  This paper introduces the generalized forward-backward splitting algorithm
for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$
has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense
that their Moreau proximity operators are easy to compute. While the
forward-backward algorithm cannot deal with more than $n = 1$ non-smooth
function, our method generalizes it to the case of arbitrary $n$. Our method
makes an explicit 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. This allows the generalized forward backward to efficiently address an
important class of convex problems. We prove its convergence in infinite
dimension, and its robustness to errors on the computation of the proximity
operators and of the gradient of $F$. Examples on inverse problems in imaging
demonstrate the advantage of the proposed methods in comparison to other
splitting algorithms.
