Analysis of the Alternating Direction Method of Multipliers for Nonconvex Problems

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 twofold. First, this work analyzes the method with local optimal 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. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.