The proximal alternating direction method of multipliers in the\n nonconvex setting: convergence analysis and rates

We propose two numerical algorithms in the fully nonconvex setting for the\nminimization of the sum of a smooth function and the composition of a nonsmooth\nfunction with a linear operator. The iterative schemes are formulated in the\nspirit of the proximal alternating direction method of multipliers and its\nlinearized variant, respectively. The proximal terms are introduced via\nvariable metrics, a fact which allows us to derive new proximal splitting\nalgorithms for nonconvex structured optimization problems, as particular\ninstances of the general schemes. Under mild conditions on the sequence of\nvariable metrics and by assuming that a regularization of the associated\naugmented Lagrangian has the Kurdyka-Lojasiewicz property, we prove that the\niterates converge to a KKT point of the objective function. By assuming that\nthe augmented Lagrangian has the Lojasiewicz property, we also derive\nconvergence rates for both the augmented Lagrangian and the iterates.\n