  In this paper, we first propose a general inertial proximal point method for
the mixed variational inequality (VI) problem. Based on our knowledge, without
stronger assumptions, convergence rate result is not known in the literature
for inertial type proximal point methods. Under certain conditions, we are able
to establish the global convergence and a $o(1/k)$ convergence rate result
(under certain measure) of the proposed general inertial proximal point method.
We then show that the linearized alternating direction method of multipliers
(ADMM) for separable convex optimization with linear constraints is an
application of a general proximal point method, provided that the algorithmic
parameters are properly chosen. As byproducts of this finding, we establish
global convergence and $O(1/k)$ convergence rate results of the linearized ADMM
in both ergodic and nonergodic sense. In particular, by applying the proposed
inertial proximal point method for mixed VI to linearly constrained separable
convex optimization, we obtain an inertial version of the linearized ADMM for
which the global convergence is guaranteed. We also demonstrate the effect of
the inertial extrapolation step via experimental results on the compressive
principal component pursuit problem.
