  The alternating direction method of multipliers (ADMM) has been widely used
for solving structured convex optimization problems. In particular, the ADMM
can solve convex programs that minimize the sum of $N$ convex functions with
$N$-block variables linked by some linear constraints. While the convergence of
the ADMM for $N=2$ was well established in the literature, it remained an open
problem for a long time whether or not the ADMM for $N \ge 3$ is still
convergent. Recently, it was shown in [3] that without further conditions the
ADMM for $N\ge 3$ may actually fail to converge. In this paper, we show that
under some easily verifiable and reasonable conditions the global linear
convergence of the ADMM when $N\geq 3$ can still be assured, which is important
since the ADMM is a popular method for solving large scale multi-block
optimization models and is known to perform very well in practice even when
$N\ge 3$. Our study aims to offer an explanation for this phenomenon.

Optimization and Control