# ADMM for monotone operators: convergence analysis and rates

## TL;DR

This paper introduces a unifying framework for various algorithms solving monotone inclusion problems, including ADMM, providing convergence analysis and rates in infinite-dimensional Hilbert spaces.

## Contribution

It unifies multiple primal-dual algorithms and ADMM variants under a single scheme, offering convergence proofs and rate derivations using variable metric and dynamic step size strategies.

## Key findings

- Convergence of the proposed scheme in infinite-dimensional spaces
- Derivation of convergence rates using variable metrics
- Unified analysis of ADMM and primal-dual algorithms

## Abstract

We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, along with some of its variants, can be embedded in this unifying scheme. While in the first part of the paper convergence results for the iterates are reported, the second part is devoted to the derivation of convergence rates obtained by combining variable metric techniques with strategies based on suitable choice of dynamical step sizes.