Primal-dual proximal bundle and conditional gradient methods for convex problems
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918147640000512 |
|---|---|
| author | Liang, Jiaming |
| author_facet | Liang, Jiaming |
| contents | This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle methods for solving convex-concave nonsmooth composite saddle-point problems and establish the iteration-complexity to find an approximate saddle-point. This paper places special emphasis on the primal-dual perspective of the proximal bundle method. In particular, we discover an interesting duality between the conditional gradient method and the cutting-plane scheme used within the proximal bundle method. Leveraging this duality, we further develop novel variants of both the conditional gradient method and the cutting-plane scheme. Additionally, we report numerical experiments to demonstrate the effectiveness and efficiency of the proposed proximal bundle methods in comparison with the subgradient method for solving a regularized matrix game. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00585 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primal-dual proximal bundle and conditional gradient methods for convex problems Liang, Jiaming Optimization and Control This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle methods for solving convex-concave nonsmooth composite saddle-point problems and establish the iteration-complexity to find an approximate saddle-point. This paper places special emphasis on the primal-dual perspective of the proximal bundle method. In particular, we discover an interesting duality between the conditional gradient method and the cutting-plane scheme used within the proximal bundle method. Leveraging this duality, we further develop novel variants of both the conditional gradient method and the cutting-plane scheme. Additionally, we report numerical experiments to demonstrate the effectiveness and efficiency of the proposed proximal bundle methods in comparison with the subgradient method for solving a regularized matrix game. |
| title | Primal-dual proximal bundle and conditional gradient methods for convex problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.00585 |