Fixing and extending some recent results on the ADMM algorithm

TL;DR

This paper analyzes convergence techniques for proximal ADMM algorithms, introduces a variant handling smooth functions with gradient evaluations, and explores variable metrics in infinite-dimensional spaces.

Contribution

It extends recent ADMM results by formulating a new variant that incorporates smooth functions and variable metrics, with convergence analysis in infinite-dimensional settings.

Findings

Convergence properties of the proposed ADMM variant are established.

The algorithm effectively handles convex problems with smooth functions.

Variable metrics are successfully integrated into the ADMM framework.

Abstract

We investigate the techniques and ideas used in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration we allow the use of variable metrics, while the investigations are carried out in the setting of infinite dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties.