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On FISTA with a relative error rule

Yunier Bello-Cruz (), Max L. N. Gonçalves () and Nathan Krislock ()
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Yunier Bello-Cruz: Northern Illinois University
Max L. N. Gonçalves: Universidade Federal de Goiás
Nathan Krislock: Northern Illinois University

Computational Optimization and Applications, 2023, vol. 84, issue 2, No 1, 295-318

Abstract: 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 $$O(k^{-1})$$ O ( k - 1 ) to the optimal complexity $$O(k^{-2})$$ 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.

Keywords: FISTA; Inexact accelerated proximal gradient method; Iteration complexity; Nonsmooth and convex optimization problems; Proximal gradient method; Relative error rule; 47H05; 47J22; 49M27; 90C25 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-022-00421-8

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