Riemannian conditional gradient methods for composite optimization problems
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914572857769984 |
|---|---|
| author | Chen, Kangming Fukuda, Ellen H. |
| author_facet | Chen, Kangming Fukuda, Ellen H. |
| contents | In this paper, we propose Riemannian conditional gradient methods for minimizing composite functions, i.e., those that can be expressed as the sum of a smooth function and a retraction-based convex function. We analyze the convergence of the proposed algorithms, utilizing three types of step-size strategies: adaptive, diminishing, and those based on the Armijo condition. We establish the convergence rate of \(\mathcal{O}(1/k)\) for the adaptive and diminishing step sizes, where \(k\) denotes the number of iterations. Additionally, we derive an iteration complexity of \(\mathcal{O}(1/ε^2)\) for the Armijo step-size strategy to achieve \(ε\)-optimality, where \(ε\) is the optimality tolerance. Finally, the effectiveness of our algorithms is validated through some numerical experiments performed on the sphere and Stiefel manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riemannian conditional gradient methods for composite optimization problems Chen, Kangming Fukuda, Ellen H. Optimization and Control In this paper, we propose Riemannian conditional gradient methods for minimizing composite functions, i.e., those that can be expressed as the sum of a smooth function and a retraction-based convex function. We analyze the convergence of the proposed algorithms, utilizing three types of step-size strategies: adaptive, diminishing, and those based on the Armijo condition. We establish the convergence rate of \(\mathcal{O}(1/k)\) for the adaptive and diminishing step sizes, where \(k\) denotes the number of iterations. Additionally, we derive an iteration complexity of \(\mathcal{O}(1/ε^2)\) for the Armijo step-size strategy to achieve \(ε\)-optimality, where \(ε\) is the optimality tolerance. Finally, the effectiveness of our algorithms is validated through some numerical experiments performed on the sphere and Stiefel manifolds. |
| title | Riemannian conditional gradient methods for composite optimization problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.19427 |