The Differential Inclusion Modeling FISTA Algorithm and Optimality of Convergence Rate in the Case b<3 - Archive ouverte HAL Access content directly
Journal Articles SIAM Journal on Optimization Year : 2018

The Differential Inclusion Modeling FISTA Algorithm and Optimality of Convergence Rate in the Case b<3

Vassilis Apidopoulos
  • Function : Author
  • PersonId : 1007811
Jean-François Aujol
Charles H Dossal
  • Function : Author
  • PersonId : 1069990

Abstract

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.
Fichier principal
Vignette du fichier
FistaInc2.pdf (616.91 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-01517708 , version 1 (03-05-2017)
hal-01517708 , version 2 (14-09-2017)

Identifiers

Cite

Vassilis Apidopoulos, Jean-François Aujol, Charles H Dossal. The Differential Inclusion Modeling FISTA Algorithm and Optimality of Convergence Rate in the Case b<3. SIAM Journal on Optimization, 2018, SIAM Journal on Optimization, 28 (1), pp.551-574. ⟨10.1137/17M1128642⟩. ⟨hal-01517708v2⟩
737 View
577 Download

Altmetric

Share

Gmail Facebook X LinkedIn More