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

翻译：我们考虑一种原始双向算法,用Fr\'echet的可变美元和1美元来尽量减少美元(Ax)l(Fr\'echet diffef f$和1美元)。这种原始双向算法在文献中有两个名称:Primal-Dual-al-Pridial-point 运算法(PDFP$2$O)和Proximal Alternational illationor-Corrector(PaPC) 。在本文中,我们证明了它在比现有阶梯化较弱的条件下的趋同。我们用其他假设来显示其线性趋同。此外,我们表明这一条件(阶梯的上限)很紧,不能被削弱。这个结果还恢复了最近提出的正-不完全线化拉格朗加法。此外,我们将这一结果应用于分散的共识算法PG-EXTRA,并得出最弱的趋同条件。