On a Class of Semi-Positive Tensors in Tensor Complementarity Problem

  • Ya-nan Zheng
  • Wei Wu


Recently, the tensor complementarity problem has been investigated in the literature. In this paper, we extend a class of structured matrices to higher-order tensors; the corresponding tensor complementarity problem has a unique solution for any nonzero nonnegative vector. We discuss their relationships with semi-positive tensors and strictly semi-positive tensors. We also study the property of such a structured tensor. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. We also give two equivalent formulations of such a structured tensor.


Tensor complementarity problem E-tensor Strictly semi-positive tensor Principal sub-tensor 

Mathematics Subject Classification

15A18 15A69 90C33 



The authors would like to thank the editors and anonymous referees 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. 11371276).


  1. 1.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)zbMATHGoogle Scholar
  2. 2.
    Pang, J.S.: On Q-matrices. Math. Program. 17, 243–247 (1979)CrossRefzbMATHGoogle Scholar
  3. 3.
    Gowda, M.S.: On Q-matrices. Math. Program. 49, 139–141 (1990)CrossRefzbMATHGoogle Scholar
  4. 4.
    Jeter, M.W., Pye, W.C.: An example of a nonregular semimonotone Q-matrix. Math. Program. 44, 351–356 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Murthy, G.S.R., Parthasarathy, T., Ravindran, G.: A copositive Q-matrix which is not \(R_0\). Math Program. 61, 131–135 (1993)CrossRefzbMATHGoogle Scholar
  6. 6.
    Eaves, B.C.: The linear complementarity problem. Manag. Sci. 17, 621–634 (1971)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Danao, R.A.: On a class of semimonotone \(Q_0\)-matrices in the linear complementarity problem. Oper. Res. Lett. 13, 121–125 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Danao, R.A.: A note on \(E^{\prime }\)-matrices. Linear Algebra Appl. 259, 299–305 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33(3), 308–323 (2017)Google Scholar
  10. 10.
    Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169(3), 1069–1078 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cottle, R.W.: Completely Q-matrices. Math. Program. 19, 347–351 (1980)CrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, H., Huang, Z., Qi, L.: Copositivity detection of tensors: theory and algorithm. J. Optim. Theory Appl. 174, 746–761 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Song, Y., Qi, L.: Necessary and sufficient conditions for copositive tensors. Linear Multilinear Algebra 63(1), 120–131 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China

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