On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”

We discuss here the convergence of the iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm,” which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semicontinuous, and proper functions (defined in a Euclidean or Hilbert space), such that one is d...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 166; no. 3; pp. 968 - 982
Main Authors: Chambolle, A., Dossal, Ch
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2015
Springer Nature B.V
Subjects:
Algorithms
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Construction
Convergence
Engineering
Euclidean geometry
Euclidean space
Hilbert space
Inverse
Iterative methods
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Optimization algorithms
Shrinkage
Studies
Theory of Computation
65B99
Forward backward splitting
90C25
Inertial algorithms
65Y20
Optimization
First-order schemes
Convergence
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:We discuss here the convergence of the iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm,” which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semicontinuous, and proper functions (defined in a Euclidean or Hilbert space), such that one is differentiable with Lipschitz gradient, and the proximity operator of the second is easy to compute. It builds a sequence of iterates for which the objective is controlled, up to a (nearly optimal) constant, by the inverse of the square of the iteration number. However, the convergence of the iterates themselves is not known. We show here that with a small modification, we can ensure the same upper bound for the decay of the energy, as well as the convergence of the iterates to a minimizer.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-015-0746-4