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

• DOI: 10.1007/s10957-014-0614-7
• Publication Date: 2014-07-10T20:37:47Z
• Authors: Kristian Bredies; Dirk A. Lorenz; Stefan Reiterer
• Keywords: 0211 other engineering and technologies; 02 engineering and technology

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.