Tight global linear convergence rate bounds for Douglas–Rachford splitting
Giselsson, Pontus LU
(2017)
In Journal of Fixed Point Theory and Applications
19(4).
p.2241-2270
- Abstract
Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions... (More)
Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.
(Less)
- author
- Giselsson, Pontus
LU
- organization
- publishing date
- 2017-12
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Douglas–Rachford splitting, Fixed-point iterations, Linear convergence, Monotone operators
- in
- Journal of Fixed Point Theory and Applications
- volume
- 19
- issue
- 4
- pages
- 2241 - 2270
- publisher
- Springer
- external identifiers
-
- wos:000414719100004
- scopus:85015059438
- ISSN
- 1661-7738
- DOI
- 10.1007/s11784-017-0417-1
- language
- English
- LU publication?
- yes
- id
- b9459797-a905-4368-8076-95ea4a2edc47
- date added to LUP
- 2017-03-23 08:40:26
- date last changed
- 2025-01-06 10:08:57
@article{b9459797-a905-4368-8076-95ea4a2edc47,
abstract = {{<p>Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.</p>}},
author = {{Giselsson, Pontus}},
issn = {{1661-7738}},
keywords = {{Douglas–Rachford splitting; Fixed-point iterations; Linear convergence; Monotone operators}},
language = {{eng}},
number = {{4}},
pages = {{2241--2270}},
publisher = {{Springer}},
series = {{Journal of Fixed Point Theory and Applications}},
title = {{Tight global linear convergence rate bounds for Douglas–Rachford splitting}},
url = {{http://dx.doi.org/10.1007/s11784-017-0417-1}},
doi = {{10.1007/s11784-017-0417-1}},
volume = {{19}},
year = {{2017}},
}