Skip to main content
SpringerLink
Log in

A partial complement method for approximating solutions of a primal dual fixed-point problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We study the convergence of the Mann Iteration applied to the partial complement of a firmly nonexpansive operator with respect to a linear subspace of a Hilbert space. A new concept considered here. A regularized version is also proposed. Furthermore, to motivate this concept, some applications to robust regression procedures and location problems are proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bauschke H.H.: A note on the paper by Eckstein and Svaiter on General projective splitting methods for sums of maximal monotone operators. SIAM J. Control Optim. 48, 2513–2515 (2009)

    Article  MathSciNet  Google Scholar 

  2. Benker H., Hamel A., Tammer C.: A proximal point algorithm for control approximation problems. Math. Methods Oper. Res. 43(3), 261–280 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bougeard M.L., Caquineau C.D.: Parallel proximal decomposition algorithms for robust estimation. Ann. Oper. Res. 90(4), 247–270 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Goebel K., Kirk W.A.: Topics in Metric Fixed-Point Theory. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  5. Groetsch C.W.: A Note on Segmating Mann Iterates. J. Math. Anal. Appl. 40, 369–372 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  6. Halpern B.: Fixed points of nonxpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MATH  Google Scholar 

  7. Idrissi H., Lefebvre O., Michelot C.: A primal-dual algoritm for a constrained Fermat-weber problem involving mixed norms. Oper. Res. 22(4), 313–330 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Lions P.-L.: Approximation de points fixe de contractions. C. R. Acad. Sci. Ser. A-B Paris 284, 1357–1359 (1977)

    MATH  Google Scholar 

  9. Moreau, J.J.: Rafle par convexe variable. Séminaire d’Analyse Convexe Montpellier exposé N 14 (1971)

  10. Nashed M.Z.: A decomposition relative to convex sets. Proc. Amer. Math. Soc. 19, 782–786 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  11. Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  13. Spingarn J.E.: Partial inverse of a monotone operator. Appl. Math. Optim. 10, 247–265 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département Scientifique Interfacultaire, CEREGMIA, Université des Antilles-Guyane, Campus de Schoelcher, 97230, Martinique Cedex (F.W.I.), France

    Abdellatif Moudafi

Authors
  1. Abdellatif Moudafi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdellatif Moudafi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moudafi, A. A partial complement method for approximating solutions of a primal dual fixed-point problem. Optim Lett 4, 449–456 (2010). https://doi.org/10.1007/s11590-009-0172-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-009-0172-3

Keywords

Search

Navigation