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

Hedy Attouch*, Juan Peypouquet

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

18 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)391-432
Number of pages42
JournalMathematical Programming
Volume174
Issue number1-2
DOIs
Publication statusPublished - 1-Mar-2019
Externally publishedYes

Keywords

  • Asymptotic stabilization
  • Damped inertial dynamics
  • Large step proximal method
  • Lyapunov analysis
  • Maximally monotone operators
  • Time-dependent viscosity
  • Vanishing viscosity
  • Yosida regularization

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