  The framework of Integral Quadratic Constraints of Lessard et al. (2014)
reduces the computation of upper bounds on the convergence rate of several
optimization algorithms to semi-definite programming (SDP). Followup work by
Nishihara et al. (2015) applies this technique to the entire family of
over-relaxed Alternating Direction Method of Multipliers (ADMM). Unfortunately,
they only provide an explicit error bound for sufficiently large values of some
of the parameters of the problem, leaving the computation for the general case
as a numerical optimization problem. In this paper we provide an exact
analytical solution to this SDP and obtain a general and explicit upper bound
on the convergence rate of the entire family of over-relaxed ADMM. Furthermore,
we demonstrate that it is not possible to extract from this SDP a general bound
better than ours. We end with a few numerical illustrations of our result and a
comparison between the convergence rate we obtain for the ADMM with known
convergence rates for the Gradient Descent.
