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Higher-degree eigenvalue complementarity problems for tensors

Author

Listed:
  • Chen Ling

    (Hangzhou Dianzi University)

  • Hongjin He

    (Hangzhou Dianzi University)

  • Liqun Qi

    (The Hong Kong Polytechnic University)

Abstract

In this paper, we introduce a unified framework of Tensor Higher-Degree Eigenvalue Complementarity Problem (THDEiCP), which goes beyond the framework of the typical Quadratic Eigenvalue Complementarity Problem for matrices. First, we study some topological properties of higher-degree cone eigenvalues of tensors. Based upon the symmetry assumptions on the underlying tensors, we then reformulate THDEiCP as a weakly coupled homogeneous polynomial optimization problem, which might be greatly helpful for designing implementable algorithms to solve the problem under consideration numerically. As more general theoretical results, we present the results concerning existence of solutions of THDEiCP without symmetry conditions. Finally, we propose an easily implementable algorithm to solve THDEiCP, and report some computational results.

Suggested Citation

  • Chen Ling & Hongjin He & Liqun Qi, 2016. "Higher-degree eigenvalue complementarity problems for tensors," Computational Optimization and Applications, Springer, vol. 64(1), pages 149-176, May.
    • Handle: RePEc:spr:coopap:v:64:y:2016:i:1:d:10.1007_s10589-015-9805-x
    DOI: 10.1007/s10589-015-9805-x
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    References listed on IDEAS

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    1. Samir Adly & Hadia Rammal, 2013. "A new method for solving Pareto eigenvalue complementarity problems," Computational Optimization and Applications, Springer, vol. 55(3), pages 703-731, July.
    2. Luís Fernandes & Joaquim Júdice & Hanif Sherali & Masao Fukushima, 2014. "On the computation of all eigenvalues for the eigenvalue complementarity problem," Journal of Global Optimization, Springer, vol. 59(2), pages 307-326, July.
    3. Luís Fernandes & Joaquim Júdice & Hanif Sherali & Maria Forjaz, 2014. "On an enumerative algorithm for solving eigenvalue complementarity problems," Computational Optimization and Applications, Springer, vol. 59(1), pages 113-134, October.
    4. A. Pinto da Costa & A. Seeger, 2010. "Cone-constrained eigenvalue problems: theory and algorithms," Computational Optimization and Applications, Springer, vol. 45(1), pages 25-57, January.
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    Cited by:

    1. Lu-Bin Cui & Yu-Dong Fan & Yi-Sheng Song & Shi-Liang Wu, 2022. "The Existence and Uniqueness of Solution for Tensor Complementarity Problem and Related Systems," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 321-334, January.
    2. Yisheng Song & Wei Mei, 2018. "Structural Properties of Tensors and Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 289-305, February.
    3. Guyan Ni & Minru Bai, 2016. "Spherical optimization with complex variablesfor computing US-eigenpairs," Computational Optimization and Applications, Springer, vol. 65(3), pages 799-820, December.
    4. Minru Bai & Jing Zhao & ZhangHui Zhang, 2020. "A descent cautious BFGS method for computing US-eigenvalues of symmetric complex tensors," Journal of Global Optimization, Springer, vol. 76(4), pages 889-911, April.
    5. Ruixue Zhao & Jinyan Fan, 2020. "Higher-degree tensor eigenvalue complementarity problems," Computational Optimization and Applications, Springer, vol. 75(3), pages 799-816, April.
    6. Zheng-Hai Huang & Liqun Qi, 2019. "Tensor Complementarity Problems—Part I: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 1-23, October.

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