Skip to main content
SpringerLink
Log in

QN-tensor and tensor complementarity problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we introduce three classes of structured tensor: QN-tensor, S-QN tensor and generalized S-QN tensor, which are proved to be nonsingular \({\mathcal {H}}\)-tensors. Moreover, we present the upper bound for the norm of the solution of TCP \(({{\mathcal {A}}},q)\) defined by a nonsingular \({\mathcal {H}}\)-tensor with positive diagonal entries. The estimation of upper bound requires a diagonal scaling matrix D such that \({{\mathcal {A}}}\cdot D\) is strictly diagonally dominant. When \({{\mathcal {A}}}\) belongs to the set of QN-tensor or S-QN tensor, a choice for the diagonal scaling matrix is given.

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

Data availability

All data included in this study are available upon request by contacting the corresponding author.

References

  1. Dai, P.F., Li, J.C., Li, Y.T., Zhang, C.Y.: Error bounds for linear complementarity problems of QN-matrices. Calcolo 53, 647–657 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Li, J.C., Li, G.: Error bounds for linear complementarity problems of S-QN matrices. Numer. Algorithms 83, 935–955 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Guo, C.H.: A new class of nonsymmetric algebraic Riccati equations. Linear Algebra Appl. 426(2–3), 636–649 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong \({\cal{H}}\)-tensors and general \({\cal{H}}\)-tensors. Linear Algebra Appl. 476, 42–55 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cui, J., Peng, G., Lu, Q., Huang, Z.: New iterative criteria for strong \(\cal{H}\)-tensors and an application. J. Inequal. Appl. 2017(1), 1–16 (2017)

    Article  MathSciNet  Google Scholar 

  6. Li, Y.T., Liu, Q.L., Qi, L.Q.: Programmable criteria for strong \({\cal{H}}\)-tensors. Numer. Algorithm 74(1), 199–221 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang, X.Z., Wei, Y.M.: \({\cal{H}}\)-tensors and nonsingular \({\cal{H}}\)-tensors. Front. Math. China 11(3), 557–575 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhao, R.J., Gao, L., Liu, Q.L., Li, Y.T.: Criterions for identifying \({\cal{H}}\)-tensors. Front. Math. China 11(3), 661–678 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, Y., Zhou, G., Caccetta, L.: Nonsingular \({\cal{H}}\)-tensor and its criteria. J. Ind. Manag. Optim. 12(4), 1173–1186 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang, J., Bu, C.: Nekrasov tensors and nonsingular \({\cal{H}}\)-tensors. Comput. Appl. Math. 37(4), 4917–4930 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding, W., Luo, Z., Qi, L.: P-Tensors, P\(_0\)-Tensors, and tensor complementarity problem (2015). arXiv preprint, arXiv:1507.06731v1

  12. Wang, Y., Huang, Z.H., Bai, X.L.: Exceptionally regular tensors and tensor complementarity problems. Optim. Method Softw. 31(4), 815–828 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170(1), 85–96 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to \({\cal{Z}}\)-tensor complementarity problems. Optim. Lett. 11(3), 471–482 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Song, Y., Qi, L.: Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim. Lett. 11(7), 1407–1426 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu, S., Huang, Z.H., Qi, L.: Strictly nonnegative tensors and nonnegative tensor partition. Sci. China Math. 57(1), 181–195 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhou, J., Sun, L., Wei, Y., Bu, C.: Some characterizations of \({\cal{M}}\)-tensors via digraphs. Linear Algebra Appl. 495, 190–198 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Kolotilina, L.Y.: Bounds for the inverses of generalized Nekrasov matrices. J. Math. Sci. 207(5), 786–794 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cvetković, L., Kostić, V., Rauši, S.: A new subclass of \(H\)-matrices. Appl. Math. Comput. 208, 206–210 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Xu, Y., Gu, W., Huang, Z.H.: Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems. Front. Math. China 14(3), 661–671 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work was supported by National Natural Science Foundation of China (12171384).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Ge Li & Jicheng Li

Authors
  1. Ge Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jicheng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jicheng Li.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

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

Li, G., Li, J. QN-tensor and tensor complementarity problem. Optim Lett 16, 2729–2751 (2022). https://doi.org/10.1007/s11590-022-01850-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-022-01850-4

Keywords

Search

Navigation