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Convergence and worst-case complexity of adaptive Riemannian trust-region methods for optimization on manifolds

Author

Listed:
  • Zhou Sheng

    (Anhui University of Technology
    Anhui Provincial Joint Key Laboratory of Disciplines for Industrial Big Data Analysis and Intelligent Decision)

  • Gonglin Yuan

    (Guangxi University)

Abstract

Trust-region methods have received massive attention in a variety of continuous optimization. They aim to obtain a trial step by minimizing a quadratic model in a region of a certain trust-region radius around the current iterate. This paper proposes an adaptive Riemannian trust-region algorithm for optimization on manifolds, in which the trust-region radius depends linearly on the norm of the Riemannian gradient at each iteration. Under mild assumptions, we establish the liminf-type convergence, lim-type convergence, and global convergence results of the proposed algorithm. In addition, the proposed algorithm is shown to reach the conclusion that the norm of the Riemannian gradient is smaller than $$\epsilon $$ ϵ within $${\mathcal {O}}(\frac{1}{\epsilon ^2})$$ O ( 1 ϵ 2 ) iterations. Some numerical examples of tensor approximations are carried out to reveal the performances of the proposed algorithm compared to the classical Riemannian trust-region algorithm.

Suggested Citation

  • Zhou Sheng & Gonglin Yuan, 2024. "Convergence and worst-case complexity of adaptive Riemannian trust-region methods for optimization on manifolds," Journal of Global Optimization, Springer, vol. 89(4), pages 949-974, August.
    • Handle: RePEc:spr:jglopt:v:89:y:2024:i:4:d:10.1007_s10898-024-01378-0
    DOI: 10.1007/s10898-024-01378-0
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    References listed on IDEAS

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    1. Xiaojing Zhu, 2017. "A Riemannian conjugate gradient method for optimization on the Stiefel manifold," Computational Optimization and Applications, Springer, vol. 67(1), pages 73-110, May.
    2. Marcio Antônio de A. Bortoloti & Teles A. Fernandes & Orizon P. Ferreira & Jinyun Yuan, 2020. "Damped Newton’s method on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 77(3), pages 643-660, July.
    3. Shenglong Hu, 2020. "An inexact augmented Lagrangian method for computing strongly orthogonal decompositions of tensors," Computational Optimization and Applications, Springer, vol. 75(3), pages 701-737, April.
    4. Shi, Zhenjun & Wang, Shengquan, 2011. "Nonmonotone adaptive trust region method," European Journal of Operational Research, Elsevier, vol. 208(1), pages 28-36, January.
    5. Zhou Sheng & Gonglin Yuan, 2018. "An effective adaptive trust region algorithm for nonsmooth minimization," Computational Optimization and Applications, Springer, vol. 71(1), pages 251-271, September.
    Full references (including those not matched with items on IDEAS)

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