  In this paper, we obtain global $\mathcal{O} (1/ \sqrt{k})$ pointwise and
$\mathcal{O} (1/ {k})$ ergodic convergence rates for a variable metric proximal
alternating direction method of multipliers(VM-PADMM) for solving linearly
constrained convex optimization problems. The VM-PADMM can be seen as a class
of ADMM variants, allowing the use of degenerate metrics (defined by
noninvertible linear operators). We first propose and study nonasymptotic
convergence rates of a variable metric hybrid proximal extragradient (VM-HPE)
framework for solving monotone inclusions. Then, the above-mentioned
convergence rates for the VM-PADMM are obtained essentially by showing that it
falls within the latter framework. To the best of our knowledge, this is the
first time that global pointwise (resp. pointwise and ergodic) convergence
rates are obtained for the VM-PADMM (resp. VM-HPE framework).
