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Structural Properties of Tensors and Complementarity Problems

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Abstract

In this paper, one of our main purposes is to prove the boundedness of the solution set of tensor complementarity problems such that the specific bounds depend only on the structural properties of such a tensor. To achieve this purpose, firstly, we prove that this class of structured tensors is strictly semi-positive. Subsequently, the strictly lower and upper bounds of operator norms are given for two positively homogeneous operators. Finally, with the help of the above upper bounds, we show that the solution set of tensor complementarity problems has the strictly lower bound. Furthermore, the upper bounds of spectral radius are obtained, which depends only on the principal diagonal entries of tensors.

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, W., Luo, Z., Qi, L.: P-Tensors, \(\text{P}_{0}\)-Tensors, and tensor complementarity problem. arXiv:1507.06731v1 (2015)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, Z., Suo, S., Wang, J.: On Q-tensors. Pac. J. Optim. arXiv:1509.03088 (2015)

  13. Song, Y., Qi, L.: Eigenvalue analysis of constrained minimization problem for homogeneous polynomial. J. Global Optim. 64(3), 563–575 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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  MATH  Google Scholar 

  15. Ling, C., He, H., Qi, L.: Higher-degree eigenvalue complementarity problems for tensors. Comput. Optim. Appl. 64(1), 149–176 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, Z., Yang, Q., Ye, L.: Generalized eigenvalue complementarity problem for tensors. Pac. J. Optim. 13(3), 527–545 (2017)

    MathSciNet  Google Scholar 

  17. Chen, T., Li, W., Wu, X., Vong, S.: Error bounds for linear complementarity problems of MB-matrices. Numer. Algorithm 70(2), 341–356 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dai, P.: Error bounds for linear complementarity problems of DB-matrices. Linear Algebra Appl. 434, 830–840 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dai, P., Li, Y., Lu, C.: Error bounds for linear complementarity problems of SB-matrices. Numer. Algorithm 61(1), 121–139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dai, P., Lu, C., Li, Y.: New error bounds for linear complementarity problems with an SB-matrices. Numer. Algorithm 64(4), 741–757 (2013)

    Article  MATH  Google Scholar 

  21. García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems for B-matrices. Appl. Math. Lett. 22(7), 1071–1075 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, H., Wang, Y.: Further discussion on the error bound for generalized linear complementarity problem over a polyhedral cone. J. Optim. Theory Appl. 159(1), 93–107 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. Society for Industrial and Applied Mathematics, Philadelphia (2017)

    Book  MATH  Google Scholar 

  25. Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM. J. Matrix Anal. Appl. 34(2), 437–452 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, C., Li, Y.: Double B-tensors and quasi-double B-tensors. Linear Algebra Appl. 466, 343–356 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Song, Y., Qi, L.: Infinite and finite dimensional Hilbert tensors. Linear Algebra Appl. 451, 1–14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Song, Y., Qi, L.: Infinite dimensional Hilbert tensors on spaces of analytic functions. Commun. Math. Sci. 15(7), 1897–1911 (2017)

    Article  MathSciNet  Google Scholar 

  30. Mei, M., Song, Y.: Infinite and finite dimensional generalized Hilbert tensors. Linear Algebra Appl. 532(1), 8–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Peña, J.M.: A class of P-matrices with applications to the localization of the eigenvalues of a real matrix. SIAM. J. Matrix Anal. Appl. 22, 1027–1037 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Peña, J.M.: On an alternative to Gerschgorin circles and ovals of cassini. Numer. Math. 95, 337–345 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yuan, P., You, L.: Some remarks on P,\(\text{ P }_{0}\), B and \(\text{ B }_{0}\) tensors. Linear Algebra Appl. 459(3), 511–521 (2014)

    Article  MathSciNet  Google Scholar 

  34. Qi, L., Song, Y.: An even order symmetric B-tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Qi, L.: Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines. J. Symb. Comput. 41, 1309–1327 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the 1st IEEE International Workshop on Computational Advances of Multi-Tensor Adaptive Processing, pp. 129-132, Dec 13–15 (2005)

  38. Song, Y., Qi, L.: Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J. Matrix Anal. Appl. 34, 1581–1595 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees/editors for their valuable suggestions which helped us to improve this manuscript. This work was supported by the National Natural Science Foundation of P.R. China (Grant Nos. 11571095, 11601134, 11701154).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Normal University, XinXiang, HeNan, People’s Republic of China

    Yisheng Song & Wei Mei

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  1. Yisheng Song
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  2. Wei Mei
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Correspondence to Yisheng Song.

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Communicated by Liqun Qi.

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Song, Y., Mei, W. Structural Properties of Tensors and Complementarity Problems. J Optim Theory Appl 176, 289–305 (2018). https://doi.org/10.1007/s10957-017-1212-2

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  • DOI: https://doi.org/10.1007/s10957-017-1212-2

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