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The alternating direction method of multipliers (ADMM) is a most widely usedoptimization scheme for solving linearly constrained separable convexoptimization problems. The convergence of the ADMM can be guaranteed when thedual step length is less than the golden ratio, while plenty of numericalevidence suggests that even larger dual step length often accelerates theconvergence. It has also been proved that the dual step length can be enlargedto less than 2 in some special cases, namely, one of the separable functions inthe objective function is linear, or both are quadratic plus some additionalassumptions. However, it remains unclear whether the golden ratio can beexceeded in the general convex setting. In this paper, the performanceestimation framework is used to analyze the convergence of the ADMM, andassisted by numerical and symbolic computations, a counter example isconstructed, which indicates that the conventional measure may losemonotonicity as the dual step length exceeds the golden ratio, ruling out thepossibility of breaking the golden ratio within this conventional analyticframework.

On the dual step length of the alternating direction method of multipliers