Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 23, Issue 3, Pages 1607-1633Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110844805
Keywords
convex optimization; accelerated forward-backward splitting; inexact proximity operator; estimate sequences; total variation
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We propose a convergence analysis of accelerated forward-backward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the 1/k(2) convergence rate for the function values can be achieved if the admissible errors are of a certain type and satisfy a sufficiently fast decay condition. Our analysis is based on the machinery of estimate sequences first introduced by Nesterov for the study of accelerated gradient descent algorithms. Furthermore, we give a global complexity analysis, taking into account the cost of computing admissible approximations of the proximal point. An experimental analysis is also presented.
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