  We expand the scope of the alternating direction method of multipliers
(ADMM). Specifically, we show that ADMM, when employed to solve problems with
multiaffine constraints that satisfy certain verifiable assumptions, converges
to the set of constrained stationary points if the penalty parameter in the
augmented Lagrangian is sufficiently large. When the Kurdyka-\L{}ojasiewicz
(K-\L{}) property holds, this is strengthened to convergence to a single
constrained stationary point. Our analysis applies under assumptions that we
have endeavored to make as weak as possible. It applies to problems that
involve nonconvex and/or nonsmooth objective terms, in addition to the
multiaffine constraints that can involve multiple (three or more) blocks of
variables. To illustrate the applicability of our results, we describe examples
including nonnegative matrix factorization, sparse learning, risk parity
portfolio selection, nonconvex formulations of convex problems, and neural
network training. In each case, our ADMM approach encounters only subproblems
that have closed-form solutions.
