A Proof of Convergence For the Alternating Direction Method of   Multipliers Applied to Polyhedral-Constrained Functions
  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]. 
Jo\~ao F. C. Mota, Jo\~ao M. F. Xavier, Pedro M. Q. Aguiar, and Markus   P\"uschel
math.OC