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

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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.

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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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Authors and Affiliations

  1. School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA, 5095, Australia

    Regina S. Burachik

  2. IMPA, Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brazil

    Alfredo N. Iusem

  3. IME/UFG, Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    Jefferson G. Melo

Authors
  1. Regina S. Burachik
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  2. Alfredo N. Iusem
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  3. Jefferson G. Melo
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Correspondence to Jefferson G. Melo.

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Burachik, R.S., Iusem, A.N. & Melo, J.G. An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians. J Optim Theory Appl 157, 108–131 (2013). https://doi.org/10.1007/s10957-012-0158-7

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  • DOI: https://doi.org/10.1007/s10957-012-0158-7

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