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Solution structures of tensor complementarity problem

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Abstract

We introduce two new types of tensors called the strictly semimonotone tensor and the range column sufficient tensor and explore their structure properties. Based on the obtained results, we make a characterization to the solution of tensor complementarity problem.

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References

  1. Bai X, Huang Z, Wang Y. Global uniqueness and solvability for tensor complementarity problems. J Optim Theory Appl, 2016, 170: 1–13

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen H, Chen Y, Li G, Qi L. A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test. Numer Linear Algebra Appl, 2017, https://doi.org/10.1002/nla.2125

    Google Scholar 

  5. Chen H, Huang Z, Qi L. Copositivity detection of tensors: theory and algorithm. J Optim Theory Appl, 2017, 174: 746–761

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen H, Huang Z, Qi L. Copositive tensor detection and its applications in physics and hypergraphs. Comput Optim Appl, 2017, https://doi.org/10.1007/s10589-017-9938-1

    Google Scholar 

  7. Chen H, Wang Y. On computing minimal H-eigenvalue of sign-structured tensors. Front Math China, 2017, 12(6): 1289–1302

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen H, Wang Y, Zhao H. A family of higher-order convergent iterative methods for computing the Moore-Penrose inverse. Appl Math Comput, 2011, 218(8): 4012–4016

    MathSciNet  MATH  Google Scholar 

  9. Cottle R, Pang J, Venkateswaran V. Sufficient matrices and linear complementarity problem. Linear Algebra Appl, 1989, 114–115: 231–249

    Article  MathSciNet  MATH  Google Scholar 

  10. Facchinei F, Pang J. Finite Dimensional Variational Inequalities and Complementarity Problems. New York: Springer, 2003

    MATH  Google Scholar 

  11. Huang Z, Qi L. Formulating an n-person noncooperative game as a tensor complementarity problem. Comput Optim Appl, 2016, 66: 1–20

    MathSciNet  Google Scholar 

  12. Jeyaraman I, Sivakumar C. Complementarity properties of singular M-matrices. Linear Algebra Appl, 2016, 510: 42–63

    Article  MathSciNet  MATH  Google Scholar 

  13. Kolda T, Bader B. Tensor decompositions and applications. SIAM Review, 2009, 51: 455–500

    Article  MathSciNet  MATH  Google Scholar 

  14. Ling C, He H, Qi L. On the cone eigenvalue complementarity problem for higher-order tensor. Comput Optim Appl, 2016, 63: 1–26

    Article  MathSciNet  MATH  Google Scholar 

  15. Luo Z, Qi L, Xiu N. The sparsest solutions to Z-tensor complementarity problem. Optim Lett, 2015, 11: 471–482

    Article  MathSciNet  MATH  Google Scholar 

  16. Qi L, Wang F, Wang Y. Z-eigenvalue methods for a global polynomial optimization problem. Math Program, 2009, 118: 301–316

    Article  MathSciNet  MATH  Google Scholar 

  17. Qi L, Wang Y, Wu Ed X. D-eigenvalues of diffusion kurtosis tensors. J Comput Appl Math, 2008, 221: 150–157

    Article  MathSciNet  MATH  Google Scholar 

  18. Qi L, Yu G, Wu E. Higher order positive semi-definite diffusion tensor imaging. SIAM J Imaging Sci, 2010, 3: 416–433

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang G, Zhou G, Caccetta L. Z-eigenvalue inclusion theorems for tensors. Discrete Contin Dyn Syst Ser B, 2017, 22(1): 187–198

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang Y, Caccetta L, Zhou G. Convergence analysis of a block improvement method for polynomial optimization over unit spheres. Numer Linear Algebra Appl, 2015, 22: 1059–1076

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang Y, Liu W, Caccetta L, Zhou G. Parameter selection for nonnegative l1 matrix/tensor sparse decomposition. Oper Res Lett, 2015, 43: 423–426

    Article  MathSciNet  Google Scholar 

  24. Wang Y, Qi L, Luo S, Xu Y. An alternative steepest direction method for the optimization in evaluating geometric discord. Pac J Optim, 2014, 10: 137–149

    MathSciNet  MATH  Google Scholar 

  25. Wang Y, Qi L, Zhang X. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numer Linear Algebra Appl, 2009, 16: 589–601

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang Y, Zhang K, Sun H. Criteria for strong H-tensors. Front Math China, 2016, 11: 577–592

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang Y, Zhou G, Caccetta L. Nonsingular H-tensor and its criteria. J Ind Manag Optim, 2016, 12: 1173–1186

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang K, Wang Y. An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. J Comput Appl Math, 2016, 305: 1–10

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang X, Ling C, Qi L. The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. SIAM J Matrix Anal Appl, 2012, 33: 806–821

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors wish to give their sincere thanks to the anonymous referees for their valuable suggestions and helpful comments, which help improve the presentation of the paper. This research was done during Xueyong Wang’s postdoctoral period in Qufu Normal University. This work was supported by the National Natural Science Foundation of China (Grants Nos. 11671228, 11601261, 11401058), the Natural Science Foundation of Shandong Province (Grant No. ZR2016AQ12), and the Science & Technology Planning Project of Qufu Normal University (XKJ201623).

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Authors and Affiliations

  1. School of Management Science, Qufu Normal University, Rizhao, 276800, China

    Xueyong Wang, Haibin Chen & Yiju Wang

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  1. Xueyong Wang
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  2. Haibin Chen
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  3. Yiju Wang
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Correspondence to Xueyong Wang.

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Wang, X., Chen, H. & Wang, Y. Solution structures of tensor complementarity problem. Front. Math. China 13, 935–945 (2018). https://doi.org/10.1007/s11464-018-0675-2

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  • DOI: https://doi.org/10.1007/s11464-018-0675-2

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