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

Abstract

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.

Convergence of damped inertial dynamics governed by regularized maximally monotone operators

“…When the operator is the subdifferential of a closed convex proper function, we estimate the rate of convergence of the values. These results are in line with the recent articles by Attouch-Cabot [3], and Attouch-Peypouquet [8]. In this last paper, the authors considered the case γ(t) = α t , which is naturally linked to Nesterov's accelerated method.…”

“…General regularization parameter λ(t). Our approach is in line with Attouch and Peypouquet [8] who studied the system (RIMS) γ,λ with a general maximally monotone operator, and in the particular case γ(t) = α/t (the importance of this system has been stressed just above). This approach can be traced back toÁlvarez-Attouch [1] and Attouch-Maingé [6] who studied the equation…”

“…To finish, let us consider the case q = −1, thus leading to a damping parameter of the form γ(t) = α t . This case was recently studied by Attouch and Peypouquet [8] in the framework of Nesterov's accelerated methods. and λ(t) = β t r for every t ≥ t 0 > 0.…”

“…So doing, in the case of a general maximal monotone operator, they were able to prove the asymptotic convergence of the trajectories to zeros of A. Our approach will consist in extending these results to the case of a general damping coefficient γ(t), taking advantage of the techniques developed in the above mentioned papers [3] and [8].…”

“…The (RIMS) γ,λ system is a natural development of some recent studies concerning rapid inertial dynamics for convex optimization and monotone equilibrium problems. We will rely heavily on the techniques developed in [3] concerning the general damping coefficient γ(t), and in [8] concerning the general Yosida regularization parameter λ(t).…”