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      A new convergence analysis and perturbation resilience of some accelerated proximal forward-backward algorithms with errors

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          Abstract

          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, similar decay rates of the errors terms yield similar 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. Taking this into account, we also re-examine the superiorization methodology and significantly extend its scope.

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          Author and article information

          Journal
          2015-08-23
          2015-08-27
          Article
          1508.05631
          0962468e-0140-49e0-a1d3-396be1848196

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          90C25, 90C31, 49K40, 49M27, 90C59
          27 pages, very slight non-mathematical modifications (mainly additional references and thanks)
          math.OC cs.NA physics.med-ph

          Numerical & Computational mathematics,Numerical methods,Medical physics
          Numerical & Computational mathematics, Numerical methods, Medical physics

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