On FISTA with a relative error rule

The fast iterative shrinkage/thresholding algorithm (FISTA) is one of the most popular first-order iterations for minimizing the sum of two convex functions. FISTA is known to improve the complexity of the classical proximal gradient method (PGM) from O ( k - 1 ) to the optimal complexity O ( k - 2...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 84; no. 2; pp. 295 - 318
Main Authors: Bello-Cruz, Yunier, Gonçalves, Max L. N., Krislock, Nathan
Format: Journal Article
Language:English
Published: New York Springer US 01.03.2023
Springer Nature B.V
Subjects:
Algorithms
Complexity
Convex and Discrete Geometry
Criteria
Errors
Iterative methods
Management Science
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Statistics
Inexact accelerated proximal gradient method
Iteration complexity
FISTA
47J22
90C25
49M27
Relative error rule
47H05
Nonsmooth and convex optimization problems
Proximal gradient method
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:The fast iterative shrinkage/thresholding algorithm (FISTA) is one of the most popular first-order iterations for minimizing the sum of two convex functions. FISTA is known to improve the complexity of the classical proximal gradient method (PGM) from O ( k - 1 ) to the optimal complexity O ( k - 2 ) in terms of the sequence of the functional values. When the evaluation of the proximal operator is hard, inexact versions of FISTA might be used to solve the problem. In this paper, we proposed an inexact version of FISTA by solving the proximal subproblem inexactly using a relative error criterion instead of exogenous and diminishing error rules. The introduced relative error rule in the FISTA iteration is related to the progress of the algorithm at each step and does not increase the computational burden per iteration. Moreover, the proposed algorithm recovers the same optimal convergence rate as FISTA. Some numerical experiments are also reported to illustrate the numerical behavior of the relative inexact method when compared with FISTA under an absolute error criterion.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-022-00421-8