Journal of Optimization Theory and Applications

, Volume 183, Issue 2, pp 365–385 | Cite as

Tensor Complementarity Problems—Part II: Solution Methods

  • Liqun QiEmail author
  • Zheng-Hai Huang
Invited Paper


This work, with its three parts, reviews the state-of-the-art of studies for the tensor complementarity problem and some related models. In the first part of this paper, we have reviewed the theoretical developments of the tensor complementarity problem and related models. In this second part, we review the developments of solution methods for the tensor complementarity problem. It has been shown that the tensor complementarity problem is equivalent to some known optimization problems, or related problems such as systems of tensor equations, systems of nonlinear equations, and nonlinear programming problems, under suitable assumptions. By solving these reformulated problems with the help of structures of the involved tensors, several numerical methods have been proposed so that a solution of the tensor complementarity problem can be found. Moreover, based on a polynomial optimization model, a semidefinite relaxation method is presented so that all solutions of the tensor complementarity problem can be found under the assumption that the solution set of the problem is finite. Further applications of the tensor complementarity problem will be given and discussed in the third part of this paper.


Tensor complementarity problem System of tensor equations System of non-smooth equations Mixed integer programming Semidefinite relaxation method 

Mathematics Subject Classification

90C33 90C30 65H10 



We are very grateful to professors Chen Ling, Yisheng Song, Shenglong Hu and Ziyan Luo for reading the first draft of this paper and putting forward valuable suggestions for revision. The first author’s work is partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717), and the second author’s work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11431002 and 11871051).


  1. 1.
    Huang, Z.H., Qi, L.: Tensor complementarity problems—part I: basic theory. J. Optim. Theory Appl. (2019).
  2. 2.
    Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to \(Z\)-tensor complementarity problems. Optim. Lett. 11(3), 471–482 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    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. 13(4), 685–694 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    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, 119–136 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ye, Y.Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)zbMATHCrossRefGoogle Scholar
  6. 6.
    Han, L.: A continuation method for tensor complementarity problems. J. Optim. Theory Appl. 180, 949–963 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Huang, Z.H., Qi, L.: Formulating an \(n\)-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66(3), 557–576 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Liu, D., Li, W., Vong, S.W.: Tensor complementarity problems: the GUS-property and an algorithm. Linear Multilinear Algebra 66(9), 1726–1749 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73, 789–800 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Zhao, X., Fan, J.: A semidefinite method for tensor complementarity problems. Optim. Method Softw. 34(4), 758–769 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Rademacher, H.: Über partielle und totale Differenzierbarkeit I. Math. Ann. 89, 340–359 (1919)zbMATHCrossRefGoogle Scholar
  13. 13.
    Nekvinda, A., Zaj́ǐcek, L.: A simple proof of Rademacher theorem. Časopis Pěst. Mat. 113, 337–341 (1988)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  15. 15.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15(6), 959–972 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ding, W., Wei, Y.: Solving multi-linear systems with \(M\)-mensors. J. Sci. Comput. 68(2), 689–715 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Li, X., Ng, M.K.: Solving sparse non-negative tensor equations: algorithms and applications. Front. Math. China 10(3), 649–680 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Han, L.: A homotopy method for solving multilinear systems with \(M\)-tensors. Appl. Math. Lett. 69, 49–54 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Xie, Z.J., Jin, X.Q., Wei, Y.: A fast algorithm for solving circulant tensor systems. Linear Multilinear Algebra 65(9), 1894–1904 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Xie, Z.J., Jin, X.Q., Wei, Y.: Tensor methods for solving symmetric \({\mathscr {M}}\)-tensor systems. J. Sci. Comput. 74(1), 412–425 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Li, D.H., Xie, S.L., Xu, H.R.: Splitting methods for tensor equations. Numer. Linear Algebra Appl. 24(5), e2102 (2017). MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Liu, D.D., Li, W., Vong, S.: The tensor splitting with application to solve multi-linear systems. J. Comput. Appl. Math. 330(1), 75–94 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    He, H., Ling, C., Qi, L., Zhou, G.: A globally and quadratically convergent algorithm for solving multilinear systems with \(M\)-tensors. J. Sci. Comput. 76, 1718–1741 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Bai, X.L., He, H., Ling, C., Zhou, G.: An efficient nonnegativity preserving algorithm for multilinear systems with nonsingular \(M\)-tensors (2018). arXiv:1811.09917
  27. 27.
    Wang, X., Che, M., Wei, Y.: Existence and uniqueness of positive solution for \({\mathscr {H}}^{+}\)-tensor equations. Appl. Math. Lett. 98, 191–198 (2019)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Todd, M.J.: Potential-reduction methods in mathematical programming. Math. Program. 76, 3–45 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168(2), 475–487 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kojima, M., Noma, T., Yoshise, A.: Global convergence in infeasible-interior-point algorithms. Math. Program. 65, 43–72 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Todd, M., Ye, Y.Y.: A centered projective algorithm for linear programming. Math. Oper. Res. 15, 508–529 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Wang, T., Monteiro, R., Pang, J.S.: An interior point potential reduction method for constrained equations. Math. Program. 74, 159–195 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Ma, F.M., Wang, Y.J., Zhao, H.: A potential reduction algorithm for generalized linear complementarity problem over a polyhedral cone. J. Ind. Manag. Optim. 6, 259–267 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kojima, M., Mizuno, M., Noma, T.: A new continuation method for complementarity problems with uniform \(P\)-functions. Math. Oper. Res. 14, 107–113 (1989)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Kojima, M., Megiddo, N., Noma, T.: Homotopy continuation method for nonlinear complementarity problems. Math. Oper. Res. 16, 754–774 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Kojima, M., Megiddo, N., Mizuno, M.: A general framework of continuation methods for complementarity problems. Math. Oper. Res. 18, 945–963 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Xu, Q., Dang, C.: A new homotopy method for solving non-linear complementarity problems. Optimization 57(5), 681–689 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Zhao, Y.B., Li, D.: On a new homotopy continuation trajectory for nonlinear complementarity problems. Math. Oper. Res. 26, 119–146 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Chen, L., Han, L., Zhou, L.: Computing tensor eigenvalues via homotopy methods. SIAM J. Matrix Anal. Appl. 37(1), 290–319 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Sun, D., Qi, L.: On NCP-functions. Comput. Optim. Appl. 13, 201–220 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Hu, S., Huang, Z.H., Chen, J.-S.: Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems. J. Comput. Appl. Math. 230, 69–82 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Chen, B., Harker, P.T.: A non-interior-point continuation method for linear complementarity problem. SIAM J. Matrix Anal. Appl. 14, 1168–1190 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67, 519–540 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Qi, L., Sun, D., Zhou, G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems. Math. Program. 87, 1–35 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Burke, J., Xu, S.: A non-interior predictor–corrector path following algorithm for the monotone linear complementarity problem. Math. Program. 87, 113–130 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Huang, Z.H., Han, J., Chen, Z.: A predictor–corrector smoothing Newton algorithm, based on a new smoothing function, for solving the nonlinear complementarity problem with a \(P_0\) function. J. Optim. Theory Appl. 117, 39–68 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Huang, Z.H., Qi, L., Sun, D.: Sub-quadratic convergence of a smoothing Newton algorithm for the \(P_0\)- and monotone LCP. Math. Program. 99, 423–441 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Huang, Z.H., Ni, T.: Smoothing algorithms for complementarity problems over symmetric cones. Comput. Optim. Appl. 45, 557–579 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Huang, Z.H.: Locating a maximally complementary solution of the monotone NCP by using noninterior-point smoothing algorithms. Math. Method Oper. Res. 61, 41–55 (2005)zbMATHCrossRefGoogle Scholar
  51. 51.
    Wang, X., Che, M., Qi, L., Wei, Y.: Modified gradient dynamic approach to the tensor complementarity problem. Optim. Method Softw. (2019).
  52. 52.
    Van Bokhoven, W.: Piecewise-Linear Modelling and Analysis. Proefschrift, Eindhoven (1981)Google Scholar
  53. 53.
    Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)zbMATHGoogle Scholar
  54. 54.
    Bai, Z.Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Zheng, H., Li, W.: The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J. Comput. Appl. Math. 288, 116–126 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Pardalos, P.M.: Linear complementarity problems solvable by integer programming. Optimization 19, 467–474 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Pardalos, P.M., Rosen, J.B.: Global optimization approach to the linear complementarity problems. SIAM J. Sci. Stat. Comput. 9, 341–353 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Lasserre, J.B., Laurent, M., Rostalski, P.: Semidefinite characterization and computation of zero-dimensional real radical ideals. Found. Comput. Math. 8, 607–647 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. 142, 485–510 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Nie, J.: The hierarchy of local minimums in polynomial optimization. Math. Program. 151, 555–583 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Hu, S., Wang, J., Huang, Z.H.: An inexact augmented Lagrangian multiplier method for solving quadratic complementary problems: an adapted algorithmic framework combining specific resolution techniques. J. Comput. Appl. Math. 361, 64–78 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Chen, C., Zhang, L.: Finding Nash equilibrium for a class of multi-person noncooperative games via solving tensor complementarity problem. Appl. Numer. Math. (2019).

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

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceHangzhou Dianzi UniversityHangzhouPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong
  3. 3.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China

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