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In this work, we present a generic step-size choice for the ADMM typeproximal algorithms. It admits a closed-form expression and is theoreticallyoptimal with respect to a worst-case convergence rate bound. It is simply givenby the ratio of Euclidean norms of the dual and primal solutions, i.e., $||{\lambda}^\star|| / ||{x}^\star||$. Numerical tests show that its practicalperformance is near-optimal in general. The only challenge is that such a ratiois not known a priori and we provide two strategies to address it. Thederivation of our step-size choice is based on studying the fixed-pointstructure of ADMM using the proximal operator. However, we demonstrate that theclassical proximal operator definition contains an input scaling issue. Thisleads to a scaled step-size optimization problem which would yield a falsesolution. Such an issue is naturally avoided by our proposed new definition ofthe proximal operator. A series of its properties is established.

A Generic Closed-form Optimal Step-size for ADMM