  We investigate the techniques and ideas used in the convergence analysis of
two proximal ADMM algorithms for solving convex optimization problems involving
compositions with linear operators. Besides this, we formulate a variant of the
ADMM algorithm that is able to handle convex optimization problems involving an
additional smooth function in its objective, and which is evaluated through its
gradient. Moreover, in each iteration we allow the use of variable metrics,
while the investigations are carried out in the setting of infinite dimensional
Hilbert spaces. This algorithmic scheme is investigated from the point of view
of its convergence properties.
