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

Kristian Bredies (), Dirk A. Lorenz () and Stefan Reiterer ()
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Kristian Bredies: University of Graz
Dirk A. Lorenz: TU Braunschweig
Stefan Reiterer: University of Graz

Journal of Optimization Theory and Applications, 2015, vol. 165, issue 1, No 5, 78-112

Abstract: 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; 49M05; 65K10 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (8)

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DOI: 10.1007/s10957-014-0614-7

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