  Recently, the alternating direction method of multipliers (ADMM) has found
many efficient applications in various areas; and it has been shown that the
convergence is not guaranteed when it is directly extended to the
multiple-block case of separable convex minimization problems where there are
$m\ge 3$ functions without coupled variables in the objective. This fact has
given great impetus to investigate various conditions on both the model and the
algorithm's parameter that can ensure the convergence of the direct extension
of ADMM (abbreviated as "e-ADMM"). Despite some results under very strong
conditions (e.g., at least $(m-1)$ functions should be strongly convex) that
are applicable to the generic case with a general $m$, some others concentrate
on the special case of $m=3$ under the relatively milder condition that only
one function is assumed to be strongly convex. We focus on extending the
convergence analysis from the case of $m=3$ to the more general case of
$m\ge3$. That is, we show the convergence of e-ADMM for the case of $m\ge 3$
with the assumption of only $(m-2)$ functions being strongly convex; and
establish its convergence rates in different scenarios such as the worst-case
convergence rates measured by iteration complexity and the asymptotically
linear convergence rate under stronger assumptions. Thus the convergence of
e-ADMM for the general case of $m\ge 4$ is proved; this result seems to be
still unknown even though it is intuitive given the known result of the case of
$m=3$. Even for the special case of $m=3$, our convergence results turn out to
be more general than the exiting results that are derived specifically for the
case of $m=3$.
