Local Convergence of the Heavy-ball Method and iPiano for Non-convex Optimization

Abstract: A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges. This result allows algorithms to exploit local properties of the objective function: The gradient of the Moreau envelope of a prox-regular functions is locally Lipschitz continuous and expressible in terms of the proximal mapping. We apply these results to establish relations between an inertial forward-backward splitting method (iPiano) and inertial averaged/alternating proximal minimization.