Skip to main content
SpringerLink
Log in

Lower bounds of the solution set of the polynomial complementarity problem

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

Abstract

In the present paper, the lower bounds of the solution set of the polynomial complementarity problem (PCP) are investigated. PCP is to find a vector \({\varvec{x}}\in {\mathbb {R}}^{n}\) such that \({\varvec{x}}\ge {\varvec{0}}\), \({\mathcal {A}}_{1}{\varvec{x}}^{m-1}+{\mathcal {A}}_{2}{\varvec{x}}^{m-2}+ \ldots +{\mathcal {A}}_{m-1}{\varvec{x}}+{\varvec{q}}\ge {\varvec{0}}\) and \({\varvec{x}}^{\top }({\mathcal {A}}_{1}{\varvec{x}}^{m-1}+ {\mathcal {A}}_{2}{\varvec{x}}^{m-2}+\ldots +{\mathcal {A}}_{m-1}{\varvec{x}}+{\varvec{q}})=0\) with \({\mathcal {A}}_{l}\) being a given \((m+1-l)\)-order n-dimensional tensor for each \(l=1,2,\ldots ,m-1\) and a given vector \({\varvec{q}}\in {\mathbb {R}}^{n}\). It is easy to see that if \({\varvec{q}}\in {\mathbb {R}}^{n}_{+}\), then 0 is a tight lower bound on the norm of the solutions of PCP. To that end, for \({\varvec{q}}\in {\mathbb {R}}^{n}\backslash {\mathbb {R}}^{n}_{+}\), two new classes of tensor tuples, that is, \(l_{0}\)-\({\varvec{q}}\) tensor tuples and \(\alpha\)-\({\varvec{q}}\) tensor tuples, are introduced. Based on the structured tensor tuples, the lower bound formulas of the solution set of PCP are derived. The formulas presented in this paper are extensions of the formula proposed by Xu and Huang (Optim. Lett. 15: 2701-2718, 2021) from the tensor complementarity problem (TCP) to PCP. A comparison between the results in this paper and the corresponding result due to Ling, He and Ling (Optimization. 67: 341-358, 2018) is made.

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.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. 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  Google Scholar 

  3. Balaji, R., Palpandi, K.: Positive definite and Gram tensor complementarity problems. Optim. Lett. 12(3), 639–648 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Che, M.L., Qi, L.Q., Wei, Y.M.: Stochastic \(R_{0}\) tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261–279 (2019)

    Article  MathSciNet  Google Scholar 

  6. Che, M.L., Qi, L.Q., Wei, Y.M.: The generalized order tensor complementarity problems. Numer. Math. Theor. Methods Appl. 13, 131–149 (2020)

    Article  MathSciNet  Google Scholar 

  7. Chen, M.L., Qi, L.Q., Song, Y.S.: Column sufficient tensors and tensor complementarity problems. Front. Math. China 13(2), 225–276 (2018)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  9. Cui, L.B., Fan, Y.D., Song, Y.S., et al.: The existence and uniqueness of solution for tensor complementarity problem and related systems. J. Optim. Theor. Appl. 192, 321–334 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Ding, W.Y., Luo, Z.Y., Qi, L.Q.: P-tensors, \(P_{0}\)-tensors, and their applications. Linear Algebra Appl. 555, 336–354 (2018)

    Article  MathSciNet  Google Scholar 

  13. Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer-Verlag, New York (2003)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  15. Guo, Q., Zheng, M.M., Huang, Z.H.: Properties of S-tensors. Linear Multilinear Algebra 67(4), 685–696 (2019)

    Article  MathSciNet  Google Scholar 

  16. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequalities and nonlinear complementarity problems: a survey of the theory, algorithms and applications. Math. Program. 48, 161–220 (1990)

    Article  Google Scholar 

  17. Hu, S.L., Wang, J., Huang, Z.H.: Error bounds for the solution sets of quadratic complementarity problems. J. Optim. Theor. Appl. 179, 983–1000 (2018)

    Article  MathSciNet  Google Scholar 

  18. Hieu, V.T.: On the \(R_{0}\)-tensors and the solution map of tensor complementarity problems. J. Optim. Theor. Appl. 181, 163–183 (2019)

    Article  Google Scholar 

  19. Hieu, V.T., Wei, Y.M., Yao, J.C.: Notes on the optimization problems corresponding to polynomial complementarity problems. J. Optim. Theor. Appl. 184, 687–695 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Huang, Z.H., Qi, L.Q.: Tensor complementarity problems-part III: application. J. Optim. Theor. Appl. 183, 771–791 (2019)

    Article  MathSciNet  Google Scholar 

  23. 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  Google Scholar 

  24. Li, G., Li, J.C.: QN-tensor and tensor complementarity problem. Optim. Lett 16(9), 2729–2751 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  26. Liu, X., Huang, G.X.: New error bounds for the tensor complementarity problem. ERA 30(6), 2196–2204 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Palpandi, K., Sharma, S.: Tensor complementarity problems with finite solution sets. J. Optim. Theor. Appl. 190, 951–965 (2021)

    Article  MathSciNet  Google Scholar 

  29. Song, Y.S., Mei, W.: Structural properties of tensor and complementarity problems. J. Optim. Theor. Appl. 176(2), 289–305 (2018)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Song, Y.S., Qi, L.Q.: Tensor complementarity problem and semi-positive tensor. J. Optim. Theor. Appl. 169, 1069–1078 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Xu, Y., Gu, W.Z., 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  Google Scholar 

  37. Wang, J., Hu, S.L., Huang, Z.H.: Solution sets of quadratic complementarity problems. J. Optim. Theor. Appl. 176, 120–136 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Wang, X.Z., Che, M.L., Wei, Y.M.: Randomized Kaczmarz methods for tensor complementarity problems. Comput. Optim. Appl. 82, 595–615 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the editor and the reviewers for their helpful comments and suggestions, which have improved the presentation of the paper. This work was partially supported by National Natural Science Foundation of China (No.11961006) and Guangxi Natural Science Foundation (No. 2020GXNSFAA159100).

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Guangxi Minzu University, Guangxi, 530006, People’s Republic of China

    Xue-liu Li

  2. School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, People’s Republic of China

    Tong-tong Shang

  3. School of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University, Guangxi, 530006, People’s Republic of China

    Guo-ji Tang

Authors
  1. Xue-liu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tong-tong Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guo-ji Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo-ji Tang.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflicts of interest.

Additional information

Publisher's Note

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

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, Xl., Shang, Tt. & Tang, Gj. Lower bounds of the solution set of the polynomial complementarity problem. Optim Lett 18, 497–511 (2024). https://doi.org/10.1007/s11590-023-02004-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-023-02004-w

Keywords

Search

Navigation