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Real eigenvalues of nonsymmetric tensors

Author

Listed:
  • Jiawang Nie

    (University of California San Diego)

  • Xinzhen Zhang

    (Tianjin University)

Abstract

This paper discusses the computation of real $$\mathtt {Z}$$ Z -eigenvalues and $$\mathtt {H}$$ H -eigenvalues of nonsymmetric tensors. A generic nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. The number of $$\mathtt {H}$$ H -eigenvalues is finite for all tensors. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every tensor that has finitely many real $$\mathtt {Z}$$ Z -eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every tensor, we can compute all its real $$\mathtt {H}$$ H -eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations.

Suggested Citation

  • Jiawang Nie & Xinzhen Zhang, 2018. "Real eigenvalues of nonsymmetric tensors," Computational Optimization and Applications, Springer, vol. 70(1), pages 1-32, May.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:1:d:10.1007_s10589-017-9973-y
    DOI: 10.1007/s10589-017-9973-y
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    References listed on IDEAS

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    1. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
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    Cited by:

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    2. Lulu Cheng & Xinzhen Zhang & Guyan Ni, 2021. "A semidefinite relaxation method for second-order cone tensor eigenvalue complementarity problems," Journal of Global Optimization, Springer, vol. 79(3), pages 715-732, March.
    3. Meilan Zeng, 2021. "Tensor Z-eigenvalue complementarity problems," Computational Optimization and Applications, Springer, vol. 78(2), pages 559-573, March.
    4. Li, Li & Yan, Xihong & Zhang, Xinzhen, 2022. "An SDP relaxation method for perron pairs of a nonnegative tensor," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    5. Lulu Cheng & Xinzhen Zhang, 2020. "A semidefinite relaxation method for second-order cone polynomial complementarity problems," Computational Optimization and Applications, Springer, vol. 75(3), pages 629-647, April.

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