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Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Author

Listed:
  • Shenglong Hu

    (The Hong Kong Polytechnic University)

  • Guoyin Li

    (University of New South Wales)

  • Liqun Qi

    (The Hong Kong Polytechnic University)

  • Yisheng Song

    (The Hong Kong Polytechnic University)

Abstract

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.

Suggested Citation

  • Shenglong Hu & Guoyin Li & Liqun Qi & Yisheng Song, 2013. "Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 717-738, September.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:3:d:10.1007_s10957-013-0293-9
    DOI: 10.1007/s10957-013-0293-9
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    References listed on IDEAS

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    1. Guoyin Li, 2012. "Global Quadratic Minimization over Bivalent Constraints: Necessary and Sufficient Global Optimality Condition," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 710-726, March.
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    Cited by:

    1. Shenglong Hu & Guoyin Li & Liqun Qi, 2016. "A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 446-474, February.
    2. Na Zhao & Qingzhi Yang & Yajun Liu, 2017. "Computing the generalized eigenvalues of weakly symmetric tensors," Computational Optimization and Applications, Springer, vol. 66(2), pages 285-307, March.
    3. Chen, Haibin & Li, Guoyin & Qi, Liqun, 2016. "Further results on Cauchy tensors and Hankel tensors," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 50-62.
    4. Gaohang Yu & Zefeng Yu & Yi Xu & Yisheng Song & Yi Zhou, 2016. "An adaptive gradient method for computing generalized tensor eigenpairs," Computational Optimization and Applications, Springer, vol. 65(3), pages 781-797, December.

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