  In this paper, we show that for a class of linearly constrained convex
composite optimization problems, an (inexact) symmetric Gauss-Seidel based
majorized multi-block proximal alternating direction method of multipliers
(ADMM) is equivalent to an {\em inexact} proximal augmented Lagrangian method
(ALM). This equivalence not only provides new perspectives for understanding
some ADMM-type algorithms but also supplies meaningful guidelines on
implementing them to achieve better computational efficiency. Even for the
two-block case, a by-product of this equivalence is the convergence of the
whole sequence generated by the classic ADMM with a step-length that exceeds
the conventional upper bound of $(1+\sqrt{5})/2$, if one part of the objective
is linear. This is exactly the problem setting in which the very first
convergence analysis of ADMM was conducted by Gabay and Mercier in 1976, but,
even under notably stronger assumptions, only the convergence of the primal
sequence was known. A collection of illustrative examples are provided to
demonstrate the breadth of applications for which our results can be used.
Numerical experiments on solving a large number of linear and convex quadratic
semidefinite programming problems are conducted to illustrate how the
theoretical results established here can lead to improvements on the
corresponding practical implementations.
