  Embedding randomization procedures in the Alternating Direction Method of
Multipliers (ADMM) has recently attracted an increasing amount of interest as a
remedy to the fact that the direct $n$-block generalization of ADMM is not
necessarily convergent ($n \geq 3$). Even if, in practice, the introduction of
such techniques could mitigate the diverging behaviour of the $n$-block
extension of ADMM, from the theoretical point of view, it can ensure just the
convergence in expectation, which may not be a good indicator of its robustness
and efficiency. In this work, analysing the strongly convex quadratic
programming case, we interpret the $n$-block Gauss-Seidel sweep performed by
ADMM in the context of the inexact Augmented Lagrangian Method. Using the
proposed analysis, we are able to outline an alternative technique to those
present in literature which, supported from stronger theoretical guarantees, is
able to ensure the convergence of the $n$-block generalization of the ADMM
method.
