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

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 $\tau \sigma \mathrm{\mid }L{\mathrm{\mid }}^2\le 4\theta (2-\theta )\mathrm{/}(1-2\theta +9{\theta }^2-4{\theta }^3)$, then the ergodic duality gap converges at rate $O(1\mathrm{/}k)$, 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 $0<\theta \le 1\mathrm{/}2$. The proof is based on a Lyapunov function that remains uniformly valid over the entire interval $0<\theta \le 1$.