Advertisement

Science China Mathematics

, Volume 61, Issue 9, pp 1695–1710 | Cite as

Tensor absolute value equations

  • Shouqiang Du
  • Liping Zhang
  • Chiyu Chen
  • Liqun Qi
Articles
  • 33 Downloads

Abstract

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.

Keywords

M-tensors absolute value equations Levenberg-Marquardt method tensor complementarity problem 

MSC(2010)

15A48 15A69 65K05 90C30 90C20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671220, 11401331, 11771244 and 11271221), the Nature Science Foundation of Shandong Province (Grant Nos. ZR2015AQ013 and ZR2016AM29), and the Hong Kong Research Grant Council (Grant Nos. PolyU 501913, 15302114, 15300715 and 15301716). The authors thank the anonymous referees for their constructive comments and suggestions which led to a significantly improved version of the paper.

References

  1. 1.
    Bader B W, Kolda T G. MATLAB Tensor Toolbox Version 2.6. https://doi.org/www.sandia.gov/~tgkolda/TensorToolbox/, 2012Google Scholar
  2. 2.
    Bai X L, Huang Z H, Wang Y. Global uniqueness and solvability for tensor complementaity problems. J Optim Theory Appls, 2016, 170: 72–84CrossRefzbMATHGoogle Scholar
  3. 3.
    Bonnans J F, Cominetti R, Shapiro A. Second order optimality conditions based on parabolic second order tangent sets. SIAM J Optim, 1999, 9: 466–493MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chang K, Qi L, Zhang T. A survey on the spectral theory of non-negative tensors. Numer Linear Algebra Appl, 2013, 20: 891–912MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Che M, Qi L, Wei Y. Positive definite tensors to nonlinear complementarity problems. J Optim Theory Appl, 2016, 168: 475–487MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen Z, Qi L. A semismooth Newton method for tensor eigenvalue complementarity problem. Comput Optim Appl, 2016, 65: 109–126MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Clarke F H. Optimization and Nonsmooth Analysis. New York: Wiley, 1983zbMATHGoogle Scholar
  8. 8.
    Cottle R W, Pang J-S, Stone R E. The Linear Complementarity Problem. Boston: Academic Press, 1992zbMATHGoogle Scholar
  9. 9.
    Ding W, Qi L, Wei Y. M-tensors and nonsingular M-tensors. Linear Algebra Appl, 2013, 439: 3264–3278MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding W, Wei Y. Solving multi-linear systems with M-Tensors. J Sci Comput, 2016, 68: 683–715MathSciNetCrossRefGoogle Scholar
  11. 11.
    Facchinei F, Kanzow C. A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Math Program, 1997, 76: 493–512MathSciNetzbMATHGoogle Scholar
  12. 12.
    Facchinei F, Pang J-S. Finite-Dimensional Variational Inequalities and Complementarity Problems. New York: Springer-Verlag, 2003zbMATHGoogle Scholar
  13. 13.
    Fischer A. A special Newton-type optimization method. Optimization, 1992, 24: 269–284MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Huang Z, Qi L. Formulating an n-person noncooperative game as a tensor complementarity problem. Comput Optim Appl, 2017, 66: 557–576MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li X, Ng M K. Solving sparse non-negative tensor equations: Algorithms and applications. Front Math China, 2015, 10: 649–680MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lim L-H. Singular values and eigenvalues of tensors: A variational approach. In: IEEE CAMSAP 2005: First In-ternational Workshop on Computational Advances in Multi-Sensor Adaptive Processing. New York: IEEE, 2005, 129–132Google Scholar
  17. 17.
    Ling C, He H, Qi L. On the cone eigenvalue complementarity problems for higher-order tensors. Comput Optim Appl, 2016, 63: 143–168MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ling C, He H, Qi L. Higher-degree eigenvalue complementarity problems for tensors. Comput Optim Appl, 2016, 64: 149–176MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Luo Z Y, Qi L, Xiu N H. The sparse solutions to Z-tensor complementarity problems. Optim Lett, 2017, 11: 471–482MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mangasarian O L. Absolute value programming. Comput Optim Appl, 2007, 36: 43–53MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mangasarian O L. Knapsack feasibility as an value equation solvable by successive linear programming. Optim Lett, 2009, 3: 161–170MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mangasarian O L, Meyer R R. Absolute value equations. Linear Algebra Appl, 2006, 419: 359–367MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Miffin R. Semismooth and semiconvex functions in constrained optimization. SIAM J Control Optim, 1997, 15: 959–972MathSciNetCrossRefGoogle Scholar
  24. 24.
    Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Qi L, Luo Z Y. Tensor Analysis: Special Theory and Special Tensors. Philadelphia: Society of Industrial and Applied Mathematics, 2017CrossRefzbMATHGoogle Scholar
  26. 26.
    Qi L, Sun D, Zhou G. A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math Program, 2000, 87: 1–35MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Qi L, Sun J. A nonsmooth version of Newton’s method. Math Program, 1993, 58: 353–367MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Qi L, Yin Y. A strongly semismooth integral function and its application. Comput Optim Appl, 2003, 25: 223–246MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Song Y, Qi L. Properties of some classes of structured tensors. J Optim Theory Appl, 2015, 165: 854–873MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Song Y, Qi L. Tensor complementarity problem and semi-positive tensors. J Optim Theory Appl, 2016, 169: 1069–1078MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Song Y, Qi L. Eigenvalue analysis of constrained minimization problem for homogeneous polynomials. J Global Optim, 2016, 64: 563–575MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Song Y, Yu G. Properties of solution set of tensor complementarity problem. J Optim Theory Appl, 2016, 170: 85–96MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sun D, Qi L. On NCP-functions. Comput Optim Appl, 1999, 13: 201–220MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sun D, Sun J. Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Math Program, 2005, 103: 575–581MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wang Y, Huang Z H, Bai X L. Exceptionally regular tensors and tensor complementarity problems. Optim Methods Softw, 2016, 31: 815–828MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wei Y, Ding W. Theory and Computation of Tensors: Multi-Dimensional Arrays. Amsterdam: Academic Press, 2016zbMATHGoogle Scholar
  37. 37.
    Xie Z, Jin X, Wei Y. Tensor methods for solving symmetric M-tensor systems. J Sci Comput, 2018, 74: 412–425MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Xu F, Ling C. Some properties on Pareto-eigenvalues of higher-order tensors. Oper Res Trans, 2015, 19: 34–41MathSciNetzbMATHGoogle Scholar
  39. 39.
    Zhang L, Qi L, Zhou G. M-tensors and some applications. SIAM J Matrix Anal Appl, 2014, 35: 437–452MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhou G, Caccetta L, Teo K L. A superlinearly convergent method for a class of complementarity problems with non-Lipschitzian functions. SIAM J Optim, 2010, 20: 1811–1827MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Shouqiang Du
    • 1
  • Liping Zhang
    • 2
  • Chiyu Chen
    • 2
  • Liqun Qi
    • 3
  1. 1.School of Mathematics and StatisticsQingdao UniversityQingdaoChina
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  3. 3.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongChina

Personalised recommendations