Skip to main content
SpringerLink
Log in

Stochastic Tensor Complementarity Problem with Discrete Distribution

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

Abstract

Stochastic tensor complementarity problem with discrete distribution is investigated, which is a kind of stochastic tensor complementarity problem with discrete probability distribution variables. First, we formulate the stochastic tensor complementarity problem with discrete distribution as a constrained minimization problem. Some properties of this reformulation are studied based on the structured tensor. Then we propose a new semismooth Newton method for solving this problem. The proposed method combines the semismooth Newton method with the Barzilai–Borwein stepsize technique. In addition, the method uses the nonmonotone linesearch technique to ensure its global convergence. Any accumulation point of the sequence generated by the proposed method approximates to a solution of the stochastic tensor complementarity problem with discrete distribution. Finally, numerical results are given to verify 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

Similar content being viewed by others

References

  1. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Birgin, E.G., Martinez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Che, M., Qi, L., Wei, Y.: Stochastic R\(_0\) tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261–279 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Che, M., Wei, Y.: Theory and Computation of Complex Tensors and Its Applications. Springer, Singapore (2020)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. 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 

  10. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  11. Ding, W., Luo, Z., Qi, L.: P-Tensors, P\(_0\)-tensors, and their application. Linear Algebra and Its Applications 555, 336–354 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Du, S., Che, M., Wei, Y.: Stochastic structured tensors to stochastic complementarity problems. Comput. Optim. Appl. 75, 649–668 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Du, S., Ding, W., Wei, Y.: Acceptable solutions and backward errors for tensor complementarity problems. J. Optim. Theory Appl. 188, 260–276 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Du, S., Zhang, L., Chen, C., Qi, L.: Tensor absolute value equations. Sci. China Math. 61, 1695–1710 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Program. 76, 513–532 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Han, L.: A continuation method for tensor complementarity problems. J. Optim. Theory Appl. 180, 949–963 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang, Z., Qi, L.: Tensor complementarity problems-part III: applications. J. Optim. Theory Appl. 183, 771–791 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jiang, H., Qi, L.: A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control Optim. 35, 178–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, D., Chen, C., Guan, H.: A lower dimensional linear equation approach to the \({\cal{M}}\)-tensor complementarity problem. Calcolo 58, Article 5 (2021). https://doi.org/10.1007/s10092-021-00397-7

  24. Li, H., Du, S., Wang, Y., Chen, M.: An inexact Levenberg–Marquardt method for tensor eigenvalue complementarity problem. Pac. J. Optim. 16, 87–99 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Liu, D., Li, W., Vong, S.W.: Tensor complementarity problem: the GUS-property and an algorithm. Linear Multilinear Algebra 66, 1726–1749 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ming, Z., Zhang, L., Qi, L.: Expected residual minimization method for monotone stochastic tensor complementarity problem. Comput. Optim. Appl. 77, 871–896 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ni, Q., Qi, L.: A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map. J. Glob. Optim. 61, 627–641 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Qi, L.: Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 439, 228–238 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Qi, L., Jiang, H.: Semismooth Karush–Kuhn–Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22, 301–325 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

  34. Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10. North-Holland, Amsterdam (2003)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 03, 92–107 (2017)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Sun, D., Han, J.: Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7, 463–480 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tan, C., Ma, S., Dai, Y., Qian, Y.: Barzilai–Borwein step size for stochastic gradient descent. In: Lee, D. D., von Luxburg, U. (eds.) NIPS’16: proceedings of the 30th international conference on neural information processing systems, pp. 685–693 (2016)

  41. Wang, X., Che, M., Qi, L., Wei, Y.: Modified gradient dynamic approach to the tensor complementarity problem. Optim. Methods Softw. 35, 394–415 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, Y., Huang, Z., Bai, X.: Exceptionally regular tensors and tensor complementarity problems. Optim. Methods Softw. 31, 815–828 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Xie, S., Li, D., Xu, H.: An iterative method for finding the least solution to the tensor complementarity problem. J. Optim. Theory Appl. 175, 119–136 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. Xu, Y., Huang, Z.: Bounds of the solution set of the tensor complementarity problem. Optim. Lett. 15, 2701–2718 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the handling editor Liqun Qi and Prof. Sanzheng Qiao for their detailed comments on the presentation of our paper. The authors are also grateful to three anonymous referees for their many helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, People’s Republic of China

    Shouqiang Du, Liyuan Cui & Yuanyuan Chen

  2. School of Mathematical Science and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

Authors
  1. Shouqiang Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liyuan Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuanyuan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yimin Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Liqun Qi.

Publisher's Note

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

S. Du: This author is supported by the National Natural Science Foundation of China under Grant 11671220. Y. Wei: This author is supported by Innovation Program of Shanghai Municipal Education Commission and the National Natural Science Foundation of China under Grant 11771099.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, S., Cui, L., Chen, Y. et al. Stochastic Tensor Complementarity Problem with Discrete Distribution. J Optim Theory Appl 192, 912–929 (2022). https://doi.org/10.1007/s10957-021-01997-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01997-7

Keywords

Mathematics Subject Classification

Search

Navigation