Journal of Global Optimization

, Volume 73, Issue 4, pp 789–800 | Cite as

A mixed integer programming approach to the tensor complementarity problem

  • Shouqiang Du
  • Liping ZhangEmail author


The tensor complementarity problem is a special instance of nonlinear complementarity problems, which has many applications. How to solve the tensor complementarity problem, via analyzing the structure of the related tensor, is one of very important research issues. In this paper, we propose a mixed integer programming approach for solving the tensor complementarity problem. We reformulate the tensor complementarity problem as an equivalent mixed integer feasibility problem. Based on the reformulation, some conditions for the solution existence and some solution properties of the tensor complementarity problem are given. We also prove that the tensor complementarity problem, corresponding to a positive definite diagonal tensor, has a unique solution. Finally, numerical results are reported to indicate the efficiency of the proposed algorithm.


Tensor complementarity problem Mixed integer programming Unique solution Positive definite 

Mathematics Subject Classification

15A69 90C11 



The authors would like to thank the editor and the anonymous referees for their helpful constructive comments and suggestions which lead to a significantly improved version of the paper.


  1. 1.
    Bai, X., Huang, Z., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chang, K.C., Pearson, K., Zhang, T.: Perron–Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6, 507–520 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Che, M., Qi, L., Wei, Y.: Positive definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)zbMATHGoogle Scholar
  5. 5.
    Ding, W., Luo, Z., Qi, L.: P-Tensors, \(\text{ P }_0\)-Tensors, and tensor complementarity problem. Linear Algebra Appl. 555, 336–354 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
  7. 7.
    Gowda, M.S., Luo, Z., Qi, L., Xiu, N.: Z-tensors and complementarity problems, arXiv: 1510.07933v2 (2016)
  8. 8.
    Huang, Z., Suo, S., Wang, J.: On Q-tensors, arXiv: 1509.03088, (2015)
  9. 9.
    Huang, Z., Qi, L.: Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lim, L.-H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Addaptive Processing (CAMSAP’05), Vol. 1, pp. 129–132. IEEE Computer Society Press, Piscataway, NJ (2005)Google Scholar
  11. 11.
    Ling, C., He, H., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. 63, 143–168 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, D., Li, W., Vong, S.W.: Tensor complementarity problems: the GUS-property and algorithm. Linear Multi Algebra 66, 1726–1749 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to Z-tensor complementarity problems. Optim. Lett 11, 471–482 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pardalos, P.M., Rosen, J.B.: Global optimization approach to the linear complementarity problem. SIAM J. Sci. Stat. Comput. 9, 341–353 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pardalos, P.M.: Linear complementarity problems solvable by integer programming. Optimization 19, 467–474 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)CrossRefzbMATHGoogle Scholar
  18. 18.
    Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelpia (2017)CrossRefzbMATHGoogle Scholar
  19. 19.
    Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33, 308–323 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170, 85–96 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Song, Y., Qi, L.: Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim. Lett. 11, 1407–1426 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069–1078 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, Y., Huang, Z., Bai, X.: Exceptionally regular tensors and tensor complementarity problems. Optim. Methods Softw. 31, 815–828 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xie, S., Li, D., Xu, H.: An iterative method for finding the least solution to the tensor complementarity problem. J. Optim. Theory Appl. 175, 119–136 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yu, W., Ling, C., He, H.: On the properties of tensor complementarity problems, arXiv:1608.01735v3 (2018)

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsQingdao UniversityQingdaoChina
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina

Personalised recommendations