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An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians

  • Regina S. Burachik
  • Alfredo N. Iusem
  • Jefferson G. Melo
Article

Abstract

We consider a primal optimization problem in a reflexive Banach space and a duality scheme via generalized augmented Lagrangians. For solving the dual problem (in a Hilbert space), we introduce and analyze a new parameterized Inexact Modified Subgradient (IMSg) algorithm. The IMSg generates a primal-dual sequence, and we focus on two simple new choices of the stepsize. We prove that every weak accumulation point of the primal sequence is a primal solution and the dual sequence converges weakly to a dual solution, as long as the dual optimal set is nonempty. Moreover, we establish primal convergence even when the dual optimal set is empty. Our second choice of the stepsize gives rise to a variant of IMSg which has finite termination.

Keywords

Banach spaces Nonsmooth optimization Nonconvex optimization Duality scheme Augmented Lagrangian Inexact modified subgradient algorithm 

Notes

Acknowledgements

Regina S. Burachik acknowledges support by the Australian Research Council Discovery Project Grant DP0556685 for this study; Jefferson G. Melo acknowledges support from CNPq, through a scholarship for a one-year long visit to University of South Australia.

We thank the reviewers for their careful reading and comments.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Regina S. Burachik
    • 1
  • Alfredo N. Iusem
    • 2
  • Jefferson G. Melo
    • 3
  1. 1.School of Mathematics and StatisticsUniversity of South AustraliaMawson LakesAustralia
  2. 2.IMPAInstituto Nacional de Matemática Pura e AplicadaRio de JaneiroBrazil
  3. 3.IME/UFG, Instituto de Matemática e EstatísticaUniversidade Federal de GoiásGoiâniaBrazil

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