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

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...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 165; no. 1; pp. 78 - 112
Main Authors: Bredies, Kristian, Lorenz, Dirk A., Reiterer, Stefan
Format: Journal Article
Language:English
Published: Boston Springer US 01.04.2015
Springer Nature B.V
Subjects:
Algebra
Algorithms
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Convergence
Convex analysis
Data recovery
Engineering
Functionals
Hilbert space
Inverse problems
Iterative methods
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Minimization
Operations Research/Decision Theory
Optimization
Partial differential equations
Projection
Splitting
Studies
Theory of Computation
Gradient projection method
65K10
Non-convex optimization
Iterative thresholding
49M05
Non-smooth optimization
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary: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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-014-0614-7