  The alternating direction method of multipliers (ADMM) is widely used in
solving structured convex optimization problems. Despite of its success in
practice, the convergence properties of the standard ADMM for minimizing the
sum of $N$ $(N\geq 3)$ convex functions with $N$ block variables linked by
linear constraints, have remained unclear for a very long time. In this paper,
we present convergence and convergence rate results for the standard ADMM
applied to solve $N$-block $(N\geq 3)$ convex minimization problem, under the
condition that one of these functions is convex (not necessarily strongly
convex) and the other $N-1$ functions are strongly convex. Specifically, in
that case the ADMM is proven to converge with rate $O(1/t)$ in a certain
ergodic sense, and $o(1/t)$ in non-ergodic sense, where $t$ denotes the number
of iterations. As a by-product, we also provide a simple proof for the $O(1/t)$
convergence rate of two-block ADMM in terms of both objective error and
constraint violation, without assuming any condition on the penalty parameter
and strong convexity on the functions.
