  In this paper, we analyze the iteration-complexity of Generalized
Forward--Backward (GFB) splitting algorithm, as proposed in \cite{gfb2011}, for
minimizing a large class of composite objectives $f + \sum_{i=1}^n h_i$ on a
Hilbert space, where $f$ has a Lipschitz-continuous gradient and the $h_i$'s
are simple (\ie their proximity operators are easy to compute). We derive
iteration-complexity bounds (pointwise and ergodic) for the inexact version of
GFB to obtain an approximate solution based on an easily verifiable termination
criterion. Along the way, we prove complexity bounds for relaxed and inexact
fixed point iterations built from composition of nonexpansive averaged
operators. These results apply more generally to GFB when used to find a zero
of a sum of $n > 0$ maximal monotone operators and a co-coercive operator on a
Hilbert space. The theoretical findings are exemplified with experiments on
video processing.
