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Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding

  • Kristian Bredies
  • Dirk A. LorenzEmail author
  • Stefan Reiterer
Article

Abstract

Convergence analysis is carried out for a forward-backward splitting/generalized gradient projection method for the minimization of a special class of non-smooth and genuinely non-convex minimization problems in infinite-dimensional Hilbert spaces. The functionals under consideration are the sum of a smooth, possibly non-convex and non-smooth, necessarily non-convex functional. For separable constraints in the sequence space, we show that the generalized gradient projection method amounts to a discontinuous iterative thresholding procedure, which can easily be implemented. In this case we prove strong subsequential convergence and moreover show that the limit satisfies strengthened necessary conditions for a global minimizer, i.e., it avoids a certain set of non-global minimizers. Eventually, the method is applied to problems arising in the recovery of sparse data, where strong convergence of the whole sequence is shown, and numerical tests are presented.

Keywords

Non-convex optimization Non-smooth optimization  Gradient projection method Iterative thresholding 

Mathematics Subject Classification

49M05 65K10 

Notes

Acknowledgments

Kristian Bredies acknowledges support by the SFB Research Center “Mathematical Optimization and Applications in Biomedical Sciences” at the University of Graz. Dirk Lorenz acknowledges support by the DFG under Grant LO 1436/2-1.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Kristian Bredies
    • 1
  • Dirk A. Lorenz
    • 2
    Email author
  • Stefan Reiterer
    • 1
  1. 1.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  2. 2.Institute for Analysis and AlgebraTU BraunschweigBraunschweigGermany

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