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Properties of Structured Tensors and Complementarity Problems

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Abstract

In this paper, we present some new results on a class of tensors, which are defined by the solvability of the corresponding tensor complementarity problem. For such structured tensors, we give a sufficient condition to guarantee the nonzero solution of the corresponding tensor complementarity problem with a vector containing at least two nonzero components and discuss their relationships with some other structured tensors. Furthermore, with respect to the tensor complementarity problem with a nonnegative such structured tensor, we obtain the upper and lower bounds of its solution set, and by the way, we show that the eigenvalues of such a tensor are closely related to this solution set.

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References

  1. Gowda, M.S.: On Q-matrices. Math. Program. 49, 139–141 (1990)

    Article  Google Scholar 

  2. Aganagic, M., Cottle, R.W.: A note on Q-matrices. Math. Program. 16, 374–377 (1979)

    Article  Google Scholar 

  3. Pang, J.S.: On Q-matrices. Math. Program. 17, 243–247 (1979)

    Article  Google Scholar 

  4. Murthy, G.S.R., Parthasarathy, T.: Some properties of fully semimonotone \(Q_{0}\)-matrices. SIAM J. Matrix Anal. Appl. 16, 1268–1286 (1995)

    Article  MathSciNet  Google Scholar 

  5. Danao, R.A.: Q-matrices and boundedness of solutions to linear complementarity problems. J. Optim. Theory Appl. 83(2), 321–332 (1994)

    Article  MathSciNet  Google Scholar 

  6. Cottle, R.W.: Completely Q-matrices. Math. Program. 19, 347–351 (1980)

    Article  Google Scholar 

  7. 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 

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

    Article  MathSciNet  Google Scholar 

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

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

  11. Murthy, G.S.R., Parthasarathy, T., Ravindran, G.: A copositive Q-matrix which is not \(R_{0}\). Math. Program. 61, 131–135 (1993)

    Article  Google Scholar 

  12. Jeter, M.W., Pye, W.C.: Some remarks on copositive Q-matrices and on a conjecture on Pang. Ind. Math. 35, 75–80 (1985)

    MathSciNet  MATH  Google Scholar 

  13. Murthy, G.S.R., Parthasarathy, T., Ravindran, G.: On copositive, semimonotone Q-matrices. Math. Program. 68, 187–203 (1995)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  18. 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 

  19. 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 

  20. 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 

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

    Article  MathSciNet  Google Scholar 

  22. 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(3), 661–671 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. 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  Google Scholar 

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

  26. Zheng, Y., Wu, W.: On a class of semi-positive tensors in tensor complementarity problem. J. Optim. Theory Appl. 177, 127–136 (2018)

    Article  MathSciNet  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 China (Grant No. 11671217).

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  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Wei Mei & Qingzhi Yang

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  1. Wei Mei
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  2. Qingzhi Yang
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Correspondence to Qingzhi Yang.

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Communicated by Liqun Qi.

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Mei, W., Yang, Q. Properties of Structured Tensors and Complementarity Problems. J Optim Theory Appl 185, 99–114 (2020). https://doi.org/10.1007/s10957-020-01631-y

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  • DOI: https://doi.org/10.1007/s10957-020-01631-y

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