Skip to main content
SpringerLink
Log in

A generalized conditional gradient method and its connection to an iterative shrinkage method

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

This article combines techniques from two fields of applied mathematics: optimization theory and inverse problems. We investigate a generalized conditional gradient method and its connection to an iterative shrinkage method, which has been recently proposed for solving inverse problems.

The iterative shrinkage method aims at the solution of non-quadratic minimization problems where the solution is expected to have a sparse representation in a known basis. We show that it can be interpreted as a generalized conditional gradient method. We prove the convergence of this generalized method for general class of functionals, which includes non-convex functionals. This also gives a deeper understanding of the iterative shrinkage method.

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. Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Math. 16, 1–3 (1966)

    MATH  MathSciNet  Google Scholar 

  2. Brezis, H.: Opérateurs maximaux monotones et Semi-groups de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies, vol. 5. North-Holland, Amsterdam (1973)

    Google Scholar 

  3. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)

    Article  MATH  Google Scholar 

  4. Demyanov, V.F., Rubinov, A.M.: Approximate Methods in Optimization Problems. Modern Analytic and Computational Methods in Science and Mathematics, vol. 32. Elsevier (1970)

  5. Dunn, J.C.: Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals. SIAM J. Control Optim. 17(2), 187–211 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dunn, J.C.: Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM J. Control Optim. 18(5), 473–487 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and Its Applications, vol. 375. Kluwer Academic, Dordrecht (1996)

    MATH  Google Scholar 

  9. Frank, M., Wolfe, P.: An algorithm for quadratic programming. Nav. Res. Log. Q. 3, 95–110 (1956)

    Article  MathSciNet  Google Scholar 

  10. Goldstein, A.A.: On steepest descent. SIAM J. Control Optim. 3, 147–151 (1965)

    Google Scholar 

  11. Lions, J.-L.: Optimal Control Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  12. Louis, A.K.: Inverse und Schlecht Gestellte Probleme. Teubner, Stuttgart (1989)

    MATH  Google Scholar 

  13. Polak, E.: An historical survey of computational methods in optimal control. SIAM Rev. 15(2), 553–584 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ramlau, R., Teschke, G.: Tikhonov replacement functionals for iteratively solving nonlinear operator equations. Inverse Probl. 21, 1571–1592 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ramlau, R., Teschke, G.: A projection iteration for nonlinear operator equations with sparsity constraints. Numer. Math. 104, 177–203 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. III. Springer, Berlin (1984)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich 03, Universität Bremen, Postfach 33 04 40, 28334, Bremen, Germany

    Kristian Bredies & Peter Maass

  2. Electrical Engineering Department, Technion—Israel Institute of Technology, 32000, Haifa, Israel

    Dirk A. Lorenz

Authors
  1. Kristian Bredies
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dirk A. Lorenz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Maass
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kristian Bredies.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bredies, K., Lorenz, D.A. & Maass, P. A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput Optim Appl 42, 173–193 (2009). https://doi.org/10.1007/s10589-007-9083-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9083-3

Keywords

Search

Navigation