Journal of Global Optimization

, Volume 66, Issue 3, pp 457–485 | Cite as

A block coordinate variable metric forward–backward algorithm

  • Emilie Chouzenoux
  • Jean-Christophe Pesquet
  • Audrey Repetti


A number of recent works have emphasized the prominent role played by the Kurdyka-Łojasiewicz inequality for proving the convergence of iterative algorithms solving possibly nonsmooth/nonconvex optimization problems. In this work, we consider the minimization of an objective function satisfying this property, which is a sum of two terms: (i) a differentiable, but not necessarily convex, function and (ii) a function that is not necessarily convex, nor necessarily differentiable. The latter function is expressed as a separable sum of functions of blocks of variables. Such an optimization problem can be addressed with the Forward–Backward algorithm which can be accelerated thanks to the use of variable metrics derived from the Majorize–Minimize principle. We propose to combine the latter acceleration technique with an alternating minimization strategy which relies upon a flexible update rule. We give conditions under which the sequence generated by the resulting Block Coordinate Variable Metric Forward–Backward algorithm converges to a critical point of the objective function. An application example to a nonconvex phase retrieval problem encountered in signal/image processing shows the efficiency of the proposed optimization method.


Nonconvex optimization Nonsmooth optimization Proximity operator Majorize–Minimize algorithm Block coordinate descent Alternating minimization Phase retrieval Inverse problems 

Mathematics Subject Classification

90C25 90C26 65K10 65F08 49M27 68U10 94A08 90C05 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Emilie Chouzenoux
    • 1
  • Jean-Christophe Pesquet
    • 1
  • Audrey Repetti
    • 2
  1. 1.Laboratoire d’Informatique Gaspard Monge and CNRS UMR 8049Université Paris-Est Marne-la-ValléeMarne-la-ValléeFrance
  2. 2.Institute of Sensors, Signals and SystemsHeriot-Watt UniversityEdinburghScotland, UK

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