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A Note on the Convergence of ADMM for Linearly Constrained Convex Optimization Problems

This note serves two purposes. Firstly, we construct a counterexample to show that the statement on the convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex optimization problems in a highly influential paper by Boyd et al. [Found. Trends Mach. Learn. 3(1) 1-122 (2011)] can be false if no prior condition on the existence of solutions to all the subproblems involved is assumed to hold. Secondly, we present fairly mild conditions to guarantee the existence of solutions to all the subproblems and provide a rigorous convergence analysis on the ADMM, under a more general and useful semi-proximal ADMM (sPADMM) setting considered by Fazel et al. [SIAM J. Matrix Anal. Appl. 34(3) 946-977 (2013)], with a computationally more attractive large step-length that can even exceed the practically much preferred golden ratio of $(1+\sqrt{5})/2$.