  Solving large scale convex semidefinite programming (SDP) problems has long
been a challenging task numerically. Fortunately, several powerful solvers
including SDPNAL, SDPNAL+ and QSDPNAL have recently been developed to solve
linear and convex quadratic SDP problems to high accuracy successfully. These
solvers are based on the augmented Lagrangian method (ALM) applied to the dual
problems with the subproblems being solved by semismooth Newton-CG methods.
Noticeably, thanks to Rockafellar's general theory on the proximal point
algorithms, the primal iteration sequence generated by the ALM enjoys an
asymptotic Q-superlinear convergence rate under a second order sufficient
condition {for the primal problem}. This second order sufficient condition
implies that the primal problem has a unique solution, which can be restrictive
in many applications. For gaining more insightful interpretations on the high
efficiency of these solvers, in this paper we conduct an asymptotic superlinear
convergence analysis of the ALM for convex SDP when the primal problem has
multiple solutions (can be unbounded). Under a fairly mild second order growth
condition, we prove that the primal iteration sequence generated by the ALM
converges asymptotically Q-superlinearly, while the dual feasibility and the
dual objective function value converge asymptotically R-superlinearly.
Moreover, by studying the metric subregularity of the Karush-Kuhn-Tucker
solution mapping, we also provide sufficient conditions to guarantee the
asymptotic R-superlinear convergence of the dual iterate.
