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Tighter bound estimation for efficient biquadratic optimization over unit spheres

Author

Listed:
  • Shigui Li

    (South China University of Technology)

  • Linzhang Lu

    (Guizhou Normal University
    Xiamen University)

  • Xing Qiu

    (University of Rochester)

  • Zhen Chen

    (Guizhou Normal University)

  • Delu Zeng

    (South China University of Technology)

Abstract

Bi-quadratic programming over unit spheres is a fundamental problem in quantum mechanics introduced by pioneer work of Einstein, Schrödinger, and others. It has been shown to be NP-hard; so it must be solve by efficient heuristic algorithms such as the block improvement method (BIM). This paper focuses on the maximization of bi-quadratic forms with nonnegative coefficient tensors, which leads to a rank-one approximation problem that is equivalent to computing the M-spectral radius and its corresponding eigenvectors. Specifically, we propose a tight upper bound of the M-spectral radius for nonnegative fourth-order partially symmetric (PS) tensors. This bound, serving as an improved shift parameter, significantly enhances the convergence speed of BIM while maintaining computational complexity aligned with the initial shift parameter of BIM. Moreover, we elucidate that the computation cost of such upper bound can be further simplified for certain sets and delve into the nature of these sets. Building on the insights gained from the proposed bounds, we derive the exact solutions of the M-spectral radius and its corresponding M-eigenvectors for certain classes of fourth-order PS-tensors and discuss the nature of this specific category. Lastly, as a practical application, we introduce a testable sufficient condition for the strong ellipticity in the field of solid mechanics. Numerical experiments demonstrate the utility of the proposed results.

Suggested Citation

  • Shigui Li & Linzhang Lu & Xing Qiu & Zhen Chen & Delu Zeng, 2024. "Tighter bound estimation for efficient biquadratic optimization over unit spheres," Journal of Global Optimization, Springer, vol. 90(2), pages 323-353, October.
    • Handle: RePEc:spr:jglopt:v:90:y:2024:i:2:d:10.1007_s10898-024-01401-4
    DOI: 10.1007/s10898-024-01401-4
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    References listed on IDEAS

    as
    1. Li, Suhua & Li, Yaotang, 2021. "Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    2. Yuning Yang & Qingzhi Yang & Liqun Qi, 2014. "Approximation Bounds for Trilinear and Biquadratic Optimization Problems Over Nonconvex Constraints," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 841-858, December.
    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. Yuning Yang & Qingzhi Yang, 2012. "On solving biquadratic optimization via semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 53(3), pages 845-867, December.
    5. 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.
    6. Ding, Weiyang & Liu, Jinjie & Qi, Liqun & Yan, Hong, 2020. "Elasticity M-tensors and the strong ellipticity condition," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    Full references (including those not matched with items on IDEAS)

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