# Analysis of the alternating direction method of multipliers for nonconvex problems

## TL;DR

This paper explores the theoretical convergence of the ADMM algorithm when applied to nonconvex problems, focusing on local solutions of subproblems and practical implications.

## Contribution

It provides the first analysis of ADMM for nonconvex problems with locally solved subproblems, establishing local convergence results.

## Key findings

- ADMM converges locally for nonconvex problems with local subproblem solutions.
- The analysis extends ADMM's theoretical understanding beyond convex cases.
- Practical relevance due to consideration of local solutions in implementation.

## Abstract

This work investigates the theoretical performance of the alternating-direction method of multipliers (ADMM) as it applies to nonconvex optimization problems, and in particular, problems with nonconvex constraint sets. The alternating direction method of multipliers is an optimization method that has largely been analyzed for convex problems. The ultimate goal is to assess what kind of theoretical convergence properties the method has in the nonconvex case, and to this end, theoretical contributions are two-fold. First, this work analyzes the method with local solution of the ADMM subproblems, which contrasts with much analysis that requires global solutions of the subproblems. Such a consideration is important to practical implementations. Second, it is established that the method still satisfies a local convergence result. The work concludes with some more detailed discussion of how the analysis relates to previous work.