Skip to main content
SpringerLink
Log in

Bounds of the solution set of the tensor complementarity problem

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

Abstract

Recently, the estimation problem of upper and lower bounds of the solution set of the tensor complementarity problem has been studied when the tensor involved is a strictly semi-positive tensor or one of its subclasses. This paper aims to study such an estimation problem in a larger scope. First, we propose a lower bound formula under the condition that the tensor complementarity problem has a solution. When the problem under consideration falls back to several types of problems that have been studied, the achieved result improves the relevant known results. Second, by means of a newly introduced quantity, we give an upper bound formula of the solution set when the problem has a solution and the tensor involved is an \(R_0\)-tensor. This formula is new even when the concerned problem falls back to several problems that have already been discussed. Several examples are also given to confirm our theoretical findings.

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Huang, Z.H., 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33(3), 308–323 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding, W., Luo, Z., Qi, L.: \(P\)-tensors, \(P_0\)-tensors, and their applications. Linear Algebra Appl. 555, 336–354 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Yu, W., Ling, C., He, H.: On the properties of tensor complementarity problems. Pac. J. Optim. 14(4), 675–691 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Bai, X.L., Huang, Z.H., Li, X.: Stability of solutions and continuity of solution maps of tensor complementarity problems. Asia Pac. J. Oper. Res. 36(2), 1940002 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hieu, V.T.: On the \(R_0\)-tensors and the solution map of tensor complementarity problems. J. Optim. Theory Appl. 181, 163–183 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Gowda, M.S., Luo, Z., Qi, L., Xiu, N.: \(Z\)-tensors and complementarity problems (2016). arXiv:1510.07933v2

  13. Liu, D.D., Li, W., Vong, S.W.: Tensor complementarity problems: the GUS-property and an algorithm. Linear Multilinear Algebra 66(9), 1726–1749 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Balaji, R., Palpandi, K.: Positive definite and Gram tensor complementarity problems. Optim. Lett. 12, 639–648 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. Chen, H.B., Qi, L., Song, Y.: Column sufficient tensors and tensor complementarity problems. Front. Math. China 13(2), 255–276 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, X., Che, M., Wei, Y.: Global uniqueness and solvability of tensor complementarity problems for \({\cal{H}}_+\)-tensors. Numer. Algor. 84, 567–590 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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  MATH  Google Scholar 

  19. Guo, Q., Zheng, M.M., Huang, Z.H.: Properties of \(S\)-tensors. Linear Multilinear Algebra 67, 685–696 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, Y., Huang, Z.H., Hu, S.: Connectedness of the solution set of the tensor complementarity problem. J. Math. Anal. Appl. 487(1), Article 123965 (2020)

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Xie, S.L., Li, D.H., Xu, H.R.: An iterative method for finding the least solution to the tensor complementarity problem. J. Optim. Theory Appl. 175(1), 119–136 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Global Optim. 73(4), 789–800 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, X., Che, M., Qi, L., Wei, Y.: Gradient dynamic approach to the tensor complementarity problem. Optim. Method Softw. 35(2), 394–415 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhao, X., Fan, J.: A semidefinite method for tensor complementarity problems. Optim. Method Softw. 34(4), 758–769 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, K.L., Chen, H.B., Zhao, P.F.: A potential reduction method for tensor complementarity problems. J. Ind. Manag. Optim. 15(2), 429–443 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xu, H.R., Li, D.H., Xie, S.L.: An equivalent tensor equation to the tensor complementarity problem with positive semi-definite \(Z\)-tensor. Optim. Lett. 13, 685–694 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hu, S., Wang, J., Huang, Z.H.: An inexact augmented Lagrangian multiplier method for solving quadratic complementary problems: an adapted algorithmic framework combining specific resolution techniques. J. Comput. Appl. Math. 361, 64–78 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Guan, H.B., Li, D.H.: Linearized methods for tensor complementarity problems. J. Optim. Theory Appl. 184, 972–987 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qi, L., Huang, Z.H.: Tensor pomplementarity problems-part II: solution methods. J. Optim. Theory Appl. 183(2), 365–385 (2019)

    Article  MathSciNet  Google Scholar 

  32. Gowda, M.S.: Polynomial complementarity problems. Pac. J. Optim. 13, 227–241 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Zheng, M.M., Huang, Z.H., Ma, X.X.: Nonemptiness and compactness of solution sets to generalized polynomial complementarity problems. J. Optim. Theory Appl. 185, 80–98 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, X.Y., Chen, H.B., Wang, Y.J.: On the solution existence of Cauchy tensor variational inequality problems. Pac. J. Optim. 14(3), 479–487 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Gowda, M.S., Sossa, D.: Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones. Math. Program. 177, 149–171 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ma, X.X., Zheng, M.M., Huang, Z.H.: A note on the nonemptiness and compactness of solution sets of weakly homogeneous variational inequalities. SIAM J. Optim. 30, 132–148 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mathias, R., Pang, J.S.: Error bounds for the linear complementarity problem with a \(P\)-matrix. Linear Algebra Appl. 132, 123–136 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  40. Chen, X.J., Xiang, S.H.: Perturbation bounds of P-matrix linear complementarity problems. SIAM J. Optim. 18(4), 1250–1265 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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 

  42. 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 

  43. Song, Y., Mei, W.: Structural properties of tensors and complementarity problems. J. Optim. Theory Appl. 176(2), 289–305 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. 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 

  45. Mei, W., Yang, Q.: Properties of structured tensors and complementarity problems. J. Optim. Theory Appl. 185, 99–114 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11871051).

Author information

Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300072, P.R. China

    Yang Xu & Zheng-Hai Huang

Authors
  1. Yang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zheng-Hai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zheng-Hai Huang.

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

Xu, Y., Huang, ZH. Bounds of the solution set of the tensor complementarity problem. Optim Lett 15, 2701–2718 (2021). https://doi.org/10.1007/s11590-020-01697-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-020-01697-7

Keywords

Search

Navigation