Continuous Newton-like Inertial Dynamics for Monotone Inclusions
- PDF / 672,470 Bytes
- 27 Pages / 439.642 x 666.49 pts Page_size
- 115 Downloads / 197 Views
Continuous Newton-like Inertial Dynamics for Monotone Inclusions ´ Csaba Laszl ´ o´ 2 Hedy Attouch1 · Szilard Received: 14 May 2020 / Accepted: 1 October 2020 / © Springer Nature B.V. 2020
Abstract In a Hilbert framework H, we study the convergence properties of a Newton-like inertial dynamical system governed by a general maximally monotone operator A : H → 2H . When A is equal to the subdifferential of a convex lower semicontinuous proper function, the dynamic corresponds to the introduction of the Hessian-driven damping in the continuous version of the accelerated gradient method of Nesterov. As a result, the oscillations are significantly attenuated. According to the technique introduced by Attouch-Peypouquet (Math. Prog. 2019), the maximally monotone operator is replaced by its Yosida approximation with an appropriate adjustment of the regularization parameter. The introduction into the dynamic of the Newton-like correction term (corresponding to the Hessian driven term in the case of convex minimization) provides a well-posed evolution system for which we will obtain the weak convergence of the generated trajectories towards the zeroes of A. We also obtain the fast convergence of the velocities towards zero. The results tolerate the presence of errors, perturbations. Then, we specialize our results to the case where the operator A is the subdifferential of a convex lower semicontinuous function, and obtain fast optimization results. Keywords Damped inertial dynamics · Hessian damping · Maximally monotone operators · Newton method · Vanishing viscosity · Yosida regularization Mathematics Subject Classification (2010) 37N40 · 46N10 · 49M30 · 65K05 · 65K10 · 90B50 · 90C25 Hedy Attouch
[email protected] Szil´ard Csaba L´aszl´o [email protected] 1
IMAG, University of Montpellier, CNRS, Montpellier, France
2
Department of Mathematics, Technical University of Cluj-Napoca, Memorandumului 28, Cluj-Napoca, Romania
H. Attouch, S. Csaba L´aszl´o
1 Introduction Let H be a real Hilbert space endowed with the scalar product ·, · and norm · . Given a general maximally monotone operator A : H → 2H , we will study the asymptotic behavior, as t → +∞, of the second-order in time evolution equation (DIN-AVD)α,β,λ,e
x(t) ¨ +
α d Aλ(t) (x(t)) + Aλ(t) (x(t)) = e(t). x(t) ˙ +β t dt
The operators JλA : H → H and Aλ : H → H which are defined by JλA = (I + λA)−1 and Aλ =
1 (I − JλA ) , λ
are respectively the resolvent of A and the Yosida regularization of A of index λ > 0. The coefficients α, β are positive damping parameters. The tuning of the time dependent parameter λ(t) which enters the Yosida regularization of A will play a crucial role in the asymptotic analysis. The second member e takes account of perturbations, errors. Without ambiguity, we refer briefly to the dynamic as (DIN-AVD). The terminology reflects the link of this dynamic with the Dynamic Inertial Newton method and to the Asymptotic Vanishing Damping, as explained in the next paragraph. According to the Lipschitz continuit
Data Loading...
