# Pointwise and ergodic convergence rates of a variable metric proximal ADMM

## TL;DR

This paper establishes the first global pointwise and ergodic convergence rates for a variable metric proximal ADMM, advancing understanding of its efficiency in solving linearly constrained convex optimization problems.

## Contribution

It introduces a novel convergence analysis for VM-PADMM, including nonasymptotic rates, by linking it to a new VM-HPE framework for monotone inclusions.

## Key findings

- Achieves $ ext{O}(1/\sqrt{k})$ pointwise convergence rate.
- Achieves $ ext{O}(1/k)$ ergodic convergence rate.
- First to establish these rates for VM-PADMM and VM-HPE framework.

## Abstract

In this paper, we obtain global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates for a variable metric proximal alternating direction method of multipliers(VM-PADMM) for solving linearly constrained convex optimization problems. The VM-PADMM can be seen as a class of ADMM variants, allowing the use of degenerate metrics (defined by noninvertible linear operators). We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient (VM-HPE) framework for solving monotone inclusions. Then, the above-mentioned convergence rates for the VM-PADMM are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the VM-PADMM (resp. VM-HPE framework).