  This paper aims to present a fairly accessible generalization of several
symmetric Gauss-Seidel decomposition based multi-block proximal alternating
direction methods of multipliers (ADMMs) for convex composite optimization
problems. The proposed method unifies and refines many constructive techniques
that were separately developed for the computational efficiency of multi-block
ADMM-type algorithms. Specifically, the majorized augmented Lagrangian
functions, the indefinite proximal terms, the inexact symmetric Gauss-Seidel
decomposition theorem, the tolerance criteria of approximately solving the
subproblems, and the large dual step-lengths, are all incorporated in one
algorithmic framework, which we named as sGS-imiPADMM. From the popularity of
convergent variants of multi-block ADMMs in recent years, especially for
high-dimensional multi-block convex composite conic programming problems, the
unification presented in this paper, as well as the corresponding convergence
results, may have the great potential of facilitating the implementation of
many multi-block ADMMs in various problem settings.
