  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 $(1+\sqrt{5})/2$.
