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Strong Convergence in Hilbert Spaces via Γ-Duality

Abstract

We analyze a primal-dual pair of problems generated via a duality theory introduced by Svaiter. We propose a general algorithm and study its convergence properties. The focus is a general primal-dual principle for strong convergence of some classes of algorithms. In particular, we give a different viewpoint for the weak-to-strong principle of Bauschke and Combettes and unify many results concerning weak and strong convergence of subgradient type methods.

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Acknowledgements

This work was completed while the first author was visiting the Institute of Mathematics and Statistics at the University of Goias, whose warm hospitality is gratefully acknowledged. The authors thank Dr. Benar Fux Svaiter for bringing this problem to our attention, for sending us a copy of [11] and [12] as well as for the criticism concerning the first version of this manuscript. The authors also thank the two anonymous referees and the editor for their valuable comments.

M. Marques Alves is partially supported by CNPq grants 305414/2011-9 and 479729/2011-5 and PRONEX-Optimization. J.G. Melo is Partially supported by Procad/NF and by PRONEX-Optimization.

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Correspondence to M. Marques Alves.

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Communicated by Alfredo N. Iusem.

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Marques Alves, M., Melo, J.G. Strong Convergence in Hilbert Spaces via Γ-Duality. J Optim Theory Appl 158, 343–362 (2013). https://doi.org/10.1007/s10957-012-0253-9

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  • DOI: https://doi.org/10.1007/s10957-012-0253-9

Keywords

  • Γ-Duality
  • Hilbert spaces
  • Convex feasibility
  • Strong convergence
  • Subgradient method