An inertial alternating direction method of multipliers

TL;DR

This paper introduces inertial effects into the classical ADMM algorithm for convex optimization in Hilbert spaces, analyzing its convergence and extending it to finite-sum problems.

Contribution

It develops and analyzes an inertial ADMM algorithm based on the inertial Douglas-Rachford splitting method, providing convergence results for the first time.

Findings

Convergence of the inertial ADMM sequences is established.

The method extends to convex problems with finite sums of functions.

Theoretical analysis confirms the effectiveness of inertial effects.

Abstract

In the context of convex optimization problems in Hilbert spaces, we induce inertial effects into the classical ADMM numerical scheme and obtain in this way so-called inertial ADMM algorithms, the convergence properties of which we investigate into detail. To this aim we make use of the inertial version of the Douglas-Rachford splitting method for monotone inclusion problems recently introduced in [12], in the context of concomitantly solving a convex minimization problem and its Fenchel dual. The convergence of both sequences of the generated iterates and of the objective function values is addressed. We also show how the obtained results can be extended to the treating of convex minimization problems having as objective a finite sum of convex functions.