The Chambolle–Pock method converges weakly with $$\theta >1/2$$ and $$\tau \sigma \Vert L\Vert ^2<4/(1+2\theta )$$

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 $$\tau $$ , $$\sigma $$ , 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 $$\theta > 1/2$$ and $$\tau \sigma \Vert L\Vert ^2<4/\mathord {\left( 1+2\theta \right) }$$ . Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.