Abstract: Recently, multi-block alternating direction method of multipliers (ADMM) has been widely used in signal processing, image processing, machine learning, engineering calculation and so on. However its convergence had been ambiguous for a long time. In 2016, Chen Caihua, et al. gave a counter-example constructed by a threedimensional linear equations to illustrate the possible convergence of multi-block ADMM. In this paper we discuss three related issues in the light of the results Chen's:1) is the divergence of the counter-example due to the initial point selection? 2) is the divergence of the counter-example due to its singleton feasible region? 3) Whether it is possible to introduce a problem-independent step-length in the dual update γ ∈ (0, 1] such that the small step-size ADMM variant converges? In theory, the paper gives negative answers to the first two questions, and we prove that when the initial point is randomly selected, there is an example where the feasible region is not a singleton such that the multi-block ADMM is divergent with probability of 1. The third problem is negated numerically and we show that even if the step-size is γ=10^-8, the multi-block ADMM can still diverge.

Key words: multi-block ADMM, divergence example, initial point, feasible region, stepsize in the dual update