The Chambolle-Pock method converges weakly with $0<\theta \le 1/2$ and $\tau \sigma \Vert L{\Vert }^2<4\theta (2-\theta )/(1-2\theta +9{\theta }^2-4{\theta }^3)$

Abstract

The Chambolle-Pock method, also known as the primal-dual hybrid gradient method, is a standard first-order algorithm for convex-concave saddle-point problems and composite convex optimization. We establish weak sequential convergence of its primal-dual iterates in real Hilbert spaces for every $0<\theta \le 1/2$ whenever $\tau \sigma \Vert L{\Vert }^2<4\theta (2-\theta )/(1-2\theta +9{\theta }^2-4{\theta }^3)$. This extends the weak-convergence theory to a previously unexplored range of extrapolation parameters.