“…We call (P) a constrained composite optimization problem because it contains set-membership constraints and a composite objective function q := f + g. Notice that the problem data, namely f , g, c and D, can be nonconvex, the nonsmooth cost term g can be discontinuous and the constraint set D can be disconnected. Thanks to their rich structure and flexibility, constrained composite problems are of interest for modeling in a variety of applications, ranging from optimal and model predictive control [21,53] to signal processing [19], low-rank and sparse approximation, compressed sensing, cardinality-constrained optimization [10] and disjunctive programming [6], such as problems with complementarity, vanishing and switching constraints [36,43]. Augmented Lagrangian methods have recently attracted revived and grown interest.…”