The Chambolle–Pock method converges weakly with θ > 1/2 and τ σ∥L∥2 < 4/(1 + 2θ)
Banert, Sebastian LU ; Upadhyaya, Manu LU
and Giselsson, Pontus
LU
(2025)
In Optimization Letters
- Abstract
- The Chambolle–Pock method is a versatile three-parameter algorithm designed to solve a broad class of composite convex optimization problems, which encompass two proper, lower semicontinuous, and convex functions, along with a linear operator L. The functions are accessed via their proximal operators, while the linear operator is evaluated in a forward manner. Among the three algorithm parameters τ , σ, and θ; τ, σ > 0 serve as step sizes for the proximal operators, and θ is an extrapolation step parameter. Previous convergence results have been based on the assumption that θ = 1. We demonstrate that weak convergence is achievable whenever θ > 1/2 and τ σ∥L∥2 < 4/(1 + 2θ). Moreover, we establish tightness of the step... (More)
- The Chambolle–Pock method is a versatile three-parameter algorithm designed to solve a broad class of composite convex optimization problems, which encompass two proper, lower semicontinuous, and convex functions, along with a linear operator L. The functions are accessed via their proximal operators, while the linear operator is evaluated in a forward manner. Among the three algorithm parameters τ , σ, and θ; τ, σ > 0 serve as step sizes for the proximal operators, and θ is an extrapolation step parameter. Previous convergence results have been based on the assumption that θ = 1. We demonstrate that weak convergence is achievable whenever θ > 1/2 and τ σ∥L∥2 < 4/(1 + 2θ). Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.
(Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/cceba969-7f55-4991-90ca-99ee38431112
- author
- Banert, Sebastian
LU
; Upadhyaya, Manu
LU
and Giselsson, Pontus
LU
- organization
- publishing date
- 2025
- type
- Contribution to journal
- publication status
- epub
- subject
- keywords
- Chambolle–Pock, Convex optimization, First-order methods
- in
- Optimization Letters
- publisher
- Springer
- external identifiers
-
- scopus:105019488119
- ISSN
- 1862-4472
- DOI
- 10.1007/s11590-025-02250-0
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: © The Author(s) 2025.
- id
- cceba969-7f55-4991-90ca-99ee38431112
- date added to LUP
- 2026-01-19 13:42:48
- date last changed
- 2026-01-19 14:48:22
@article{cceba969-7f55-4991-90ca-99ee38431112,
abstract = {{The Chambolle–Pock method is a versatile three-parameter algorithm designed to solve a broad class of composite convex optimization problems, which encompass two proper, lower semicontinuous, and convex functions, along with a linear operator L. The functions are accessed via their proximal operators, while the linear operator is evaluated in a forward manner. Among the three algorithm parameters τ , σ, and θ; τ, σ > 0 serve as step sizes for the proximal operators, and θ is an extrapolation step parameter. Previous convergence results have been based on the assumption that θ = 1. We demonstrate that weak convergence is achievable whenever θ > 1/2 and τ σ∥L∥<sup>2</sup> < 4/(1 + 2θ). Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.<br/>}},
author = {{Banert, Sebastian and Upadhyaya, Manu and Giselsson, Pontus}},
issn = {{1862-4472}},
keywords = {{Chambolle–Pock; Convex optimization; First-order methods}},
language = {{eng}},
publisher = {{Springer}},
series = {{Optimization Letters}},
title = {{The Chambolle–Pock method converges weakly with θ > 1/2 and τ σ∥L∥<sup>2</sup> < 4/(1 + 2θ)}},
url = {{http://dx.doi.org/10.1007/s11590-025-02250-0}},
doi = {{10.1007/s11590-025-02250-0}},
year = {{2025}},
}