Journal of Global Optimization

, Volume 46, Issue 3, pp 347–361 | Cite as

A primal dual modified subgradient algorithm with sharp Lagrangian

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


We apply a modified subgradient algorithm (MSG) for solving the dual of a nonlinear and nonconvex optimization problem. The dual scheme we consider uses the sharp augmented Lagrangian. A desirable feature of this method is primal convergence, which means that every accumulation point of a primal sequence (which is automatically generated during the process), is a primal solution. This feature is not true in general for available variants of MSG. We propose here two new variants of MSG which enjoy both primal and dual convergence, as long as the dual optimal set is nonempty. These variants have a very simple choice for the stepsizes. Moreover, we also establish primal convergence when the dual optimal set is empty. Finally, our second variant of MSG converges in a finite number of steps.


Nonsmooth optimization Nonconvex optimization Duality scheme Sharp Lagrangian Modified subgradient algorithm 

Mathematics Subject Classification (2000)

90C26 49M29 49M37 


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Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Regina S. Burachik
    • 1
  • Alfredo N. Iusem
    • 2
  • Jefferson G. Melo
    • 2
  1. 1.School of Mathematics and StatisticsUniversity of South AustraliaMawson LakesAustralia
  2. 2.IMPA, Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil

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