  Due to the possible lack of primal-dual-type error bounds, the superlinear
convergence for the Karush-Kuhn-Tucker (KKT) residues of the sequence generated
by augmented Lagrangian method (ALM) for solving convex composite conic
programming (CCCP) has long been an outstanding open question. In this paper,
we aim to resolve this issue by first conducting convergence rate analysis for
the ALM with Rockafellar's stopping criteria under only a mild quadratic growth
condition on the dual of CCCP. More importantly, by further assuming that the
Robinson constraint qualification holds, we establish the R-superlinear
convergence of the KKT residues of the iterative sequence under
easy-to-implement stopping criteria {for} the augmented Lagrangian subproblems.
Equipped with this discovery, we gain insightful interpretations on the
impressive numerical performance of several recently developed semismooth
Newton-CG based ALM solvers for solving linear and convex quadratic
semidefinite programming.
