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

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 involving two proper, lower semicontinuous, convex functions and a bounded linear operator $L$. We study its convergence in real Hilbert spaces for step sizes $\tau,\sigma>0$ and relaxation parameter $0<\theta\le 1$. We prove that, if $\tau\sigma|L|^{2} \leq 4\theta(2-\theta)/(1 - 2\theta + 9\theta^{2} - 4\theta^{3})$, then the ergodic duality gap converges at rate $O(1/k)$, and that, when the inequality is strict, the primal-dual iterates converge weakly to a KKT point. In particular, this extends the weak-convergence theory to the previously unexplored regime $0<\theta\le 1/2$. The proof is based on a Lyapunov function that remains uniformly valid over the entire interval $0<\theta\le 1$.