Fast convergence of dynamical ADMM via time scaling of damped inertial dynamics

TL;DR

This paper introduces a second-order dynamic system with time-varying parameters that achieves rapid convergence for convex optimization problems with affine constraints, laying the groundwork for accelerated ADMM algorithms.

Contribution

It proposes a novel damped inertial dynamic system with time scaling in a Hilbert space, providing theoretical convergence guarantees and insights for developing faster optimization algorithms.

Findings

Lyapunov analysis confirms fast convergence of the proposed dynamics.

Parameters tuning leads to improved convergence rates.

Framework facilitates the development of accelerated ADMM algorithms.

Abstract

In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are respectively associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization.