  In this paper, we propose a new primal-dual algorithm for minimizing $f(x) +
g(x) + h(Ax)$, where $f$, $g$, and $h$ are proper lower semi-continuous convex
functions, $f$ is differentiable with a Lipschitz continuous gradient, and $A$
is a bounded linear operator. The proposed algorithm has some famous
primal-dual algorithms for minimizing the sum of two functions as special
cases. E.g., it reduces to the Chambolle-Pock algorithm when $f = 0$ and the
proximal alternating predictor-corrector when $g = 0$. For the general convex
case, we prove the convergence of this new algorithm in terms of the distance
to a fixed point by showing that the iteration is a nonexpansive operator. In
addition, we prove the $O(1/k)$ ergodic convergence rate in the primal-dual
gap. With additional assumptions, we derive the linear convergence rate in
terms of the distance to the fixed point. Comparing to other primal-dual
algorithms for solving the same problem, this algorithm extends the range of
acceptable parameters to ensure its convergence and has a smaller per-iteration
cost. The numerical experiments show the efficiency of this algorithm.
