  The alternating direction method of multipliers (ADMM) is a most widely used
optimization scheme for solving linearly constrained separable convex
optimization problems. The convergence of the ADMM can be guaranteed when the
dual step length is less than the golden ratio, while plenty of numerical
evidence suggests that even larger dual step length often accelerates the
convergence. It has also been proved that the dual step length can be enlarged
to less than 2 in some special cases, namely, one of the separable functions in
the objective function is linear, or both are quadratic plus some additional
assumptions. However, it remains unclear whether the golden ratio can be
exceeded in the general convex setting. In this paper, the performance
estimation framework is used to analyze the convergence of the ADMM, and
assisted by numerical and symbolic computations, a counter example is
constructed, which indicates that the conventional measure may lose
monotonicity as the dual step length exceeds the golden ratio, ruling out the
possibility of breaking the golden ratio within this conventional analytic
framework.
