Computational Optimization and Applications

, Volume 69, Issue 1, pp 133–158 | Cite as

Copositive tensor detection and its applications in physics and hypergraphs

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Abstract

Copositivity of tensors plays an important role in vacuum stability of a general scalar potential, polynomial optimization, tensor complementarity problem and tensor generalized eigenvalue complementarity problem. In this paper, we propose a new algorithm for testing copositivity of high order tensors, and then present applications of the algorithm in physics and hypergraphs. For this purpose, we first give several new conditions for copositivity of tensors based on the representative matrix of a simplex. Then a new algorithm is proposed with the help of a proper convex subcone of the copositive tensor cone, which is defined via the copositivity of Z-tensors. Furthermore, by considering a sum-of-squares program problem, we define two new subsets of the copositive tensor cone and discuss their convexity. As an application of the proposed algorithm, we prove that the coclique number of a uniform hypergraph is equivalent to an optimization problem over the completely positive tensor cone, which implies that the proposed algorithm can be applied to compute an upper bound of the coclique number of a uniform hypergraph. Then we study another application of the proposed algorithm on particle physics in testing copositivity of some potential fields. At last, various numerical examples are given to show the performance of the algorithm.

Keywords

Symmetric tensor Strictly copositive tensor Positive semi-definiteness Simplicial partition Particle physics Hypergraphs 

AMS Subject Classification

65H17 15A18 90C30 
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Notes

Acknowledgements

We are very grateful to the editor and the two reviewers for their constructive comments and valuable suggestions on our manuscript. Furthermore, the authors are thankful to Kristjan Kannike for the discussion on the vacuum stability model.

References

  1. 1.
    Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bundfuss, S., Dür, M.: Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl. 428(7), 1511–1523 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Che, M., Qi, L., Wei, Y.: Positive definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, H., Chen, Y., Li, G., Qi, L.: Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs (2016). arXiv:1511.02328v2
  5. 5.
    Chen, H., Huang, Z.H., Qi, L.: Copositivity detection of tensors: theory and algorithm. J. Optim. Theory Appl. (2017). doi: 10.1007/s10957-017-1131-2
  6. 6.
    Chen, H., Li, G., Qi, L.: SOS tensor decomposition: theory and applications. Commun. Math. Sci. 8, 2073–2100 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, H., Qi, L.: Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J. Ind. Manag. Optim. 11, 1263–1274 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cooper, J., Dutle, A.: Spectra of uniform hypergraphs. Linear Algebra Appl. 436, 3268–3292 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dickinson, P.J.C., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57, 403–415 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ding, W., Luo, Z., Qi, L.: P-tensors, \(P _0 \)-tensors, and tensor complementarity problem (2015). arXiv:1507.06731
  11. 11.
    Hillar, C.J., Lim, L.H.: Most tensor problems are NP-hard. J. ACM (JACM) 60, 45 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Huang, Z.H., Qi, L.: Formulating an n-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong \(H\)-tensors and general \(H\)-tensors. Linear Algebra Appl. 476, 42–55 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kannike, K.: Vacuum stability of a general scalar potential of a few fields. Eur. Phys. J. C 76, 324 (2016)CrossRefGoogle Scholar
  15. 15.
    Kolda, T.: Numerical optimization for symmetric tensor decomposition. Math. Program. Ser. B 151, 225–248 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li, C., Li, Y.: Double B tensors and quasi-double B tensors. Linear Algebra Appl. 466, 343–356 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Li, C., Qi, L., Li, Y.: MB-tensors and MB\(_0\)-tensors. Linear Algebra Appl. 484, 141–153 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Li, C., Wang, F., Zhao, J., Zhu, Y., Li, Y.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1–14 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ling, C., He, H., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. 63(1), 143–168 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Löfberg, J.: YALMIP : a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, (2004)Google Scholar
  21. 21.
    Löfberg, J.: Pre- and post-processing sums-of-squares programs in practice. IEEE Trans. Autom. Control 54, 1007–1011 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Luo, Z., Qi, L.: Completely positive tensors: properties, easily checkable subclasses and tractable relaxations. SIAM J. Matrix Anal. Appl. 37, 1675–1698 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pena, J., Vera, J.C., Zuluaga, L.F.: Completely positive reformulations for polynomial optimization. Math. Program. 151(2), 405–431 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Qi, L.: Eigenvalue of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Qi, L.: Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 439, 228–238 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Qi, L.: H\(^+\)-eigenvalues of Laplacian and signless Laplacian tensors. Commun. Math. Sci. 12, 1045–1064 (2014)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Qi, L., Song, Y.: An even order symmetric B tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Qi, L., Xu, C., Xu, Y.: Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm. SIAM J. Matrix Anal. Appl. 35(4), 1227–1241 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Shao, J.Y.: A general product of tensors with applications. Linear Algebra Appl. 439, 2350–2366 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Song, Y., Qi, L.: Necessary and sufficient conditions for copositive tensors. Linear and Multilinear Algebra 63(1), 120–131 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Song, Y., Qi, L.: Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim. Lett. (2016). doi: 10.1007/s11590-016-1104-7 MATHGoogle Scholar
  33. 33.
    Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069–1078 (2016)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Song, Y., Qi, L.: Eigenvalue analysis of constrained minimization problem for homogeneous polynomials. J. Global Optim. 64, 563–575 (2016)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sponsel, J., Bundfuss, S., Dür, M.: An improved algorithm to test copositivity. J. Global Optim. 52(3), 537–551 (2012)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(12), 625–653 (1999)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Tanaka, A., Yoshise, A.: Tractable subcones and LP-based algorithms for testing copositivity (2016). arXiv:1601.06878
  38. 38.
    Wang, Y., Huang, Z.H., Bai, X.L.: Exceptionally regular tensors and complementarity problems. Optim. Methods Softw. 31(4), 815–828 (2016)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Xu, J., Yao, Y.: An algorithm for determining copositive matrices. Linear Algebra Appl. 435(11), 2784–2792 (2011)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Yoshise, A., Matsukawa, Y.: On optimization over the doubly nonnegative cone. In: 2010 IEEE International Symposium on Computer-Aided Control System Design (CACSD), IEEE pp. 13–18 (2010)Google Scholar
  41. 41.
    Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM J. Matrix Anal. Appl. 35, 437–452 (2014)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Management ScienceQufu Normal UniversityRizhaoPeople’s Republic of China
  2. 2.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China
  3. 3.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong

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