The Chambolle--Pock method converges weakly with $\theta >1\mathrm{/}2$ and $\tau \sigma \mathrm{\parallel }L{\mathrm{\parallel }}^2<4\mathrm{/}(1+2\theta )$

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 $\tau$, $\sigma$, and $\theta$; $\tau ,\sigma >0$ serve as step sizes for the proximal operators, and $\theta$ is an extrapolation step parameter. Previous convergence results have been based on the assumption that $\theta =1$. We demonstrate that weak convergence is achievable whenever $\theta >1\mathrm{/}2$ and $\tau \sigma \mathrm{\parallel }L{\mathrm{\parallel }}^2<4\mathrm{/}(1+2\theta )$. Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.