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An efficient alternating minimization method for fourth degree polynomial optimization

Author

Listed:
  • Haibin Chen

    (Qufu Normal University)

  • Hongjin He

    (Ningbo University)

  • Yiju Wang

    (Qufu Normal University)

  • Guanglu Zhou

    (Curtin University)

Abstract

In this paper, we consider a class of fourth degree polynomial problems, which are NP-hard. First, we are concerned with the bi-quadratic optimization problem (Bi-QOP) over compact sets, which is proven to be equivalent to a multi-linear optimization problem (MOP) when the objective function of Bi-QOP is concave. Then, we introduce an augmented Bi-QOP (which can also be regarded as a regularized Bi-QOP) for the purpose to guarantee the concavity of the underlying objective function. Theoretically, both the augmented Bi-QOP and the original problem share the same optimal solutions when the compact sets are specified as unit spheres. By exploiting the multi-block structure of the resulting MOP, we accordingly propose a proximal alternating minimization algorithm to get an approximate optimal value of the problem under consideration. Convergence of the proposed algorithm is established under mild conditions. Finally, some preliminary computational results on synthetic datasets are reported to show the efficiency of the proposed algorithm.

Suggested Citation

  • Haibin Chen & Hongjin He & Yiju Wang & Guanglu Zhou, 2022. "An efficient alternating minimization method for fourth degree polynomial optimization," Journal of Global Optimization, Springer, vol. 82(1), pages 83-103, January.
    • Handle: RePEc:spr:jglopt:v:82:y:2022:i:1:d:10.1007_s10898-021-01060-9
    DOI: 10.1007/s10898-021-01060-9
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    References listed on IDEAS

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    1. P. Tseng, 2001. "Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 475-494, June.
    2. Haibin Chen & Zheng-Hai Huang & Liqun Qi, 2018. "Copositive tensor detection and its applications in physics and hypergraphs," Computational Optimization and Applications, Springer, vol. 69(1), pages 133-158, January.
    3. Xinzhen Zhang & Chen Ling & Liqun Qi, 2011. "Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints," Journal of Global Optimization, Springer, vol. 49(2), pages 293-311, February.
    4. Haibin Chen & Zheng-Hai Huang & Liqun Qi, 2017. "Copositivity Detection of Tensors: Theory and Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 746-761, September.
    5. Yuning Yang & Qingzhi Yang, 2012. "On solving biquadratic optimization via semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 53(3), pages 845-867, December.
    6. Sheng-Long Hu & Zheng-Hai Huang, 2011. "Alternating direction method for bi-quadratic programming," Journal of Global Optimization, Springer, vol. 51(3), pages 429-446, November.
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