Journal of Mathematical Imaging and Vision

, Volume 51, Issue 2, pp 311–325 | Cite as

An Inertial Forward-Backward Algorithm for Monotone Inclusions

  • Dirk A. LorenzEmail author
  • Thomas Pock


In this paper, we propose an inertial forward-backward splitting algorithm to compute a zero of the sum of two monotone operators, with one of the two operators being co-coercive. The algorithm is inspired by the accelerated gradient method of Nesterov, but can be applied to a much larger class of problems including convex-concave saddle point problems and general monotone inclusions. We prove convergence of the algorithm in a Hilbert space setting and show that several recently proposed first-order methods can be obtained as special cases of the general algorithm. Numerical results show that the proposed algorithm converges faster than existing methods, while keeping the computational cost of each iteration basically unchanged.


Convex optimization Forward-backward splitting Monotone inclusions Primal-dual algorithms Saddle-point problems Image restoration 



Thomas Pock acknowledges support from the Austrian science fund (FWF) under the project “Efficient algorithms for nonsmooth optimization in imaging”, No. I1148 and the FWF-START project Bilevel optimization for Computer Vision, No. Y729. The authors wish to thank Antonin Chambolle for very helpful discussions.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Analysis and AlgebraTU BraunschweigBraunschweigGermany
  2. 2.Institute for Computer Graphics and VisionGraz University of TechnologyGrazAustria
  3. 3.The Safety & Security DepartmentAIT Austrian Institute of Technology GmbHViennaAustria

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