Skip to main content

Advertisement

SpringerLink
Log in

Quantitative analysis for a class of two-stage stochastic linear variational inequality problems

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

Abstract

This paper considers a class of two-stage stochastic linear variational inequality problems whose first stage problems are stochastic linear box-constrained variational inequality problems and second stage problems are stochastic linear complementary problems having a unique solution. We first give conditions for the existence of solutions to both the original problem and its perturbed problems. Next we derive quantitative stability assertions of this two-stage stochastic problem under total variation metrics via the corresponding residual function. Moreover, we study the discrete approximation problem. The convergence and the exponential rate of convergence of optimal solution sets are obtained under moderate assumptions respectively. Finally, through solving a non-cooperative game in which each player’s problem is a parameterized two-stage stochastic program, we numerically illustrate our theoretical results.

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

Fig. 1

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Chen, X., Xiang, S.: Newton iterations in implicit time-stepping scheme for differential linear complementarity systems. Math. Program. 138, 579–606 (2013)

    Article  MathSciNet  Google Scholar 

  5. Chen, X., Ye, Y.: On homotopy-smoothing methods for box-constrained variational inequalities. SIAM J. Control Optim. 37, 589–616 (1999)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  8. King, A.J., Rockafellar, R.T.: Sensitivity analysis for nonsmooth generalized equations. Math. Program. 55, 193–212 (1992)

    Article  MathSciNet  Google Scholar 

  9. Lei, J., Shanbhag, U.V., Pang, J.S., Sen, S.: On synchronous, asynchronous, and randomized best-response schemes for computing equilibria in stochastic nash games. Math. Oper. Res. 45, 157–190 (2020)

    Article  Google Scholar 

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

    Book  Google Scholar 

  11. Pang, J.S., Sen, S., Shanbhag, U.V.: Two-stage non-cooperative games with risk-averse players. Math. Program. 165, 235–290 (2017)

    Article  MathSciNet  Google Scholar 

  12. Rachev, S.T., Klebanov, L., Stoyanov, S.V., Fabozzi, F.: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York (2013)

    Book  Google Scholar 

  13. Ravat, U., Shanbhag, U.V.: On the characterization of solution sets of smooth and nonsmooth convex stochastic nash games. SIAM J. Optim. 21, 1168–1199 (2011)

    Article  MathSciNet  Google Scholar 

  14. Ravat, U., Shanbhag, U.V.: On the existence of solutions to stochastic quasi-variational inequality and complementarity problems. Math. Program. 165, 291–330 (2017)

    Article  MathSciNet  Google Scholar 

  15. Robinson, S.M.: Generalized equations and their solutions, part I: Basic theory. In: Huard, P. (ed.) Point-to-Set Maps and Mathematical Programming, pp. 128–141. Springer, Berlin (1979)

    Chapter  Google Scholar 

  16. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Rockafellar, R.T., Wets, R.J.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16, 119–147 (1991)

    Article  MathSciNet  Google Scholar 

  19. Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, Berlin (2009)

    MATH  Google Scholar 

  20. Rockafellar, R.T., Wets, R.J.: Stochastic variational inequalities: single-stage to multistage. Math. Program. 165, 331–360 (2016)

    Article  MathSciNet  Google Scholar 

  21. Römisch, W.: Stability of stochastic programming problems. Handb. Oper. Res. Manag. Sci. 10, 483–554 (2003)

    MathSciNet  Google Scholar 

  22. Shapiro, A.: Monte Carlo sampling methods. Handb. Oper. Res. Manag. Sci. 10, 353–425 (2003)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Zhang, J., Xu, H., Zhang, L.: Quantitative stability analysis for distributionally robust optimization with moment constraints. SIAM J. Optim. 26, 1855–1882 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank Dr. Hailin Sun for his help in the numerical experiment. They would like to thank the editor and three anonymous referees for their insightful comments which help us to improve the content and presentation of this manuscript.

Funding

Xiaojun Chen’s work was partially supported by Hong Kong Research Grant Council Grant PolyU153000/17P. Zhiping Chen’s work was partially supported by the National Natural Science Foundation of China Nos. 11991023, 11991020, 11735011.

Author information

Authors and Affiliations

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

    Jie Jiang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Jie Jiang & Xiaojun Chen

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, China

    Zhiping Chen

Authors
  1. Jie Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaojun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhiping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaojun Chen.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, J., Chen, X. & Chen, Z. Quantitative analysis for a class of two-stage stochastic linear variational inequality problems. Comput Optim Appl 76, 431–460 (2020). https://doi.org/10.1007/s10589-020-00185-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-020-00185-z

Keywords

Search

Navigation