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

• 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
• DOI: 10.1007/s10957-015-0746-4

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.