The Chambolle--Pock method converges weakly with $θ>1/2$ and $τσ\|L\|^2<4/(1+2θ)$

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 $τσ\|L\|^2<4/(1+2θ)$. Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.