SPLITTING ALGORITHMS FOR NONCONVEX OPTIMIZATION: UNIFIED ANALYSIS AND NEWTON-TYPE ACCELERATION

We provide a unified interpretation of splitting algorithms for nonconvex optimization through the lens of majorization-minimization. Possibly under assumptions to compensate the lack of convexity, this setting is general enough to cover ADMM as well as forward-backward, Douglas-Rachford and Davis-Yin splittings. Proximal envelopes, a generalization of the Moreau envelope, are shown to be natural merit functions for establishing convergence results. Their regularity properties also enable the integration of fast direction of quasi-Newton-type, that differently from any other approach for nonsmooth optimization preserve the same operation complexity of the original splitting scheme.