Pointwise and ergodic iteration-complexity results for the proximal alternating direction method of multipliers (ADMM) for any stepsize in $(0,(1+\sqrt{5})/2)$ have been recently established in the literature. In addition to giving alternative proofs of these results, this paper also extends the ergodic iteration-complexity result to include the case in which the stepsize is equal to $(1+\sqrt{5})/2$. As far as we know, this is the first ergodic iteration-complexity for the stepsize $(1+\sqrt{5})/2$ obtained in the ADMM literature. These results are obtained by showing that the proximal ADMM is an instance of a non-Euclidean hybrid proximal extragradient framework whose pointwise and ergodic convergence rate are also studied.
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Institute of Mathematics and Statistics, Federal University of Goias, Campus II- Caixa Postal 131, CEP 74001-970, Goiania-GO, Brazil