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A proximal alternating minimization algorithm for the largest C-eigenvalue of piezoelectric-type tensors

Author

Listed:
  • Wenjie Wang

    (Qufu Normal University)

  • Haibin Chen

    (Qufu Normal University)

  • Yiju Wang

    (Qufu Normal University)

  • Guanglu Zhou

    (Curtin University)

Abstract

C-eigenvalues of piezoelectric-type tensors play an important role in piezoelectric effect and converse piezoelectric effect. While the largest C-eigenvalue of a given piezoelectric-type tensor has concrete physical meaning which determines the highest piezoelectric coupling constant. In this paper, we focus on computing the maximum C-eigenvalue of piezoelectric-type tensors which is a third degree polynomial problem. To do that, we first establish the equivalence between the proposed polynomial optimization problem (POP) and a multi-linear optimization problem (MOP) under conditions that the original objective function is concave. Then, an augmented POP (which can also be regarded as a regularized POP) is introduced for the purpose to guarantee the concavity of the underlying objective function. Theoretically, both the augmented POP 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 maximum C-eigenvalue. Furthermore, convergence of the proposed algorithm is established under mild conditions. Finally, some preliminary computational results on synthetic data sets are reported to show the efficiency of the proposed algorithm.

Suggested Citation

  • Wenjie Wang & Haibin Chen & Yiju Wang & Guanglu Zhou, 2023. "A proximal alternating minimization algorithm for the largest C-eigenvalue of piezoelectric-type tensors," Journal of Global Optimization, Springer, vol. 87(2), pages 405-422, November.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01180-w
    DOI: 10.1007/s10898-022-01180-w
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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. Li, Chaoqian & Liu, Yajun & Li, Yaotang, 2019. "C-eigenvalues intervals for piezoelectric-type tensors," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 244-250.
    3. 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.
    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. 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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