Further study on tensor absolute value equations


In this paper, we consider the tensor absolute value equations (TAVEs), which is a newly introduced problem in the context of multilinear systems. Although the system of the TAVEs is an interesting generalization of matrix absolute value equations (AVEs), the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of the TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of the TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of the TAVEs for some special cases. To find a solution to the TAVEs, we employ the generalized Newton method and report some preliminary results.

This is a preview of subscription content, access via your institution.


  1. 1

    Alexandroff P, Hopf H. Topologie. Chelsea: New York, 1965

    MATH  Google Scholar 

  2. 2

    Bader B, Kolda T. MATLAB Tensor Toolbox Version 2.6. Available online 2015. http://www.sandia.gov/~tgkolda/TensorToolbox/

  3. 3

    Caccetta L, Qu B, Zhou G. A globally and quadratically convergent method for absolute value equations. Comput Optim Appl, 2011, 48: 45–58

    MathSciNet  MATH  Google Scholar 

  4. 4

    Chen H, Chen Y, Li G, et al. A semidefinite 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: e2125

    MathSciNet  MATH  Google Scholar 

  5. 5

    Chen H, Qi L. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J Ind Manag Optim, 2015, 11: 1263–1274

    MathSciNet  MATH  Google Scholar 

  6. 6

    Cottle R, Pang J, Stone R. The Linear Complementarity Problem. Boston: Boston Academic Press, 1992

    MATH  Google Scholar 

  7. 7

    Ding W, Luo Z, Qi L. P-tensors, P0-tensors, and tensor complementarity problem. ArXiv:1507.06731, 2015

  8. 8

    Ding W, Wei Y. Solving multilinear systems with M-tensors. J Sci Comput, 2016, 68: 689–715

    MathSciNet  MATH  Google Scholar 

  9. 9

    Du S Q, Zhang L P, Chen C Y, et al. Tensor absolute value equations. Sci China Math, 2018, 61: 1695–1710

    MathSciNet  MATH  Google Scholar 

  10. 10

    Fonseca G, Fonseca I, Gangbo W. Degree Theory in Analysis and Applications. Oxford Lecture Series in Mathematics and Its Applications, vol. 2. Oxford: Oxford University Press, 1995

    MATH  Google Scholar 

  11. 11

    Gleich D, Lim L, Yu Y. Multilinear PageRank. SIAM J Matrix Anal Appl, 2015, 36: 1507–1541

    MathSciNet  MATH  Google Scholar 

  12. 12

    He H, Ling C, Qi L, et al. A globally and quadratically convergent algorithm for solving multilinear systems with M-tensors. J Sci Comput, 2018, 76: 1718–1741

    MathSciNet  MATH  Google Scholar 

  13. 13

    Hu S, Huang Z, Zhang Q. A generalized Newton method for absolute value equations associated with second order cones. J Comput Appl Math, 2011, 235: 1490–1501

    MathSciNet  MATH  Google Scholar 

  14. 14

    Iqbal J, Iqbal A, Arif M. Levenberg-Marquardt method for solving systems of absolute value equations. J Comput Appl Math, 2015, 282: 134–138

    MathSciNet  MATH  Google Scholar 

  15. 15

    Isac G. Leray-Schauder Type Alternatives, Complementarity Problems and Variational Inequalities. New York: Springer, 2006

    MATH  Google Scholar 

  16. 16

    Li D, Xie S, Xu H. Splitting methods for tensor equations. Numer Linear Algebra Appl, 2017, 24: e2102

    MathSciNet  MATH  Google Scholar 

  17. 17

    Li X, Ng M. Solving sparse non-negative tensor equations: Algorithms and applications. Front Math China, 2015, 10: 649–680

    MathSciNet  MATH  Google Scholar 

  18. 18

    Liu D, Li W, Vong S. The tensor splitting with application to solve multi-linear systems. J Comput Appl Math, 2018, 330: 75–94

    MathSciNet  MATH  Google Scholar 

  19. 19

    Liu W, Li W. On the inverse of a tensor. Linear Algebra Appl, 2016, 495: 199–205

    MathSciNet  MATH  Google Scholar 

  20. 20

    Lloyd N. Degree Theory. Cambridge: Cambridge University Press, 1978

    MATH  Google Scholar 

  21. 21

    Luo Z, Qi L. Doubly nonnegative tensors, completely positive tensors and applications. ArXiv:1504.07806, 2015

  22. 22

    Mangasarian O. Absolute value programming. Comput Optim Appl, 2007, 36: 43–53

    MathSciNet  MATH  Google Scholar 

  23. 23

    Mangasarian O. A generalized Newton method for absolute value equations. Optim Lett, 2009, 3: 101–108

    MathSciNet  MATH  Google Scholar 

  24. 24

    Mangasarian O, Meyer R. Absolute value equations. Linear Algebra Appl, 2006, 419: 359–367

    MathSciNet  MATH  Google Scholar 

  25. 25

    Miao X, Yang J, Saheya B, et al. A smoothing Newton method for absolute value equation associated with second-order cone. Appl Numer Math, 2017, 120: 82–96

    MathSciNet  MATH  Google Scholar 

  26. 26

    Pearson K. Essentially positive tensors. Int J Algebra, 2010, 4: 421–427

    MathSciNet  MATH  Google Scholar 

  27. 27

    Prokopyev O. On equivalent reformulations for absolute value equations. Comput Optim Appl, 2009, 44: 363–372

    MathSciNet  MATH  Google Scholar 

  28. 28

    Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324

    MathSciNet  MATH  Google Scholar 

  29. 29

    Qi L. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439: 228–238

    MathSciNet  MATH  Google Scholar 

  30. 30

    Qi L, Chen H, Chen Y. Tensor Eigenvalues and Their Applications. Singapore: Springer, 2018

    MATH  Google Scholar 

  31. 31

    Qi L, Sun D. A survey of some nonsmooth equations and smoothing Newton methods. In: Progress in Optimization, vol. 30. Netherlands: Kluwer Acad Publ, 1999: 121–146

    Google Scholar 

  32. 32

    Qi L, Xu C, Xu Y. Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm. SIAM J Matrix Anal Appl, 2014, 35: 1227–1241

    MathSciNet  MATH  Google Scholar 

  33. 33

    Rheinboldt W. Methods for solving systems of nonlinear equations. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 70. Philadelphia: SIAM, 1998, 125–125

    Google Scholar 

  34. 34

    Rohn J. A theorem of the alternatives for the equation Ax + B|x| = b. Linear Algebra Appl, 2004, 52: 421–426

    MathSciNet  MATH  Google Scholar 

  35. 35

    Shao J. A general product of tensors with applications. Linear Algebra Appl, 2013, 439: 2350–2366

    MathSciNet  MATH  Google Scholar 

  36. 36

    Shao J, You L. On some properties of three different types of triangular blocked tensors. Linear Algebra Appl, 2016, 511: 110–140

    MathSciNet  MATH  Google Scholar 

  37. 37

    Song Y, Qi L. Infinite and finite dimensional Hilbert tensors. Linear Algebra Appl, 2014, 451: 1–14

    MathSciNet  MATH  Google Scholar 

  38. 38

    Song Y, Qi L. Necessary and sufficient conditions for copositive tensors. Linear Multilinear Algebra, 2015, 63: 120–131

    MathSciNet  MATH  Google Scholar 

  39. 39

    Song Y, Qi L. Properties of some classes of structured tensors. J Optim Theory Appl, 2015, 165: 854–873

    MathSciNet  MATH  Google Scholar 

  40. 40

    Wang X, Wei Y. -tensors and nonsingular -tensors. Front Math China, 2016, 11: 557–575

    MathSciNet  Google Scholar 

  41. 41

    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

    MathSciNet  MATH  Google Scholar 

  42. 42

    Xie Z, Jin X, Wei Y. Tensor methods for solving symmetric M-tensor systems. J Sci Comput, 2018, 74: 412–425

    MathSciNet  MATH  Google Scholar 

  43. 43

    Yuan P, You L. Some remarks on P, P0, B and B0 tensors. Linear Algebra Appl, 2014, 459: 511–521

    MathSciNet  MATH  Google Scholar 

  44. 44

    Zhang C, Wei Q. Global and finite convergence of a generalized Newton method for absolute value equations. J Optim Theory Appl, 2009, 143: 391–403

    MathSciNet  MATH  Google Scholar 

Download references


The first author and the third author were supported by National Natural Science Foundation of China (Grant Nos. 11571087 and 11771113) and Natural Science Foundation of Zhejiang Province (Grant No. LY17A010028). The fourth author was supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717).

Author information



Corresponding author

Correspondence to Hongjin He.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ling, C., Yan, W., He, H. et al. Further study on tensor absolute value equations. Sci. China Math. 63, 2137–2156 (2020). https://doi.org/10.1007/s11425-018-9560-3

Download citation


  • tensor absolute value equations
  • H+-tensor
  • P-tensor
  • copositive tensor
  • generalized Newton method


  • 15A48
  • 15A69
  • 65K05
  • 90C30
  • 90C20