Skip to main content
Log in

A Note on Stability for Risk-Averse Stochastic Complementarity Problems

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

Abstract

In this paper, we propose a new approach to stochastic complementarity problems which allows to take into account various notions of risk aversion. Our model enhances the expected residual minimization formulation of Chen and Fukushima (Math Oper Res 30:1022–1038, 2005) by replacing the expectation with a more general convex, nondecreasing and law-invariant risk functional. Relevant examples of such risk functionals include the Expected Excess and the Conditional Value-at-Risk. We examine qualitative stability of the resulting one-stage stochastic optimization problem with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the decision vector and the entering probability measure. By a classical result from parametric optimization, this implies upper semicontinuity of the optimal value function. Throughout the analysis, we assume for building the model that a nonlinear complementarity function fulfills a certain polynomial growth condition. We conclude the paper demonstrating that this assumption holds in the vast majority of all practically relevant cases.

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

Access this article

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. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  2. Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)

    MATH  Google Scholar 

  3. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hamatani, K., Fukushima, M.: Pricing American options with uncertain volatility through stochastic linear complementarity models. Comput. Optim. Appl. 50, 263–286 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lin, G.H., Fukushima, M.: Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: a survey. Pac. J. Optim. 6, 455–482 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Zhang, C., Chen, X., Sumalee, A.: Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: expected residual minimization approach. Transp. Res. Part B Methodol. 45, 534–552 (2011)

    Article  Google Scholar 

  7. Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang, H., Chen, X., Fukushima, M.: Stochastic \(R_0\) matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Claus, M., Krätschmer, V., Schultz, R.: Weak continuity of risk functionals with applications to stochastic programming. https://www.uni-due.de/imperia/md/content/mathematik/2015_preprint_790 (2015)

  11. Galantai, A.: Properties and construction of NCP functions. Comput. Optim. Appl. 52, 805–824 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Luo, M.-J., Lu, Y.: Properties of expected residual minimization model for a class of stochastic complementarity problems. J. Appl. Math., Article ID 497586 (2013). doi:10.1155/2013/497586

  13. Chen, B., Harker, P.T.: Smooth approximations to nonlinear complementarity problems. SIAM J. Optim. 7(2), 403–420 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  15. Lang, R.: A note on the measurability of convex sets. Arch. Math. 47, 90–92 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Föllmer, H., Schied, A.: Stochastic Finance, 3rd edn. de Gruyter, Berlin (2011)

    Book  MATH  Google Scholar 

  18. Krätschmer, V., Schied, A., Zähle, H.: Domains of weak continuity of statistical functionals with a view on robust statistics. http://arxiv.org/pdf/1511.08677v1 (2015)

  19. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  20. Schultz, R.: Some aspects of stability in stochastic programming. Ann. Oper. Res. 100, 55–84 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. Technical Rep. 00-11, Department of Computational and Applied Mathematics, Rice University (2000)

  22. Mangasarian, O.L.: Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math. 31, 89–92 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank Prof. R. Schultz for helpful discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matthias Claus.

Additional information

Communicated by René Henrion.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burtscheidt, J., Claus, M. A Note on Stability for Risk-Averse Stochastic Complementarity Problems. J Optim Theory Appl 172, 298–308 (2017). https://doi.org/10.1007/s10957-016-1020-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-1020-0

Keywords

Mathematics Subject Classification

Navigation