# Alternating Direction Method of Multipliers with Variable Metric Indefinite Proximal Terms for Convex Optimization

## TL;DR

This paper introduces a variable metric indefinite proximal ADMM for convex optimization, providing convergence conditions and a new BFGS-based proximal term that enhances algorithm speed and applicability.

## Contribution

It develops a globally convergent variable metric indefinite proximal ADMM and proposes a novel BFGS-based indefinite proximal term.

## Key findings

- The proposed method converges globally under certain conditions.
- A new BFGS-based indefinite proximal term satisfies convergence criteria.
- Numerical experiments show improved performance over fixed positive semidefinite proximal terms.

## Abstract

This paper studies a proximal alternating direction method of multipliers (ADMM) with variable metric indefinite proximal terms for linearly constrained convex optimization problems. The proximal ADMM plays an important role in many application areas, since the subproblems of the method are easy to solve. Recently, it is reported that the proximal ADMM with a certain fixed indefinite proximal term is faster than that with a positive semidefinite term, and still has the global convergence property. On the other hand, Gu and Yamashita studied a variable metric semidefinite proximal ADMM whose proximal term is generated by the BFGS update. They reported that a slightly indefinite matrix also makes the algorithm work well in their numerical experiments. Motivated by this fact, we consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, we propose a new indefinite proximal term based on the BFGS update which can satisfy the conditions for the global convergence.