The Dual Step Size Of The Alternating Direction Method Can Be Larger Than 1.618 When One Function Is Strongly Convex

The alternating direction method of multipliers (ADMM) is one of the most well-known optimization scheme for solving linearly constrained separable convex problem. In the literature, Fortin and Glowinski proved that the step size for updating the Lagrange multiplier of the ADMM can be chosen in the open interval of zero to the golden ratio. But, it is still unknown whether the dual step size can be larger than the golden ratio. In this paper, for the case where one function term is strongly convex and the associate coefficient matrix is full column rank, we present an affirmative answer to the above question. We then derive an exact relationship between the modulus and the dual step size. Our analysis deepens the understanding of the convergence properties of the ADMM.