Skip to main content
SpringerLink
Log in

Expected residual minimization method for monotone stochastic tensor complementarity problem

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

Abstract

In this paper, we first introduce a new class of structured tensors, named strictly positive semidefinite tensors, and show that a strictly positive semidefinite tensor is not an \(R_0\) tensor. We focus on the stochastic tensor complementarity problem (STCP), where the expectation of the involved tensor is a strictly positive semidefinite tensor. We denote such an STCP as a monotone STCP. Based on three popular NCP functions, the min function, the Fischer–Burmeister (FB) function and the penalized FB function, as well as the special structure of the monotone STCP, we introduce three new NCP functions and establish the expected residual minimization (ERM) formulation of the monotone STCP. We show that the solution set of the ERM problem is nonempty and bounded if the solution set of the expected value (EV) formulation for such an STCP is nonempty and bounded. Moreover, an approximate regularized model is proposed to weaken the conditions for nonemptiness and boundedness of the solution set of the ERM problem in practice. Numerical results indicate that the performance of the ERM method is better than that of the EV method.

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. Bai, X., Huang, Z., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)

    Article  MathSciNet  Google Scholar 

  2. Che, M., Qi, L., Wei, Y.: Positive-definite tensor to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)

    Article  MathSciNet  Google Scholar 

  3. Che, M., Qi, L., Wei, Y.: Stochastic \(R_0\) tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261–279 (2019)

    Article  MathSciNet  Google Scholar 

  4. Chen, B., Chen, X., Kanzow, C.: A penalized Fischer–Burmeister NCP-function. Math. Program. 88, 211–216 (2000)

    Article  MathSciNet  Google Scholar 

  5. Chen, C., Zhang, L.: Finding Nash equilibrium for a class of multi-person noncooperative games via solving tensor complementarity problem. Appl. Numer. Math. 145, 458–468 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. 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 

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

    Article  MathSciNet  Google Scholar 

  11. Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73, 789–800 (2019)

    Article  MathSciNet  Google Scholar 

  12. Du, S., Che, M., Wei, Y.: Stochastic structured tensors to stochastic complementarity problems. Comput. Optim. Appl. (2019). https://doi.org/10.1007/s10589-019-00144-3

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

    MATH  Google Scholar 

  14. Fang, H., Chen, X., Fukushima, M.: Stochastic \(R_0\) matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)

    Article  MathSciNet  Google Scholar 

  15. Han, J., Xiu, N., Qi, H.: Nonlinear Complementarity Theory and Algorithms. Shanghai Science and Technology Press, Shanghai (2006). (in Chinese)

    Google Scholar 

  16. Huang, Z., Qi, L.: Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017)

    Article  MathSciNet  Google Scholar 

  17. Huang, Z., Qi, L.: Tensor complementarity problems—part I: basic theory. J. Optim. Theory Appl. 183, 1–23 (2019)

    Article  MathSciNet  Google Scholar 

  18. Huang, Z., Qi, L.: Tensor complementarity problems—part III: applications. J. Optim. Theory Appl. (2019). https://doi.org/10.1007/s10957-019-01573-0

  19. Huang, Z., Suo, Y., Wang, J.: On Q-tensors. To appear in Pac. J. Optim. (2018)

  20. Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to Z-tensor complementarity problems. Optim. Lett. 11, 471–482 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Marti, K.: Stochastic Optimization Methods. Springer, Berlin (2005)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Qi, L., Huang, Z.: Tensor complementarity problems-part II: solutionmethods. J. Optim. Theory Appl. 183, 365–385 (2019)

    Article  MathSciNet  Google Scholar 

  25. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. 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 

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

    Book  Google Scholar 

  29. Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)

    Article  MathSciNet  Google Scholar 

  30. Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170, 85–96 (2016)

    Article  MathSciNet  Google Scholar 

  31. Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069–1078 (2016)

    Article  MathSciNet  Google Scholar 

  32. Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Anna. Appl. Math. 33, 308–323 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Song, Y., Qi, L.: Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim. Lett. 11, 1407–1426 (2017)

    Article  MathSciNet  Google Scholar 

  34. Tseng, P.: Growth behavior of a class of merit functions for the nonlinear complementarity problem. J. Optim. Theory Appl. 89, 17–37 (1996)

    Article  MathSciNet  Google Scholar 

  35. Wang, Y., Huang, Z., Qi, L.: Global uniqueness and solvability of tensor variational inequalities. J. Optim. Theory Appl. 177, 137–152 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Our deepest gratitude goes to two anonymous reviewers for their meaningful suggestions that have helped improve this paper substantially.

Funding

Liping Zhang’s work was supported by the National Natural Science Foundation of China (Grant No. 11771244). Liqun Qi’s work was supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15301716 and 15300717).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Zhenyu Ming & Liping Zhang

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

    Liqun Qi

Authors
  1. Zhenyu Ming
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liping Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liqun Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liping Zhang.

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

Ming, Z., Zhang, L. & Qi, L. Expected residual minimization method for monotone stochastic tensor complementarity problem. Comput Optim Appl 77, 871–896 (2020). https://doi.org/10.1007/s10589-020-00222-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-020-00222-x

Keywords

Mathematics Subject Classification

Search

Navigation