HAL : hal-00613637, version 5

Fiche détaillée

Récupérer au format BibTeX - EndNote - TEI - RefWorks -

Versions disponibles :

v1 (06-08-2011)

v2 (22-08-2011)

v3 (11-09-2011)

v4 (12-01-2012)

v5 (02-09-2012)

A Generalized Forward-Backward Splitting

Hugo Raguet 1, Jalal Fadili 2, Gabriel Peyré 1

(01/07/2011)

This paper introduces a generalized forward-backward splitting algorithm for finding a zero of a sum of maximal monotone operators $B + \sum_{i=1}^{n} A_i$, where $B$ is cocoercive. It involves the computation of $B$ in an explicit (forward) step and of the parallel computation of the resolvents of the $A_i$'s in a subsequent implicit (backward) step. We prove its convergence in infinite dimension, and robustness to summable errors on the computed operators in the explicit and implicit steps. In particular, this allows efficient minimization of the sum of convex functions $f + \sum_{i=1}^n g_i$, where $f$ has a Lipschitz-continuous gradient and each $g_i$ is simple in the sense that its proximity operator is easy to compute. The resulting method makes use of the regularity of $f$ in the forward step, and the proximity operators of the $g_i$'s are applied in parallel in the backward step. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, we generalize it to the case of arbitrary $n$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.

1 :	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine
2 :	Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen (GREYC)
	CNRS : UMR6072 – Université de Caen Basse-Normandie – Ecole Nationale Supérieure d'Ingénieurs de Caen

Équipe de recherche : image

Domaine

:

Mathématiques/Optimisation et contrôle Sciences de l'ingénieur/Traitement du signal et de l'image Informatique/Traitement du signal et de l'image

Mots Clés : Forward-backward algorithm – splitting – proximal – convex optimization – image processing – total variation – wavelets

hal-00613637, version 5

http://hal.archives-ouvertes.fr/hal-00613637

oai:hal.archives-ouvertes.fr:hal-00613637

Contributeur : Gabriel Peyré <>

Soumis le : Dimanche 2 Septembre 2012, 11:12:57

Dernière modification le : Dimanche 2 Septembre 2012, 18:45:01