Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

  • Max L. N. Gonçalves
  • Maicon Marques Alves
  • Jefferson G. Melo


In this paper, we obtain global pointwise and ergodic convergence rates for a variable metric proximal alternating direction method of multipliers for solving linearly constrained convex optimization problems. We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient framework for solving monotone inclusions. Then, the convergence rates for the former method are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the variable metric proximal alternating direction method of multipliers (resp. variable metric hybrid proximal extragradient framework).


Alternating direction method of multipliers Variable metric Pointwise and ergodic convergence rates Hybrid proximal extragradient method Convex program 

AMS subject classification

90C25 90C60 49M27 47H05 47J22 65K10 



The work of these authors was supported in part by CNPq Grants 406250/2013-8, 444134/2014-0, 309370/2014-0 and 406975/2016-7. We thank the reviewers for their careful reading and comments.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Max L. N. Gonçalves
    • 1
  • Maicon Marques Alves
    • 2
  • Jefferson G. Melo
    • 1
  1. 1.IMEUniversidade Federal de GoiásGoiâniaBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Santa CatarinaFlorianópolisBrazil

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