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Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

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Abstract

In this paper, the concepts of Pareto H-eigenvalue and Pareto Z-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto H-eigenvalue (Pareto Z-eigenvalue). Furthermore, the minimum Pareto H-eigenvalue (or Pareto Z-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor \({\mathcal {A}}\) is strictly copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is positive, and \({\mathcal {A}}\) is copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is non-negative.

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References

  1. Chang, K.C.: A nonlinear Krein Rutman theorem. J. Syst. Sci. Com. 22(4), 542–554 (2009)

    Article  MATH  Google Scholar 

  2. Chang, K.C., Pearson, K., Zhang, K.: Perron–Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6, 507–520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, K.C., Pearson, K., Zhang, K.: On eigenvalue problems of real symmetric tensors. J. Math. Anal. Appl. 350, 416–422 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems: vol. I. Springer, New York (2011)

    Google Scholar 

  5. Hiriart-Urruty, J.B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52(4), 593–629 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Han, J.Y., Xiu, N.H., Qi, H.D.: Nonlinear Omplementary Theory and Algorithm. Shanghai Science and Technology Press, Shanghai (2006). (in Chinese)

    Google Scholar 

  7. Ling, C., He, H., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. (2015). doi:10.1007/s10589-015-9767-z

  8. Li, C., Wang, F., Zhao, J., Zhu, Y., Li, Y.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comp. Appl. Math. 255, 1–14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lim, L.H.: Singular values and eigenvalues of tensors: A variational approach. In: Proc. 1st IEEE International workshop on computational advances of multi-tensor adaptive processing, December 13–15, pp 129–132 (2005)

  10. Motzkin, T.S.: Quadratic Forms. National Bureau of Standards Report 1818, pp. 11–12 (1952)

  11. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MATH  Google Scholar 

  12. Qi, L.: Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines. J. Symb. Comput. 41, 1309–1327 (2006)

    Article  MATH  Google Scholar 

  13. Qi, L.: Eigenvalues and invariants of tensors. J. Math. Anal. Appl. 325, 1363–1377 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Qi, L.: \(H^+\)-eigenvalues of Laplacian and signless Laplacian tensors. Commun. Math. Sci. 12, 1045–1064 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Qi, L.: Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 439(1), 228–238 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Qi, L., Song, Y.: An even order symmetric B tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Song, Y., Qi, L.: Necessary and sufficient conditions for copositive tensors. Linear Multilinear Algebra 63(1), 120–131 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Song, Y., Qi, L.: Positive eigenvalue-eigenvector of nonlinear positive mappings. Front. Math. China 9(1), 181–199 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Song, Y., Qi, L.: The existence and uniqueness of eigenvalues for monotone homogeneous mapping pairs. Nonlinear Anal. 75(13), 5283–5293 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Song, Y., Qi, L.: Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J. Matrix Anal. Appl. 34, 1581–1595 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165(3), 854–873 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Seeger, A.: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions. Linear Algebra Appl. 292, 1–14 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, Y., Yang, Q.: Further results for Perron–Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31(5), 2517–2530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, Q., Yang, Y.: Further results for Perron–Frobenius theorem for nonnegative tensors II. SIAM J. Matrix Anal. Appl. 32(4), 1236–1250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank the anonymous referee for his valuable suggestions which helped us to improve this manuscript.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Normal University, XinXiang, 453007, Henan, People’s Republic of China

    Yisheng Song

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Yisheng Song & Liqun Qi

Authors
  1. Yisheng Song
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  2. Liqun Qi
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Correspondence to Yisheng Song.

Additional information

Yisheng Song work was partially supported by the National Natural Science Foundation of P.R. China (Grant No. 11171094, 11271112, 61262026), NCET Programm of the Ministry of Education (NCET 13-0738), science and technology programm of Jiangxi Education Committee (LDJH12088).

Liqun Qi work was supported by the Hong Kong Research Grant Council (Grant No. PolyU 502111, 501212, 501913 and 15302114).

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Song, Y., Qi, L. Eigenvalue analysis of constrained minimization problem for homogeneous polynomial. J Glob Optim 64, 563–575 (2016). https://doi.org/10.1007/s10898-015-0343-y

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  • DOI: https://doi.org/10.1007/s10898-015-0343-y

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