Applied Mathematics & Optimization

, Volume 80, Issue 3, pp 547–598 | Cite as

Convergence of a Relaxed Inertial Forward–Backward Algorithm for Structured Monotone Inclusions

  • Hedy AttouchEmail author
  • Alexandre Cabot


In a Hilbert space \({{\mathcal {H}}}\), we study the convergence properties of a class of relaxed inertial forward–backward algorithms. They aim to solve structured monotone inclusions of the form \(Ax + Bx \ni 0\) where \(A:{{\mathcal {H}}}\rightarrow 2^{{\mathcal {H}}}\) is a maximally monotone operator and \(B:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\) is a cocoercive operator. We extend to this class of problems the acceleration techniques initially introduced by Nesterov, then developed by Beck and Teboulle in the case of structured convex minimization (FISTA). As an important element of our approach, we develop an inertial and parametric version of the Krasnoselskii–Mann theorem, where joint adjustment of the inertia and relaxation parameters plays a central role. This study comes as a natural extension of the techniques introduced by the authors for the study of relaxed inertial proximal algorithms. An illustration is given to the inertial Nash equilibration of a game combining non-cooperative and cooperative aspects.


Structured monotone inclusions Inertial forward–backward algorithms Cocoercive operators Relaxation Convergence rate Inertial Krasnoselskii–Mann iteration Nash equilibration 

Mathematics Subject Classification

49M37 65K05 65K10 90C25 



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Authors and Affiliations

  1. 1.Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRSUniversité MontpellierMontpellier Cedex 5France
  2. 2.Institut de Mathématiques de Bourgogne, UMR 5584 CNRSUniversité Bourgogne Franche-ComtéDijonFrance

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