[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

The Existence and Uniqueness of Solution for Tensor Complementarity Problem and Related Systems

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

Abstract

In the paper, we are concerned with the existence and uniqueness of solution for tensor complementarity problem (TCP) and tensor absolute value equations (TAVEs) with special structure. First, we give a sufficient condition of P tensors by using the nonsingularity of the relevant tensors and the properties of eigenvalues. In addition, we give the conditions for the existence and uniqueness of solution for the TCP and prove that the TAVEs with a special structure are equivalent to the TCP. Based on these, using the equivalence relationship with the TCP, some sufficient conditions for the existence and uniqueness of solution for the TAVEs with special structure are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  2. Bu, F., Ma, C.: The tensor splitting methods for solving tensor absolute value equation. Comput. Appl. Math. 39, 178 (2020)

    Article  MathSciNet  Google Scholar 

  3. Bu, C., Zhang, X., Zhou, J., Wang, W., Wei, Y.: The inverse, rank and product of tensors. Linear Algebra Appl. 446, 269–280 (2014)

    Article  MathSciNet  Google Scholar 

  4. Caccetta, L., Qu, B., Zhou, G.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45–58 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Gowda, M., Luo, Z., Qi, L., Xiu, N.: \({\cal{Z}}\)-tensors and complementarity problems. arXiv:1510.07933v2 (2016)

  9. Hu, S., Huang, Z., Zhang, Q.: A generalized Newton method for absolute value equations associated with second order cones. J. Comput. Appl. Math. 235, 1490–1501 (2011)

    Article  MathSciNet  Google Scholar 

  10. Hu, S., Huang, Z., Ling, C., Qi, L.: On determinants and eigenvalue theory of tensors. J Symb. Comput. 50, 508–531 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Iqbal, J., Iqbal, A., Arif, M.: Levenberg–Marquardt method for solving systems of absolute value equations. J. Comput. Appl. Math. 282, 134–138 (2015)

    Article  MathSciNet  Google Scholar 

  15. Ke, R., Li, W., Ng, M.: Numerical ranges of tensor. Linear Algebra Appl. 508, 100–132 (2016)

    Article  MathSciNet  Google Scholar 

  16. Ling, C., He, H., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. 63, 143–168 (2016)

    Article  MathSciNet  Google Scholar 

  17. Ling, C., He, H., Qi, L.: Higher-degree eigenvalue complementarity problems for tensors. Comput. Optim. Appl. 64, 149–176 (2016)

    Article  MathSciNet  Google Scholar 

  18. Ling, C., Yan, W., He, H., Qi, L.: Further study on tensor absolute value equations. Sci. China Math. 63, 2137–2156 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Luo, Z., Qi, L., Xiu, X.: The sparsest solutions to Z-tensor complementarity problems. Optim. Lett. 11, 471–482 (2015)

    Article  MathSciNet  Google Scholar 

  21. Mangasarian, O., Meyer, R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)

    Book  Google Scholar 

  24. Rohn, J.: A theorem of the alternatives for the equation \(Ax + B|x| = b\). Linear Multilinear Algebra 52, 421–426 (2004)

    Article  MathSciNet  Google Scholar 

  25. Rohn, J.: An algorithm for solving the absolute value equations. Electron. J. Linear Algebra 18, 589–599 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Sun, M., Liu, J.: General six-step discrete-time Zhang neural network for time-varying tensor absolute value equations. Discrete Dyn. Nat. Soc. 2019, 1–12 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Salkuyeh, D.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8, 2191–2202 (2014)

    Article  MathSciNet  Google Scholar 

  33. Wang, X., Che, M., Wei, Y.: Neural network approach for solving nonsingular multi-linear tensor systems. J. Comput. Appl. Math. 368, 112569 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Wu, S., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169, 705–712 (2016)

    Article  MathSciNet  Google Scholar 

  37. Wu, S., Li, C.: The unique solution of the absolute value equations. Appl. Math. Lett. 76, 195–200 (2018)

    Article  MathSciNet  Google Scholar 

  38. Wu, S., Shen, S.: On the unique solution of the generalized absolute value equation. Optim. Lett. (2020). https://doi.org/10.1007/s11590-020-01672-2

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Zhang, C., Wei, Q.: Global and finite convergence of a generalized newton method for absolute value equations. J. Optim. Theory Appl. 143, 391–403 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the handling Associate Editor and anonymous referees for useful comments and suggestions that contributed to improving the quality of the manuscript.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Lu-Bin Cui & Yu-Dong Fan

  2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Yi-Sheng Song

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

    Shi-Liang Wu

Authors
  1. Lu-Bin Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu-Dong Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi-Sheng Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shi-Liang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shi-Liang Wu.

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 document is the results of the research project funded by National Natural Science Foundations of China (Nos. 12171064, 11601134, 11961082, 17HASTIT012), Foundation of Henan Educational Committee (No. 21A110013), Foundation of Henan Normal University (No. 2021PL03).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cui, LB., Fan, YD., Song, YS. et al. The Existence and Uniqueness of Solution for Tensor Complementarity Problem and Related Systems. J Optim Theory Appl 192, 321–334 (2022). https://doi.org/10.1007/s10957-021-01972-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01972-2

Keywords

Mathematics Subject Classification

Search

Navigation