APIDOPOULOS, Vassilis; AUJOL, Jean-François; DOSSAL, Charles

doi:10.1137/17M1128642

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APIDOPOULOS, Vassilis
Institut de Mathématiques de Bordeaux [IMB]

AUJOL, Jean-François
Institut de Mathématiques de Bordeaux [IMB]

DOSSAL, Charles
Institut de Mathématiques de Bordeaux [IMB]

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Article de revue

Ce document a été publié dans

SIAM Journal on Optimization. 2018-02-22, vol. 28, n° 1, p. 551–574

Society for Industrial and Applied Mathematics

Résumé en anglais

In this paper we are interested in the differential inclusion 0 ∈x ¨(t)+ b /t x _(t)+∂F (x(t)) in a finite dimensional Hilbert space Rd, where F is a proper, convex, lower semi-continuous function. The motivation of this study is that the differential inclusion models the FISTA algorithm as considered in [18]. In particular we investigate the different asymptotic properties of solutions for this inclusion for b > 0. We show that the convergence rate of F (x(t)) towards the minimum of F is of order of O(t− 2b/3) when 0 < b < 3, while for b > 3 this order is of o(t−2) and the solution-trajectory converges to a minimizer of F. These results generalize the ones obtained in the differential setting ( where F is differentiable ) in [6], [7], [11] and [31]. In addition we show that order of the convergence rate O(t− 2b/3) of F(x(t)) towards the minimum is optimal, in the case of low friction b < 3, by making a particular choice of F.< Réduire

Mots clés en anglais

asymptotic behavior

fast minimization

differential inclusion

FISTA algorithm

Convex optimization

URI

https://oskar-bordeaux.fr/handle/20.500.12278/193564

DOI

http://dx.doi.org/10.1137/17M1128642

Project ANR

Generalized Optimal Transport Models for Image processing - ANR-16-CE33-0010

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251