TY - JOUR

T1 - Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators

AU - Attouch, Hedy

AU - Peypouquet, Juan

PY - 2019/3/1

Y1 - 2019/3/1

N2 - We study the behavior of the trajectories of a second-order differential equation with vanishing damping, governed by the Yosida regularization of a maximally monotone operator with time-varying index, along with a new Regularized Inertial Proximal Algorithm obtained by means of a convenient finite-difference discretization. These systems are the counterpart to accelerated forward–backward algorithms in the context of maximally monotone operators. A proper tuning of the parameters allows us to prove the weak convergence of the trajectories to zeroes of the operator. Moreover, it is possible to estimate the rate at which the speed and acceleration vanish. We also study the effect of perturbations or computational errors that leave the convergence properties unchanged. We also analyze a growth condition under which strong convergence can be guaranteed. A simple example shows the criticality of the assumptions on the Yosida approximation parameter, and allows us to illustrate the behavior of these systems compared with some of their close relatives.

AB - We study the behavior of the trajectories of a second-order differential equation with vanishing damping, governed by the Yosida regularization of a maximally monotone operator with time-varying index, along with a new Regularized Inertial Proximal Algorithm obtained by means of a convenient finite-difference discretization. These systems are the counterpart to accelerated forward–backward algorithms in the context of maximally monotone operators. A proper tuning of the parameters allows us to prove the weak convergence of the trajectories to zeroes of the operator. Moreover, it is possible to estimate the rate at which the speed and acceleration vanish. We also study the effect of perturbations or computational errors that leave the convergence properties unchanged. We also analyze a growth condition under which strong convergence can be guaranteed. A simple example shows the criticality of the assumptions on the Yosida approximation parameter, and allows us to illustrate the behavior of these systems compared with some of their close relatives.

KW - Asymptotic stabilization

KW - Damped inertial dynamics

KW - Large step proximal method

KW - Lyapunov analysis

KW - Maximally monotone operators

KW - Time-dependent viscosity

KW - Vanishing viscosity

KW - Yosida regularization

UR - http://www.scopus.com/inward/record.url?scp=85042919710&partnerID=8YFLogxK

U2 - 10.1007/s10107-018-1252-x

DO - 10.1007/s10107-018-1252-x

M3 - Article

AN - SCOPUS:85042919710

VL - 174

SP - 391

EP - 432

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -