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Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 255–276 | Cite as

Column sufficient tensors and tensor complementarity problems

  • Haibin Chen
  • Liqun Qi
  • Yisheng Song
Research Article

Abstract

Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.

Keywords

Column sufficient tensor H-eigenvalue tensor complementarity problems handicap 

MSC

65H17 15A18 90C30 

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Notes

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11601261, 11571095, 11601134), the Hong Kong Research Grant Council (Grant No.PolyU 502111, 501212, 501913, 15302114), the Natural Science Foundation of Shandong Province (No. ZR2016AQ12), and the China Postdoctoral Science Foundation (Grant No. 2017M622163).

References

  1. 1.
    Bai X, Huang Z, Wang Y. Global uniqueness and solvability for tensor complementarity problems. J Optim Theory Appl, 2016, 170: 1–13MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bu C, Zhang X, Zhou J, Wang W, Wei Y. The inverse, rank and product of tensors. Linear Algebra Appl, 2014, 446: 269–280MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Che M, Qi L, Wei Y. Positive definite tensors to nonlinear complementarity problems. J Optim Theory Appl, 2016, 168(2): 475–487MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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, 2018, 25: e2125CrossRefzbMATHGoogle Scholar
  5. 5.
    Chen H, Huang Z, Qi L. Copositivity detection of tensors: theory and algorithm. J Optim Theory Appl, 2017, 174: 746–761MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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/s10589-017-9938-1Google Scholar
  7. 7.
    Chen H, Li G, Qi L. SOS tensor decomposition: theory and applications. Commun Math Sci, 2016, 14(8): 2073–2100MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen H, Qi L. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J Ind Manag Optim, 2015, 11: 1263–1274MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen H, Wang Y. On computing minimal H-eigenvalue of sign-structured tensors. Front Math China, 2017, 12(6): 1289–1302MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen Z, Qi L. Circulant tensors with applications to spectral hypergraph theory and stochastic process. J Ind Manag Optim, 2016, 12: 1227–1247MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3292MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cottle R W, Guu S M. Are P *-matrices just sufficient? Presented at the 36th Joint National Meeting of Operations Research Society of America and the Institute of Management Science, Phoenix, AZ, 1 Nov, 1993Google Scholar
  13. 13.
    Cottle R W, Pang J S, Stone R E. The Linear Complementarity Problem. Boston: Academic Press, 1992zbMATHGoogle Scholar
  14. 14.
    Cottle R W, Pang J S, Venkateswaran V. Sufficient matrices and the linear complementarity problem. Linear Algebra Appl, 1989, 114: 231–249MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ding W, Qi L, Wei Y. M-Tensors and nonsingular M-tensors. Linear Algebra Appl, 2013, 439: 3264–3278MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ding W, Qi L, Wei Y. Fast Hankel tensor-vector products and application to exponential data fitting. Numer Linear Algebra Appl, 2015, 22: 814–832MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gowda M S, Luo Z, Qi L, Xiu N. Z-tensors and complementarity problems. 2015, arXiv: 1510.07933Google Scholar
  18. 18.
    Guu S M, Cottle R W. On a subclass of P 0: Linear Algebra Appl, 1995, 223: 325–335Google Scholar
  19. 19.
    Han J Y, Xiu N H, Qi H D. Nonlinear Complementary Theory and Algorithm. Shanghai: Shanghai Science and Technology Press, 2006Google Scholar
  20. 20.
    Huang Z, Qi L. Formulating an n-person noncooperative game as a tensor complementarity problems. Comput Optim Appl, 2017, 66: 557–576MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kannan M R, Shaked-Monderer N, Berman A. Some properties of strong H-tensors and general H-tensors. Linear Algebra Appl, 2015, 476: 42–55MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Klerk E D, Nagy M E. On the complexity of computing the handicap of a sufficient matrix. Math Program, 2011, 129(2): 383–402MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kojima M, Megiddo N, Noma T, Yoshise A. A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. Berlin: Springer-Verlag, 1991CrossRefzbMATHGoogle Scholar
  24. 24.
    Li C, Li Y. Double B tensors and quasi-double B tensors. Linear Algebra Appl, 2015, 466: 343–356MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li C, Wang F, Zhao J, Zhu Y, Li Y. Criterions for the positive definiteness of real supersymmetric tensors. J Comput Appl Math, 2014, 255: 1–14MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP05), Vol 1. 2005, 129–132Google Scholar
  27. 27.
    Luo Z, Qi L, Xiu N. The sparsest solutions to Z-tensor complementarity problems. Optim Lett, 2017, 11(3): 471–482MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ng M, Qi L, Zhou G. Finding the largest eigenvalue of a non-negative tensor. SIAM J Matrix Anal Appl, 2009, 31: 1090–1099MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ni Q, Qi L. A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map. J Global Optim, 2015, 61: 627–641MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Qi L. Eigenvalue of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Qi L. H+-eigenvalues of Laplacian and signless Laplacian tensors. Commun Math Sci, 2014, 12: 1045–1064MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Qi L. Hankel tensors: associated Hankel matrices and Vandermonde decomposition. Commun Math Sci, 2015, 13: 113–125MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Qi L, Song Y. An even order symmetric B tensor is positive definite. Linear Algebra Appl, 2014, 457: 303–312MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Qi L, Wei Y, Xu C, Zhang T. Linear algebra and multi-linear algebra. Front Math China, 2016, 11(3): 509–510MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Qi L, Yu G, Wu E X. Higher order positive semi-definite difiusion tensor imaging. SIAM J Imaging Sci, 2010, 3: 416–433MathSciNetCrossRefGoogle Scholar
  36. 36.
    Shao J Y. A general product of tensors with applications. Linear Algebra Appl, 2013, 439: 2350–2366MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Shao J Y, Shan H Y, Zhang L. On some properties of the determinants of tensors. Linear Algebra Appl, 2013, 439: 3057–3069MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Song Y, Qi L. Infinite and finite dimensional Hilbert tensors. Linear Algebra Appl, 2014, 451: 1–14MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Song Y, Qi L. Properties of some classes of structured tensors. J Optim Theory Appl, 2015, 165: 854–873MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Song Y, Qi L. Tensor complementarity problems and semi-positive tensors. J Optim Theory Appl, 2016, 169(3): 1069–1078MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Song Y, Qi L. Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim Lett, 2017, 11(7): 1407–1426MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Song Y, Qi L. Properties of tensor complementarity problem and some classes of structured tensors. Ann of Appl Math, 2017, 33(3): 308–323Google Scholar
  43. 43.
    Sun J, Huang Z. A smoothing Newton algorithm for the LCP with a sufficient matrix that terminates finitely at a maximally complementary solution. Optim Methods Softw, 2006, 21: 597–615MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Väliaho H. P *-matrices are just sufficient. Linear Algebra Appl, 1996, 239: 103–108MathSciNetzbMATHGoogle Scholar
  45. 45.
    Zhang L, Qi L, Zhou G. M-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 35: 437–452MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Management ScienceQufu Normal UniversityRizhaoChina
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongChina
  3. 3.School of Mathematics and Information ScienceHenan Normal UniversityXinxiangChina

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