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

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 $τ,σ>0$ and relaxation parameter $0<θ\le 1$. We prove that, if $τσ|L|^{2} \leq 4θ(2-θ)/(1 - 2θ+ 9θ^{2} - 4θ^{3})$, 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 $0<θ\le 1/2$. The proof is based on a Lyapunov function that remains uniformly valid over the entire interval $0<θ\le 1$.