Skip to main content

Advertisement

SpringerLink
Log in

Monotonicity and Complexity of Multistage Stochastic Variational Inequalities

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

Abstract

In this paper, we consider multistage stochastic variational inequalities (MSVIs). First, we give multistage stochastic programs and multistage multi-player noncooperative game problems as source problems. After that, we derive the monotonicity properties of MSVIs under less restrictive conditions. Finally, the polynomial rate of convergence with respect to sample sizes between the original problem and its sample average approximation counterpart has been established.

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. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2013)

    MATH  Google Scholar 

  2. Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30(4), 1022–1038 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Chen, X., Pong, T.K., Wets, R.J.B.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Program. 165(1), 71–111 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Chen, X., Shapiro, A., Sun, H.: Convergence analysis of sample average approximation of two-stage stochastic generalized equations. SIAM J. Optim. 29(1), 135–161 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Chen, X., Sun, H., Xu, H.: Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems. Math. Program. 177(1–2), 255–289 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Chen, X., Wets, R.J.B., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22(2), 649–673 (2012)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Cui, X., Sun, J., Zhang, L.: Solvability of multistage pseudomonotone stochastic variational inequalities. arXiv:2201.01454 (2022)

  9. Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2007)

    MATH  Google Scholar 

  10. Gürkan, G., Yonca Özge, A., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84(2), 313–333 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Jiang, J., Chen, X., Chen, Z.: Quantitative analysis for a class of two-stage stochastic linear variational inequality problems. Comput. Optim. Appl. 76(2), 431–460 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Jiang, J., Li, S.: On complexity of multistage stochastic programs under heavy tailed distributions. Oper. Res. Lett. 49(2), 265–269 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Jiang, J., Li, S.: Regularized sample average approximation approach for two-stage stochastic variational inequalities. J. Optim. Theory Appl. 190(2), 650–671 (2021)

    MathSciNet  MATH  Google Scholar 

  14. Jiang, J., Shi, Y., Wang, X., Chen, X.: Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty. J. Comput. Math. 37(6), 813–842 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Jiang, J., Sun, H., Zhou, B.: Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage. Numer. Algorithms 89(1), 167–194 (2022)

    MathSciNet  MATH  Google Scholar 

  16. Kyparisis, J.: Solution differentiability for variational inequalities. Math. Program. 48(1–3), 285–301 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Li, M., Zhang, C.: Two-stage stochastic variational inequality arising from stochastic programming. J. Optim. Theory Appl. 186(1), 324–343 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Li, M., Zhang, C., Ding, M., Lv, R.: A two-stage stochastic variational inequality model for storage and dynamic distribution of medical supplies in epidemic management. Appl. Math. Model. 102, 35–61 (2022)

    MathSciNet  Google Scholar 

  19. Liu, J., Li, S., Jiang, J.: Quantitative stability of two-stage stochastic linear variational inequality problems with fixed recourse. Appl. Anal. 101(8), 3122–3138 (2022)

    MathSciNet  MATH  Google Scholar 

  20. Pham, K.D., Bui, N.M.: Error bounds for strongly monotone and Lipshitz continuous variational inequalities. Optim. Lett. 12(5), 971–984 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Rockafellar, R.T., Sun, J.: Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174(1–2), 453–471 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, New York (2009)

    MATH  Google Scholar 

  23. Rockafellar, R.T., Wets, R.J.B.: Stochastic variational inequalities: single-stage to multistage. Math. Program. 165(1), 331–360 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Shanbhag, U.V.: Stochastic variational inequality problems: Applications, analysis, and algorithms. In: INFORMS TutORials in Operations Research, pp. 71–107. INFORMS (2013)

  25. Shapiro, A.: On complexity of multistage stochastic programs. Oper. Res. Lett. 34(1), 1–8 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, 2nd edn. SIAM, Philadelphia (2014)

    MATH  Google Scholar 

  27. Sun, H., Chen, X.: Two-stage stochastic variational inequalities: theory, algorithms and applications. J. Oper. Res. Soc. China 9(1), 1–32 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Wang, X., Chen, X.: Solving two-stage stochastic variational inequalities by a projection semismooth Newton algorithm. Working paper (2022)

  29. Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia-Pacific J. Oper. Res. 27(01), 103–119 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Zhang, M., Sun, J., Xu, H.: Two-stage quadratic games under uncertainty and their solution by progressive hedging algorithms. SIAM J. Optim. 29(3), 1799–1818 (2019)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by China Postdoctoral Science Foundation (Grant No. 2020M673117) and National Natural Science Foundation of China (Grant Nos. 11871276 and 12122108).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Jie Jiang

  2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Hailin Sun

Authors
  1. Jie Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hailin Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hailin Sun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by René Henrion.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, J., Sun, H. Monotonicity and Complexity of Multistage Stochastic Variational Inequalities. J Optim Theory Appl 196, 433–460 (2023). https://doi.org/10.1007/s10957-022-02099-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-022-02099-8

Keywords

Mathematics Subject Classification

Search

Navigation