Skip to main content
SpringerLink
Log in

Some Perturbation Bounds of the Tensor Complementarity Problem

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

Abstract

In this paper, we introduce some constants with the tensors of special structures and present their some useful properties. Furthermore, some perturbation bounds of the tensor complementarity problem are obtained on the base of these constants.

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

Data Availability

We do not analyze or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

References

  1. Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170(1), 72–84 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Chen, X.J., Xiang, S.H.: Perturbation bounds of P-matrix linear complementarity problems. SIAM J. Optim. 18(4), 1250–1265 (2008)

    Article  MathSciNet  Google Scholar 

  4. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)

    Google Scholar 

  5. Dai, P.F., Wu, S.L.: The GUS-property and modulus-based methods for tensor complementarity problems. J. Optim. Theory Appl. 195, 976–1006 (2022)

    Article  MathSciNet  Google Scholar 

  6. Ding, W.Y., Qi, L., Wei, Y.M.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439(10), 3264–3278 (2013)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. Gowda, M.S., Luo, Z.Y., Qi, L., Xiu, N.H.: Z-tensors and complementarity problems. arXiv preprint arXiv:1510.07933 (2015)

  9. Gabriel, S.A., Moré, J.J.: Smoothing of mixed complementarity problems. Complement. Var. Probl. State Art 92, 105–116 (1997)

    MathSciNet  Google Scholar 

  10. Huang, Z.H., Qi, L.: Formulating an n-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66(3), 557–576 (2017)

    Article  MathSciNet  Google Scholar 

  11. Huang, Z.H., Suo, Y.Y., Wang, J.: On Q-tensors. Pac. J. Optim. 16(1), 67–86 (2020)

    MathSciNet  Google Scholar 

  12. Ling, L.Y., He, H.J., Ling, C.: On error bounds of polynomial complementarity problems with structured tensors. Optimization 67(2), 341–358 (2018)

    Article  MathSciNet  Google Scholar 

  13. Liu, D.D., Li, W., Vong, S.W.: The tensor splitting with application to solve multi-linear systems. J. Comput. Appl. Math. 330, 7–94 (2018)

    Article  MathSciNet  Google Scholar 

  14. Luo, Z.Y., Qi, L., Xiu, N.H.: The sparsest solutions to Z-tensor complementarity problems. Optim. Lett. 11(3), 471–482 (2017)

    Article  MathSciNet  Google Scholar 

  15. Pearson, K.: Essentially positive tensors. Int. J. Algebra 4(9–12), 421–427 (2010)

    MathSciNet  Google Scholar 

  16. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  Google Scholar 

  17. Shao, J.Y.: A general product of tensors with applications. Linear Algebra Appl. 439(8), 2350–2366 (2013)

    Article  MathSciNet  Google Scholar 

  18. Song, Y.S., Qi, L.: Error bound of P-tensor nonlinear complementarity problem. arXiv preprint arXiv:1508.02005 (2015)

  19. Song, Y.S., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165(3), 854–873 (2015)

    Article  MathSciNet  Google Scholar 

  20. Song, Y.S., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169(3), 1069–1078 (2016)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  22. Wang, X.Z., Che, M.L., Wei, Y.M.: Global uniqueness and solvability of tensor complementarity problems for \(H_{+}\)-tensors. Numer. Algorithms 84, 567–590 (2020)

    Article  MathSciNet  Google Scholar 

  23. Wang, X.Z., Che, M.L., Wei, Y.M.: Preconditioned tensor splitting AOR iterative methods for H-tensor equations. Numer. Linear Algebra Appl. 27(6), e2329 (2020)

    Article  MathSciNet  Google Scholar 

  24. Wu, S.L., Li, W., Wang, H.H.: The perturbation bound of the extended vertical linear complementarity problem. J. Oper. Res. Soc. China (2023). https://doi.org/10.1007/s40305-023-00456-6

    Article  Google Scholar 

  25. Yuan, P.Z., You, L.H.: Some remarks on P, \({\rm P_{0}}\), B and \({\rm B_{0}}\) tensors. Linear Algebra Appl. 459, 511–521 (2014)

    Article  MathSciNet  Google Scholar 

  26. Zheng, M.M., Zhang, Y., Huang, Z.H.: Global error bounds for the tensor complementarity problem with a P-tensor. J. Ind. Manag. Optim. 15(2), 933–946 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank four referees, whose opinions and comments improved the presentation of the paper greatly.

Author information

Authors and Affiliations

  1. School of Mathematics, Yunnan Normal University, Kunming, 650500, Yunnan, People’s Republic of China

    Li-Ming Li & Shi-Liang Wu

  2. Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming, 650500, Yunnan, People’s Republic of China

    Li-Ming Li & Shi-Liang Wu

  3. School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, Guangdong, People’s Republic of China

    Ping-Fan Dai

Authors
  1. Li-Ming Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shi-Liang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ping-Fan Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shi-Liang Wu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Liqun Qi.

Publisher's Note

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

This research was supported by National Natural Science Foundations of China (No. 11961082).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) 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

Li, LM., Wu, SL. & Dai, PF. Some Perturbation Bounds of the Tensor Complementarity Problem. J Optim Theory Appl 201, 825–842 (2024). https://doi.org/10.1007/s10957-024-02420-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-024-02420-7

Keywords

Mathematics Subject Classification

Search

Navigation