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Strong Convergence in Hilbert Spaces via Γ-Duality

Journal of Optimization Theory and Applications volume 158, pages 343–362 (2013)Cite this article

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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References

  1. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    MATH  Book  Google Scholar 

  3. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  4. Vinter, R.: Optimal Control. Modern Birkhäuser Classics, Secaucus (2000)

    MATH  Google Scholar 

  5. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996)

    MATH  Book  Google Scholar 

  6. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  7. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  8. Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  10. Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, New Jersey (1963)

    MATH  Google Scholar 

  11. Svaiter, B.F.: Uma nova abordagem para dualidade em programação não linear e aplicações. Ph.D. Thesis, IMPA (1994)

  12. Svaiter, B.F.: A new duality theory for mathematical programming. Optimization 60, 1209–1231 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  13. Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent method for nonsmooth convex minimization in Hilbert spaces. Numer. Funct. Anal. Optim. 32, 1009–1018 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  14. Fan, Ky.: On systems of linear inequalities. Linear inequalities and related systems. In: Annals of Math. Studies, vol. 38, pp. 99–156. Princeton University Press, New Jersey (1956)

    Google Scholar 

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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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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, Brazil

    M. Marques Alves

  2. Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiânia, Brazil

    J. G. Melo

Authors
  1. M. Marques Alves
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  2. J. G. Melo
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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