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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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
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Abstract	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.
AbstractList	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. 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.
Author	Bello-Cruz, Yunier Gonçalves, Max L. N. Krislock, Nathan
Author_xml	– sequence: 1 givenname: Yunier orcidid: 0000-0002-7877-5688 surname: Bello-Cruz fullname: Bello-Cruz, Yunier email: yunierbello@niu.edu organization: Department of Mathematical Sciences, Northern Illinois University – sequence: 2 givenname: Max L. N. surname: Gonçalves fullname: Gonçalves, Max L. N. organization: IME, Universidade Federal de Goiás – sequence: 3 givenname: Nathan surname: Krislock fullname: Krislock, Nathan organization: Department of Mathematical Sciences, Northern Illinois University
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Snippet	The fast iterative shrinkage/thresholding algorithm (FISTA) is one of the most popular first-order iterations for minimizing the sum of two convex functions....
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SubjectTerms	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
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Title	On FISTA with a relative error rule
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