  In this paper we provide a generalization of the Douglas-Rachford splitting
(DRS) for solving monotone inclusions in a real Hilbert space involving a
general linear operator. The proposed method activates the linear operator
separately from the monotone operators appearing in the inclusion and, in the
simplest case when the linear operator is the identity, it reduces to classical
DRS. Moreover, the weak convergence of primal-dual sequences to a point in the
extended solution set is guaranteed, generalizing the main result in Svaiter
(2011). Inspired in Gabay (1983), we derive a new Split-ADMM (SADMM) by
applying our method to the dual of a convex optimization problem involving a
linear operator which can be expressed as the composition of two linear
operators. The proposed SADMM activates one linear operator implicitly and the
other explicitly and we recover ADMM when the latter is set as the identity.
Connections and comparisons of our theoretical results with respect to the
literature are provided for the main algorithm and SADMM and the flexibility
and efficiency of SADMM is illustrated via a numerical simulation in a sparse
minimization problem.
