  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.
