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

Peter Ochs (ochs***at***math.uni-sb.de)

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 within this neighborhood. This result allows algorithms to exploit local properties of the objective function. In particular, the abstract theory in this paper applies to the inertial forward--backward splitting method: iPiano---a generalization of the Heavy-ball method. Moreover, it reveals an equivalence between iPiano and inertial averaged/alternating proximal minimization and projection methods. Key for this equivalence is the attraction to a local minimum within a common neighborhood and the fact that, for a prox-regular function, the gradient of the Moreau envelope is locally Lipschitz continuous and expressible in terms of the proximal mapping. In a numerical feasibility problem, the inertial alternating projection method significantly outperforms its non-inertial variants.

Keywords: inertial forward--backward splitting, non-convex feasibility, prox-regularity, gradient of Moreau envelopes, Heavy-ball method, alternating projection, averaged projection, iPiano

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: arXiv:1606.09070v3

Download: [PDF]

Entry Submitted: 06/09/2017
Entry Accepted: 06/09/2017
Entry Last Modified: 01/29/2018

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