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A Deflected Subgradient Method Using a General Augmented Lagrangian Duality with Implications on Penalty Methods

In: Variational Analysis and Generalized Differentiation in Optimization and Control

Author

Listed:
  • Regina S. Burachik

    (University of South Australia)

  • C. Yalçın Kaya

    (University of South Australia)

Abstract

We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primal–dual method with strong duality, that is, with zero duality gap. We prove that our method converges to a dual solution if and only if a dual solution exists. We also prove that all accumulation points of an auxiliary primal sequence are primal solutions. Our results apply, in particular, to classical penalty methods, since the penalty functions associated with these methods can be recovered as a special case of our augmented Lagrangians. Besides the classical augmenting terms given by the ℓ 1- or ℓ 2-norm forms, terms of many other forms can be used in our Lagrangian function. Using a practical selection of the step-size parameters, as well as various choices of the augmenting term, we demonstrate the method on test problems. Our numerical experiments indicate that it is more favourable to use an augmenting term of an exponential form rather than the classical ℓ 1- or ℓ 2-norm forms.

Suggested Citation

  • Regina S. Burachik & C. Yalçın Kaya, 2010. "A Deflected Subgradient Method Using a General Augmented Lagrangian Duality with Implications on Penalty Methods," Springer Optimization and Its Applications, in: Regina S. Burachik & Jen-Chih Yao (ed.), Variational Analysis and Generalized Differentiation in Optimization and Control, pages 109-132, Springer.
  • Handle: RePEc:spr:spochp:978-1-4419-0437-9_5
    DOI: 10.1007/978-1-4419-0437-9_5
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    Cited by:

    1. Regina Burachik & Wilhelm Freire & C. Kaya, 2014. "Interior Epigraph Directions method for nonsmooth and nonconvex optimization via generalized augmented Lagrangian duality," Journal of Global Optimization, Springer, vol. 60(3), pages 501-529, November.
    2. Regina S. Burachik & Alfredo N. Iusem & Jefferson G. Melo, 2013. "An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 108-131, April.

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