A Splitting Algorithm for Coupled System of Primal–Dual Monotone Inclusions

We propose a splitting algorithm for solving a coupled system of primal–dual monotone inclusions in real Hilbert spaces. The proposed algorithm has a structure identical to that of the forward–backward algorithm with variable metric. The operators involved in the problem formulation are used separately in the sense that single-valued operators are used individually and approximately in the forward steps and multi-valued operators are used individually via their generalization resolvent in the backward steps. The weak convergence of the algorithm proposed is proved. Applications to coupled system of monotone inclusions in duality and minimization problems, and multi-dictionary signal representation are demonstrated.