Convergence rate of inertial Forward–Backward algorithm beyond Nesterov’s rule

  • Vassilis ApidopoulosEmail author
  • Jean-François Aujol
  • Charles Dossal
Full Length Paper Series A


In this paper we study the convergence of an Inertial Forward–Backward algorithm, with a particular choice of an over-relaxation term. In particular we show that for a sequence of over-relaxation parameters, that do not satisfy Nesterov’s rule, one can still expect some relatively fast convergence properties for the objective function. In addition we complement this work by studying the convergence of the algorithm in the case where the proximal operator is inexactly computed with the presence of some errors and we give sufficient conditions over these errors in order to obtain some convergence properties for the objective function.


Convex optimization Proximal operator Inertial FB algorithm Nesterov’s rule Rate of convergence 

Mathematics Subject Classification

49M20 46N10 90C25 65K10 



The authors would like to thank the anonymous reviewers for all their useful commentaries and advices and for pointing out some important references.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.IMB, UMR 5251Université de BordeauxTalenceFrance

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