New convergence analysis of a primal-dual algorithm with large stepsizes

We consider a primal-dual algorithm for minimizing $f(x)+h\square l(Ax)$ with Fr\'echet differentiable $f$ and $l^*$. This has two names in literature: Primal-Dual Fixed-Point based on the Proximity Operator (PDFP$^2$O) Proximal Alternating Predictor-Corrector (PAPC). In this paper, we prove its convergence under weaker condition stepsizes than existing ones. With additional assumptions, show linear convergence. addition, that (the upper bound of stepsize) is tight can not be weakened. result also recovers recently proposed positive-indefinite linearized augmented Lagrangian method. apply to decentralized consensus PG-EXTRA derive weakest condition.