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An equivalent tensor equation to the tensor complementarity problem with positive semi-definite Z-tensor

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Abstract

We are concerned with the tensor complementarity problem with positive semi-definite Z-tensor. Under the assumption that the problem has a solution at which the strict complementarity holds, we show that the problem is equivalent to a system of lower dimensional tensor equations. It provides a way to get a solution of the complementarity problem via solving a system of lower dimensional tensor equations. The results obtained in the paper improve the existing results even for the linear complementarity problem. Our preliminary numerical results positively support the results of the paper.

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References

  1. 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 

  2. Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algorithms 62, 59–77 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Ding, W., Luo, Z., Qi, L.: \(P\)-tensors, \(P_0\)-tensors, and tensor complementarity problem. arXiv preprint, arXiv:1507.06731 (2015)

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, W., Wei, Y.: Solving multi-linear systems with \({{\cal{M}}}\)-tensors. J. Sci. Comput. 68, 689–715 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gowda, M.S., Luo, Z., Qi, L., Xiu, N.: Z-tensors and complementarity problems. arXiv preprint, arXiv:1510.07933 (2015)

  9. Harker, P.T., Xiao, B.: Newton’s method for the nonlinear complementarity problem: a B-differentiable equation approach. Math. Program. 48, 339–357 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. Huang, Z.H., Suo, Y., Wang, J.: On \(Q\)-tensors. arXiv preprint, arXiv:1509.03088 (2015)

  12. Liu, D., Li, W., Vong, S.-W.: Tensor complementarity problems: the GUS-property and an algorithm. Linear Multilinear Algebra (2017). https://doi.org/10.1080/03081087.2017.1369929

  13. Luca, T.D., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Luo, Z.Y., Qi, L., Xiu, N.H.: The sparse solutions to Z-tensor complementarity problems. Optim. Lett. 11, 471–482 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Song, Y., Qi, L.: Eigenvalues and structured properties of P-tensors. arXiv preprint, arXiv:1508.02005v3 (2015)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Wei, Y., Ding, W.: Theory and Computation of Tensors: Multi-Dimensional Arrays. Academic Press, Amsterdam (2016)

    MATH  Google Scholar 

  23. Zhang, L., Qi, L., Zhou, G.: \(M\)-tensors and some applications. SIAM J. Matrix Anal. Appl. 35, 437–452 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, L.-L.: Two-step modulus based matrix iteration for linear complementarity problems. Numer. Algorith. 57, 83–99 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, L.-L.: Two-stage multisplitting iteration methods using modulus-based matrix splitting as inner iteration for linear complementarity problems. J. Optim. Theory Appl. 160, 189–203 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zheng, H., Li, W., Qu, W.: A non-modulus linear method for solving the linear complementarity problem. Linear Algebra Appl. 495, 38–50 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zheng, H., Li, W.: The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J. Comput. Appl. Math. 288, 116–126 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Supported by the Chinese NSF Grant 11371154, 11601188, 11771157, by training program for outstanding young teachers in Guangdong Province (Grant No. 20140202), and by Educational Commission of Guangdong Province, China (Grant No. 2014KQNCX210).

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Authors and Affiliations

  1. School of Mathematics, Jiaying University, Meizhou, 514015, China

    Hong-Ru Xu & Shui-Lian Xie

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Dong-Hui Li & Shui-Lian Xie

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  1. Hong-Ru Xu
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  2. Dong-Hui Li
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Correspondence to Shui-Lian Xie.

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Xu, HR., Li, DH. & Xie, SL. An equivalent tensor equation to the tensor complementarity problem with positive semi-definite Z-tensor. Optim Lett 13, 685–694 (2019). https://doi.org/10.1007/s11590-018-1268-4

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