  The convex minimization of
$f(\mathbf{x})+g(\mathbf{x})+h(\mathbf{A}\mathbf{x})$ over $\mathbb{R}^n$ with
differentiable $f$ and linear operator $\mathbf{A}: \mathbb{R}^n\rightarrow
\mathbb{R}^m$, has been well-studied in the literature. By considering the
primal-dual optimality of the problem, many algorithms are proposed from
different perspectives such as monotone operator scheme and fixed point theory.
In this paper, we start with a base algorithm to reveal the connection between
several algorithms such as AFBA, PD3O and Chambolle-Pock. Then, we prove its
convergence under a relaxed assumption associated with the linear operator and
characterize the general constraint on primal and dual stepsizes. The result
improves the upper bound of stepsizes of AFBA and indicates that
Chambolle-Pock, as the special case of the base algorithm when $f=0$, can take
the stepsize of the dual iteration up to $4/3$ of the previously proven one.
