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Linearized Methods for Tensor Complementarity Problems

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Abstract

In this paper, we first propose a linearized method for solving the tensor complementarity problem. The subproblems of the method can be solved by solving linear complementarity problems with a constant matrix. We show that if the initial point is appropriately chosen, then the generated sequence of iterates converges to a solution of the problem monotonically. We then propose a lower-dimensional equation method and establish its monotone convergence. The subproblems of the method are lower-dimensional systems of linear equations. At last, we do numerical experiments to test the proposed methods. The results show the efficiency of the proposed methods.

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References

  1. Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33(03), 308–323 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Huang, Z.H., Qi, L.: Tensor Complementarity Problems-Part III: Applications. J. Optim. Theory Appl. 183(3), 771–791 (2019)

    Article  MathSciNet  Google Scholar 

  4. 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 

  5. Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169(3), 1069–1078 (2016)

    Article  MathSciNet  Google Scholar 

  6. Wang, Y., Huang, Z., Bai, X.: Exceptionally regular tensors and tensor complementarity peoblems. Optim. Methods Softw. 31(4), 815–828 (2016)

    Article  MathSciNet  Google Scholar 

  7. Wang, X.Y., Chen, H.B., Wang, Y.J.: On the solution existence of Cauchy tensor variational inequality problems. Pac. J. Optim. 14(3), 479–487 (2018)

    MathSciNet  Google Scholar 

  8. Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168(2), 475–487 (2016)

    Article  MathSciNet  Google Scholar 

  9. Wang, X., Che, M., Wei, Y.: Existence and uniqueness of positive solution for \(H^+\)-tensor equations. Appl. Math. Lett. 98, 191–198 (2019)

    Article  MathSciNet  Google Scholar 

  10. 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 

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

  12. Liu, 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 

  13. Wang, X., Che, M., Wei, Y.: Global uniqueness and solvability of tensor complementarity problems for \(H^+\)-tensors. Numer. Algorithms (2019). https://doi.org/10.1007/s11075-019-00769-9

    Article  Google Scholar 

  14. Mohamed, T.A., Rahmati, S.: Complementarity problems over a hypermatrix (tensor) set. Optim. Lett. 12(6), 1443–1454 (2018)

    Article  MathSciNet  Google Scholar 

  15. Ling, L., He, H., Ling, C.: On error bounds of polynomial complementarity problems with structured tensors. Optimization 67(2), 341–358 (2018)

    Article  MathSciNet  Google Scholar 

  16. Song, Y., Qi, L.: Error bound of P-tensor nonlinear complementarity problem (2015). arXiv:1508.02005v2

  17. Yu, W., Ling, C., He, H.: On the properties of tensor complementarity problems. Pac. J. Optim. 14(4), 675–691 (2018)

    MathSciNet  Google Scholar 

  18. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and their Applications, vol. 39 of Advances in Mechanics and Mathematics. Springer, Singapore (2018)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Xie, S.L., Li, D.H., Xu, H.R.: An iterative method for finding the least solution to the tensor complementarity problem. J. Optim. Theory Appl. 175(1), 119–136 (2017)

    Article  MathSciNet  Google Scholar 

  21. Han, L.: A continuation method for tensor complementarity problems. J. Optim. Theory Appl. (2018). https://doi.org/10.1007/s10957-018-1422-2

    Article  Google Scholar 

  22. Du, S., Zhang, L., Chen, C., Qi, L.: Tensor absolute value equations. Sci. China Math. (2018). https://doi.org/10.1007/s11425-017-9238-6

    Article  MathSciNet  MATH  Google Scholar 

  23. Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Global Optim. 73(4), 789–800 (2019)

    Article  MathSciNet  Google Scholar 

  24. Wang, X., Che, M., Qi, L., Wei, Y.: Gradient dynamic approach to the tensor complementarity problem. Opt. Methods Softw. (2019). https://doi.org/10.1080/10556788.2019.1578766

    Article  MATH  Google Scholar 

  25. Zhao, X., Fan, J.: A semidefinite method for tensor complementarity problems. Optim. Methods Softw. (2018). https://doi.org/10.1080/10556788.2018.1439489

    Article  MATH  Google Scholar 

  26. Zhang, K.L., Chen, H.B., Zhao, P.F.: A potential reduction method for tensor complementarity problems. J. Ind. Manag. Optim. 15(2), 429–443 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Bai, X., He, H., Ling, C., Zhou, G.: An efficient nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors (2018). arXiv:1811.09917v1

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

    Article  MathSciNet  Google Scholar 

  29. Berman, A., Plemmons, R.J.: Non-negative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  30. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  31. Ding, W., Qi, L., Wei, Y.: M-Tensors and Nonsingular M-Tensors. Linear Algebra Appl. 439(10), 3264–3278 (2013)

    Article  MathSciNet  Google Scholar 

  32. Huang, Z.H., Qi, L.: Tensor Complementarity Problems-Part II: Solution Methods. J. Optim. Theory Appl. 183(2), 365–385 (2019)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  34. Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM J. Matrix Anal. Appl. 35(2), 437–452 (2014)

    Article  MathSciNet  Google Scholar 

  35. 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 

  36. Dai, P.F., Li, J.C., Li, Y.T., Bai, J.: A general preconditioner for linear complementarity problem with an M-matrix. J. Comput. Appl. Math. 317, 100–112 (2017)

    Article  MathSciNet  Google Scholar 

  37. Dehghan, M., Hajarian, M.: Two class of synchronous matrix multisplitting schemes for solving linear complementarity problems. J. Comput. Appl. Math. 235(15), 4325–4336 (2011)

    Article  MathSciNet  Google Scholar 

  38. Li, L., Kobayashi, Y.: A block recursive algorithm for the linear complementarity problem with an M-matrix. Information 2(6), 287–293 (2006)

    Google Scholar 

  39. Li, D.H., Guan, H.B., Wang, X.Z.: Finding a nonnegative solution to an M-tensor equation (2018). arXiv:1811.11343

  40. Bader, B.W., Kolda, T.G., et al.: MATLAB Tensor Toolbox Version 2.6 (2015)

  41. Li, D.H., Xie, S., Xu, H.R.: Splitting methods for tensor equations. Numer. Linear Algebra Appl. (2017). https://doi.org/10.1002/nla.2102

    Article  MathSciNet  MATH  Google Scholar 

  42. Ding, W., Wei, Y.: Solving multi-linear systems with M-Tensors. J. Sci. Comput. 68(2), 683–715 (2016)

    Article  MathSciNet  Google Scholar 

  43. Xie, Z.J., Jin, X.Q., Wei, Y.M.: Tensor methods for solving symmetric M-tensor systems. J. Sci. Comput. 74(1), 412–425 (2017)

    Article  MathSciNet  Google Scholar 

  44. Xu, H.R., Li, D.H., Xie, S.L.: An equivalent tensor equation to the tensor complementarity problem with positive semi-definite Z-tensor. Optim. Lett. (2018). https://doi.org/10.1080/10556788.2018.1439489

    Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank two anonymous referees for their valuable comments. This paper was supported by the Chinese NSF Grant 11771157 and Hunan Provincial Education Department of China Grant 15C0359.

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Authors and Affiliations

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

    Hong-Bo Guan & Dong-Hui Li

  2. School of Mathematics, Physics and Energy Engineering, Hunan Institute of Technology, Hengyang, 421002, Hunan, China

    Hong-Bo Guan

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  1. Hong-Bo Guan
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  2. Dong-Hui Li
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Correspondence to Dong-Hui Li.

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Communicated by Communicated by Liqun Qi.

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Guan, HB., Li, DH. Linearized Methods for Tensor Complementarity Problems. J Optim Theory Appl 184, 972–987 (2020). https://doi.org/10.1007/s10957-019-01627-3

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