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Article Dans Une Revue SIAM Journal on Imaging Sciences Année : 2013

A Generalized Forward-Backward Splitting

Résumé

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.
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Dates et versions

hal-00613637 , version 1 (05-08-2011)
hal-00613637 , version 2 (22-08-2011)
hal-00613637 , version 3 (11-09-2011)
hal-00613637 , version 4 (12-01-2012)
hal-00613637 , version 5 (02-09-2012)

Identifiants

Citer

Hugo Raguet, Jalal M. Fadili, Gabriel Peyré. A Generalized Forward-Backward Splitting. SIAM Journal on Imaging Sciences, 2013, 6 (13), pp.1199-1226. ⟨10.1137/120872802⟩. ⟨hal-00613637v5⟩
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