  In this paper, we aim to provide a comprehensive analysis on the linear rate
convergence of the alternating direction method of multipliers (ADMM) for
solving linearly constrained convex composite optimization problems. Under a
certain error bound condition, we establish the global linear rate of
convergence for a more general semi-proximal ADMM with the dual steplength
being restricted to be in the open interval $(0, (1+\sqrt{5})/2)$. In our
analysis, we assume neither the strong convexity nor the strict complementarity
except an error bound condition, which holds automatically for convex composite
quadratic programming. This semi-proximal ADMM, which includes the classic
ADMM, not only has the advantage to resolve the potentially non-solvability
issue of the subproblems in the classic ADMM but also possesses the abilities
of handling multi-block convex optimization problems efficiently. We shall use
convex composite quadratic programming and quadratic semi-definite programming
as important applications to demonstrate the significance of the obtained
results. Of its own novelty in second-order variational analysis, a complete
characterization is provided on the isolated calmness for the nonlinear convex
semi-definite optimization problem in terms of its second order sufficient
optimality condition and the strict Robinson constraint qualification for the
purpose of proving the linear rate convergence of the semi-proximal ADMM when
applied to two- and multi-block convex quadratic semi-definite programming.
