  The primal-dual algorithm recently proposed by Chambolle & Pock (abbreviated
as CPA) for structured convex optimization is very efficient and popular. It
was shown by Chambolle & Pock in \cite{CP11} and also by Shefi & Teboulle in
\cite{ST14} that CPA and variants are closely related to preconditioned
versions of the popular alternating direction method of multipliers
(abbreviated as ADM). In this paper, we further clarify this connection and
show that CPAs generate exactly the same sequence of points with the so-called
linearized ADM (abbreviated as LADM) applied to either the primal problem or
its Lagrangian dual, depending on different updating orders of the primal and
the dual variables in CPAs, as long as the initial points for the LADM are
properly chosen. The dependence on initial points for LADM can be relaxed by
focusing on cyclically equivalent forms of the algorithms. Furthermore, by
utilizing the fact that CPAs are applications of a general weighted proximal
point method to the mixed variational inequality formulation of the KKT system,
where the weighting matrix is positive definite under a parameter condition, we
are able to propose and analyze inertial variants of CPAs. Under certain
conditions, global point-convergence, nonasymptotic $O(1/k)$ and asymptotic
$o(1/k)$ convergence rate of the proposed inertial CPAs can be guaranteed,
where $k$ denotes the iteration index. Finally, we demonstrate the profits
gained by introducing the inertial extrapolation step via experimental results
on compressive image reconstruction based on total variation minimization.
