Variants of the A-HPE and large-step A-HPE algorithms for strongly convex problems with applications to accelerated high-order tensor methods

For solving strongly convex optimization problems, we propose and study the global convergence of variants of the accelerated hybrid proximal extragradient (A-HPE) and large-step A-HPE algorithms of R.D.C. Monteiro and B.F. Svaiter [An accelerated hybrid proximal extragradient method for convex opti...

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Published in:Optimization methods & software Vol. 37; no. 6; pp. 2007 - 2037
Main Author: Marques Alves, M.
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 02.11.2022
Taylor & Francis Ltd
Subjects:
accelerated methods
Algorithms
Computational geometry
Convergence
Convex optimization
Convexity
high-order tensor methods
large-step
Mathematical analysis
Optimization
proximal-Newton method
proximal-point algorithm
strongly convex
superlinear convergence
Tensors
ISSN:1055-6788, 1029-4937
Online Access:Get full text
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Summary:For solving strongly convex optimization problems, we propose and study the global convergence of variants of the accelerated hybrid proximal extragradient (A-HPE) and large-step A-HPE algorithms of R.D.C. Monteiro and B.F. Svaiter [An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods, SIAM J. Optim. 23 (2013), pp. 1092-1125.]. We prove linear and the superlinear $ \mathcal {O}\left (k^{\,-k\left (\frac {p-1}{p+1}\right )}\right ) $ O ( k − k ( p − 1 p + 1 ) ) global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter $ p\geq ~2 $ p ≥ 2 appears in the (high-order) large-step condition of the new large-step A-HPE algorithm. We apply our results to high-order tensor methods, obtaining a new inexact (relative-error) tensor method for (smooth) strongly convex optimization with iteration-complexity $ \mathcal {O}\left (k^{\,-k\left (\frac {p-1}{p+1}\right )}\right ) $ O ( k − k ( p − 1 p + 1 ) ) . In particular, for p = 2, we obtain an inexact proximal-Newton algorithm with fast global $ \mathcal {O}\left (k^{\,-k/3}\right ) $ O ( k − k / 3 ) convergence rate.
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ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2021.2022148