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

On the R-superlinear convergence of the KKT residues generated by the augmented Lagrangian method for convex composite conic programming