On Glowinski’s Open Question on the Alternating Direction Method of Multipliers

Abstract

The alternating direction method of multipliers was proposed by Glowinski and Marrocco in 1974, and it has been widely used in a broad spectrum of areas, especially in some sparsity-driven application domains. In 1982, Fortin and Glowinski suggested to enlarge the range of the dual step size for updating the multiplier from 1 to the open interval of zero to the golden ratio, and this strategy immediately accelerates the convergence of alternating direction method of multipliers for most of its applications. Meanwhile, Glowinski raised the question of whether or not the range can be further enlarged to the open interval of zero to 2; this question remains open with nearly no progress in the past decades. In this paper, we answer this question affirmatively for the case where both the functions in the objective function are quadratic. Thus, Glowinski’s open question is partially answered. We further establish the global linear convergence of the alternating direction method of multipliers with this enlarged step size range for the quadratic programming under a tight condition.