SPLIT-DOUGLAS--RACHFORD ALGORITHM FOR COMPOSITE MONOTONE INCLUSIONS AND SPLIT-ADMM.

In this paper we provide a generalization of the Douglas--Rachford splitting (DRS) and the primal-dual algorithm [L. Condat, J. Optim. Theory Appl., 158 (2013), pp. 460-479; B. C. V\~u, Adv. Comput. Math., 38 (2013), pp. 667-681] for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual nonstandard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn--Tucker point is guaranteed, generalizing the main result in [B. F. Svaiter, SIAM J. Control Optim., 49 (2011), pp. 280-287]. Inspired by [D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, M. Fortin and R. Glowinski, eds., Stud. Math. Appl. 15, North-Holland, Amsterdam, 1983, pp. 299-331], we also derive a new split alternating direction method of multipliers (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 one 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. The flexibility and efficiency of both methods is illustrated via numerical simulations in total variation image restoration and a sparse minimization problem. [ABSTRACT FROM AUTHOR]