  In this paper, we study a general optimization model, which covers a large
class of existing models for many applications in imaging sciences. To solve
the resulting possibly nonconvex, nonsmooth and non-Lipschitz optimization
problem, we adapt the alternating direction method of multipliers (ADMM) with a
general dual step-size to solve a reformulation that contains three blocks of
variables, and analyze its convergence. We show that for any dual step-size
less than the golden ratio, there exists a computable threshold such that if
the penalty parameter is chosen above such a threshold and the sequence thus
generated by our ADMM is bounded, then the cluster point of the sequence gives
a stationary point of the nonconvex optimization problem. We achieve this via a
potential function specifically constructed for our ADMM. Moreover, we
establish the global convergence of the whole sequence if, in addition, this
special potential function is a Kurdyka-{\L}ojasiewicz function. Furthermore,
we present a simple strategy for initializing the algorithm to guarantee
boundedness of the sequence. Finally, we perform numerical experiments
comparing our ADMM with the proximal alternating linearized minimization (PALM)
proposed in [5] on the background/foreground extraction problem with real data.
The numerical results show that our ADMM with a nontrivial dual step-size is
efficient.
