Journal of Global Optimization

, Volume 73, Issue 2, pp 331–348 | Cite as

Iteration-complexity analysis of a generalized alternating direction method of multipliers

  • V. A. Adona
  • M. L. N. GonçalvesEmail author
  • J. G. Melo


This paper analyzes the iteration-complexity of a generalized alternating direction method of multipliers (G-ADMM) for solving separable linearly constrained convex optimization problems. This ADMM variant, first proposed by Bertsekas and Eckstein, introduces a relaxation parameter \(\alpha \) into the second ADMM subproblem in order to improve its computational performance. It is shown that, for a given tolerance \(\varepsilon >0\), the G-ADMM with \(\alpha \in (0, 2)\) provides, in at most \({\mathcal {O}}(1/\varepsilon ^2)\) iterations, an approximate solution of the Lagrangian system associated to the optimization problem under consideration. It is further demonstrated that, in at most \({\mathcal {O}}(1/\varepsilon )\) iterations, an approximate solution of the Lagrangian system can be obtained by means of an ergodic sequence associated to a sequence generated by the G-ADMM with \(\alpha \in (0, 2]\). Our approach consists of interpreting the G-ADMM as an instance of a hybrid proximal extragradient framework with some special properties. Some preliminary numerical experiments are reported in order to confirm that the use of \(\alpha >1\) can lead to a better numerical performance than \(\alpha =1\) (which corresponds to the standard ADMM).


Generalized alternating direction method of multipliers Hybrid extragradient method Convex program Pointwise iteration-complexity Ergodic iteration-complexity 

Mathematics Subject Classification

47H05 49M27 90C25 90C60 65K10 


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Copyright information

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

Authors and Affiliations

  • V. A. Adona
    • 1
  • M. L. N. Gonçalves
    • 1
    Email author
  • J. G. Melo
    • 1
  1. 1.IMEUniversidade Federal de GoiasGoiâniaBrazil

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