  Many problems in science and engineering involve, as part of their solution
process, the consideration of a separable function which is the sum of two
convex functions, one of them possibly non-smooth. Recently a few works have
discussed inexact versions of several accelerated proximal methods aiming at
solving this minimization problem. This paper shows that inexact versions of a
method of Beck and Teboulle (FISTA) preserve, in a Hilbert space setting, the
same (non-asymptotic) rate of convergence under some assumptions on the decay
rate of the error terms. The notion of inexactness discussed here seems to be
rather simple, but, interestingly, when comparing to related works, closely
related decay rates of the errors terms yield closely related convergence
rates. The derivation sheds some light on the somewhat mysterious origin of
some parameters which appear in various accelerated methods. A consequence of
the analysis is that the accelerated method is perturbation resilient, making
it suitable, in principle, for the superiorization methodology. By taking this
into account, we re-examine the superiorization methodology and significantly
extend its scope.
