The Pivoting Framework: Frank-Wolfe Algorithms with Active Set Size Control
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908478885330944 |
|---|---|
| author | Wirth, Elias Besançon, Mathieu Pokutta, Sebastian |
| author_facet | Wirth, Elias Besançon, Mathieu Pokutta, Sebastian |
| contents | We propose the pivoting meta algorithm (PM) to enhance optimization algorithms that generate iterates as convex combinations of vertices of a feasible region $C\subseteq \mathbb{R}^n$, including Frank-Wolfe (FW) variants. PM guarantees that the active set (the set of vertices in the convex combination) of the modified algorithm remains as small as $\mathrm{dim}(C)+1$ as stipulated by Carathéodory's theorem. PM achieves this by reformulating the active set expansion task into an equivalent linear program, which can be efficiently solved using a single pivot step akin to the primal simplex algorithm; the convergence rate of the original algorithms are maintained. Furthermore, we establish the connection between PM and active set identification, in particular showing under mild assumptions that PM applied to the away-step Frank-Wolfe algorithm or the blended pairwise Frank-Wolfe algorithm bounds the active set size by the dimension of the optimal face plus $1$. We provide numerical experiments to illustrate practicality and efficacy on active set size reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11760 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Pivoting Framework: Frank-Wolfe Algorithms with Active Set Size Control Wirth, Elias Besançon, Mathieu Pokutta, Sebastian Optimization and Control We propose the pivoting meta algorithm (PM) to enhance optimization algorithms that generate iterates as convex combinations of vertices of a feasible region $C\subseteq \mathbb{R}^n$, including Frank-Wolfe (FW) variants. PM guarantees that the active set (the set of vertices in the convex combination) of the modified algorithm remains as small as $\mathrm{dim}(C)+1$ as stipulated by Carathéodory's theorem. PM achieves this by reformulating the active set expansion task into an equivalent linear program, which can be efficiently solved using a single pivot step akin to the primal simplex algorithm; the convergence rate of the original algorithms are maintained. Furthermore, we establish the connection between PM and active set identification, in particular showing under mild assumptions that PM applied to the away-step Frank-Wolfe algorithm or the blended pairwise Frank-Wolfe algorithm bounds the active set size by the dimension of the optimal face plus $1$. We provide numerical experiments to illustrate practicality and efficacy on active set size reduction. |
| title | The Pivoting Framework: Frank-Wolfe Algorithms with Active Set Size Control |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2407.11760 |