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Communication Dans Un Congrès Année : 2015

Activity Identification and Local Linear Convergence of Douglas–Rachford/ADMM under Partial Smoothness

Résumé

Convex optimization has become ubiquitous in most quantitative disciplines of science, including variational image processing. Prox-imal splitting algorithms are becoming popular to solve such structured convex optimization problems. Within this class of algorithms, Douglas– Rachford (DR) and ADMM are designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local convergence behaviour of DR (resp. ADMM) when the involved functions (resp. their Legendre-Fenchel conjugates) are moreover partly smooth. More precisely, when both of the two functions (resp. their conjugates) are partly smooth relative to their respective manifolds, we show that DR (resp. ADMM) identifies these manifolds in finite time. Moreover, when these manifolds are affine or linear, we prove that DR/ADMM is locally linearly convergent with a rate in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified manifolds. This is illustrated by several concrete examples and supported by numerical experiments.
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Dates et versions

hal-01201596 , version 1 (17-09-2015)

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Jingwei Liang, Jalal M. Fadili, Gabriel Peyré, Russell Luke. Activity Identification and Local Linear Convergence of Douglas–Rachford/ADMM under Partial Smoothness. SSVM 2015 - International Conference on Scale Space and Variational Methods in Computer Vision, May 2015, Lège-Cap Ferret, France. pp.642-653, ⟨10.1007/978-3-319-18461-6_51⟩. ⟨hal-01201596⟩
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