The Chambolle--Pock method converges weakly with $θ>1/2$ and $τσ\|L\|^2<4/(1+2θ)$
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| Format: | Preprint |
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2023
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| _version_ | 1866909871716171776 |
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| author | Banert, Sebastian Upadhyaya, Manu Giselsson, Pontus |
| author_facet | Banert, Sebastian Upadhyaya, Manu Giselsson, Pontus |
| contents | 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 $τσ\|L\|^2<4/(1+2θ)$. Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03998 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Chambolle--Pock method converges weakly with $θ>1/2$ and $τσ\|L\|^2<4/(1+2θ)$ Banert, Sebastian Upadhyaya, Manu Giselsson, Pontus Optimization and Control 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 $τσ\|L\|^2<4/(1+2θ)$. Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated. |
| title | The Chambolle--Pock method converges weakly with $θ>1/2$ and $τσ\|L\|^2<4/(1+2θ)$ |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2309.03998 |