W.: Metric regularity and quantitative stability in stochastic programs with probabilistic constraints. Math. Program. 84, 55–88 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Rachev, S.T., Römisch, W.: Quantitative stability in stochastic programming: the method of probability metrics. Math. Oper. Res. 27, 792–818 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Römisch, W.: Stability of stochastic programming problems. In: Ruszczynski, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, vol. 10, pp. 483–554. Elsevier, Amsterdam (2003)

    Chapter  Google Scholar 

  14. Dentcheva, D., Henrion, R., Ruszczynski, A.: Stability and sensitivity of optimization problems with first order stochastic dominance constraints. SIAM J. Optim. 18, 322–333 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 843–962 (2000)

    Article  Google Scholar 

  17. Klatte, D.: On the stability of local and global optimal solutions in parametric problems of nonlinear programming. Part I: Basic results. Prepr. - Humboldt-Univ. Berl. Fachbereich Math. 75, 1–21 (1985)

    Google Scholar 

  18. Klatte, D.: A note on quantitative stability results in nonlinear optimization. Prepr. - Humboldt-Univ. Berl. Fachbereich Math. 90, 77–86 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Lin, G.-H., Guo, L., Ye, J.J.: Solving mathematical programs with equilibrium constraints and constrained equations. Submitted

  20. Chen, B., Harker, P.T.: Smooth approximations to nonlinear complementarity problems. SIAM J. Optim. 7, 403–420 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  22. Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pang, J.S.: Error bound in mathematical programming. Math. Program. 79, 299–332 (1997)

    MATH  Google Scholar 

  24. Zhang, J., Xiu, N.: Global s-type error bound for the extended linear complementarity problem and applications. Math. Program. 88, 391–410 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Luo, Z.Q., Tseng, P.: Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM J. Optim. 2, 43–54 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. 14, 206–214 (1981)

    MATH  Google Scholar 

  27. Mangasarian, O.L.: Error bounds for nondegenerate monotone linear complementarity problems. Math. Program. 48, 435–455 (1990)

    Article  MathSciNet  Google Scholar 

  28. Mangasarian, O.L., Shiau, T.-H.: Error bounds for monotone linear complementarity problems. Math. Program. 36, 81–89 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mangasarian, O.L., Ren, J.: New improved error bounds for the linear complementarity problem. Math. Program. 66, 241–255 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen, B.: Error bounds for R0-type and monotone nonlinear complementarity problems. J. Optim. Theory Appl. 108, 297–316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I–II. Springer, Berlin (2003)

    Google Scholar 

  32. Xu, H.: Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming. J. Math. Anal. Appl. 368, 692–710 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ruszczynski, A., Shapiro, A.: Stochastic Programming. Handbook in Operations Research and Management Science. Amsterdam, Elsevier (2003)

    MATH  Google Scholar 

  34. Balaji, R., Xu, H.: Approximating stationary points of stochastic optimization problems in Banach space. J. Math. Anal. Appl. 347, 333–343 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, China

    Yongchao Liu & Gui-Hua Lin

  2. School of Mathematics, University of Southampton, Southampton, UK

    Huifu Xu

Authors
  1. Yongchao Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huifu Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gui-Hua Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongchao Liu.

Additional information

Communicated by Johannes O. Royset.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Y., Xu, H. & Lin, GH. Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints. J Optim Theory Appl 152, 537–555 (2012). https://doi.org/10.1007/s10957-011-9903-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9903-6

Keywords

Search

Navigation