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

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

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