[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We reformulate a stochastic nonlinear complementarity problem as a stochastic programming problem which minimizes an expected residual defined by a restricted NCP function with nonnegative constraints and CVaR constraints which guarantee the stochastic nonlinear function being nonnegative with a high probability. By applying smoothing technique and penalty method, we propose a penalized smoothing sample average approximation algorithm to solve the CVaR-constrained stochastic programming. We show that the optimal solution of the penalized smoothing sample average approximation problem converges to the solution of the corresponding nonsmooth CVaR-constrained stochastic programming problem almost surely. Finally, we report some preliminary numerical test results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1

Similar content being viewed by others

References

  1. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, X., Lin, G.H.: CVaR-based formulation and approximation method for stochastic variational inequalities. Numer. Algebra Control Optim. 1, 35–48 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, X., Wets, R.J.B., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22, 649–673 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementarity problems vol. 1. Springer, New York (2003)

    Google Scholar 

  8. Fang, H., Chen, X., Fukushima, M.: Stochastic R 0 matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 66–713 (1997)

    Article  MathSciNet  Google Scholar 

  10. Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program., Ser. B 48, 161–220 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Kalos, M.H., Whitlock, P.A.: Monte Carlo Methods. Wiley, New York (1986)

    Book  MATH  Google Scholar 

  14. Lin, G.H., Chen, X., Fukushima, M.: New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC. Optimization 56, 641–653 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lin, G.H., Fukushima, M.: New reformulations for stochastic nonlinear complementarity problems. Optim. Methods Softw. 21, 551–564 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lin, G.H.: Combined Monte Carlo sampling and penalty method for stochastic nonlinear complementarity problems. Math. Comput. 78, 1671–1686 (2009)

    Article  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  18. Ling, C., Qi, L., Zhou, G., Caccetta, L.: The SC 1 property of an expected residual function arising from stochastic complementarity problems. Oper. Res. Lett. 36, 456–460 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Heidelberg (2004)

    Google Scholar 

  21. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Google Scholar 

  22. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  23. Wang, M., Ali, M.M.: Stochastic nonlinear complementarity problems: stochastic programming reformulation and penalty-based approximation method. J. Optim. Theory Appl. 144, 597–614 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wu, D., Han, J.Y., Zhu, J.H.: Robust solutions to uncertain linear complementarity problems. Acta Math. Appl. Sin. 27, 339–352 (2011)

    Article  MathSciNet  Google Scholar 

  25. Yamashita, N.: Properties of restricted NCP functions for nonlinear complementarity problems. J. Optim. Theory Appl. 98, 701–717 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang, C., Chen, X.: Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty. J. Optim. Theory Appl. 137, 277–295 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Zhou, G.L., Caccetta, L.: Feasible semismooth Newton method for a class of stochastic linear complementarity problems. J. Optim. Theory Appl. 139, 379–392 (2008)

    Article  MathSciNet  Google Scholar 

  28. Zhou, J.C., Xiu, N.H., Chen, J.S.: Solution properties and error bounds for semi-infinite complementarity problems. J. Ind. Manag. Optim. 9, 99–115 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research was supported by the National Natural Science Foundation of China (11171051, 91230103).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, 116025, China

    Liyan Xu & Bo Yu

  2. College of Science, Harbin Engineering University, Harbin, Heilongjiang, 150001, China

    Liyan Xu

Authors
  1. Liyan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo Yu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, L., Yu, B. CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems. Comput Optim Appl 58, 483–501 (2014). https://doi.org/10.1007/s10589-013-9625-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-013-9625-9

Keywords

Search

Navigation