In this paper, we aim to prove the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a mild calmness condition, which holds automatically for convex composite piecewise linear-quadratic programming, we establish the global Q-linear rate of convergence for a general semi-proximal ADMM with the dual step-length being taken in (0, (1 + √ 5)/2). This semi-proximal ADMM, which covers the

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... ssic one, has the advantage to resolve the potentially non-solvability issue of the subproblems in the classic ADMM and possesses the abilities of handling the multi-block cases efficiently. We demonstrate the usefulness of the obtained results when applied to two-and multi-block convex quadratic (semidefinite) programming.