A Proof of Convergence For the Alternating Direction Method of Multipliers Applied to Polyhedral-Constrained Functions

TL;DR

This paper provides a general convergence proof for the ADMM algorithm when applied to problems with polyhedral constraints, ensuring the generated sequence converges to an optimal solution.

Contribution

It extends existing convergence proofs of ADMM to include functions constrained on polyhedral sets, broadening its theoretical foundation.

Findings

Proves convergence of ADMM for polyhedral-constrained functions

Shows the sequence generated by ADMM converges to an optimal solution

Extends prior proofs to more general constrained problems

Abstract

We give a general proof of convergence for the Alternating Direction Method of Multipliers (ADMM). ADMM is an optimization algorithm that has recently become very popular due to its capabilities to solve large-scale and/or distributed problems. We prove that the sequence generated by ADMM converges to an optimal primal-dual optimal solution. We assume the functions f and g, defining the cost f(x) + g(y), are real-valued, but constrained to lie on polyhedral sets X and Y. Our proof is an extension of the proofs from [Bertsekas97, Boyd11].