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

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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 \(\epsilon \) within \({\mathcal {O}}(\frac{1}{\epsilon ^2})\) 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.

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Notes

  1. This package was downloaded from www.manopt.org.

  2. It is easy to ensure that such a condition is true from (A6) of Sect. 3.2.

  3. This toolbox was download from http://www.sandia.gov/~tgkolda/TensorToolbox/.

  4. This package was download from www.tensorlab.net.

  5. We thank one of the referees, who suggested us this more efficient way than our original one. In the original version, we used the default values \(\kappa =0.1\) and \(\theta =1\) in the stopping criterion [2, Equation (10)] of tCG. By tuning the parameters, we looked for a set of most efficient parameters (i.e., producing the fewest number of iterations for solving trust-region subproblems) in each example. With this new way, we can observe that both Algorithms 1 and 2 have lower computational costs compared to the original version.

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Acknowledgements

We are grateful to the associate editor and the two anonymous reviewers for their useful suggestions and comments, which helped to improve the presentation of the paper. We would like to thank Dr. Jianze Li for carefully reading the original manuscript and for providing constructive comments.

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Authors and Affiliations

  1. Department of Data Science, Anhui University of Technology, Maanshan, 243032, Anhui, China

    Zhou Sheng

  2. Anhui Provincial Joint Key Laboratory of Disciplines for Industrial Big Data Analysis and Intelligent Decision, Maanshan, Anhui, 243032, China

    Zhou Sheng

  3. School of Mathematics and Information Science, Center for Applied Mathematics of Guangxi (Guangxi University), Guangxi University, Nanning, 530004, Guangxi, China

    Gonglin Yuan

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  1. Zhou Sheng
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Correspondence to Gonglin Yuan.

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The work of the first author was supported by the Youth Foundation of Anhui University of Technology (No. QZ202114), and Anhui Provincial Natural Science Foundation (No. 2208085QA07). The work of the second author was partially supported by Guangxi Science and Technology Base and Talent Project (No. AD22080047), the Special Funds for Local Science and Technology Development Guided by the Central Government (No. ZY20198003), High Level Innovation Teams and Excellent Scholars Program in Guangxi institutions of higher education (No. [2019]52), and National Natural Science Foundation of China (No. 11661009).

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Sheng, Z., Yuan, G. Convergence and worst-case complexity of adaptive Riemannian trust-region methods for optimization on manifolds. J Glob Optim 89, 949–974 (2024). https://doi.org/10.1007/s10898-024-01378-0

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