  In this paper, we conduct a convergence rate analysis of the augmented
Lagrangian method with a practical relative error criterion designed in
Eckstein and Silva [Math. Program., 141, 319--348 (2013)] for convex nonlinear
programming problems. We show that under a mild local error bound condition,
this method admits locally a Q-linear rate of convergence. More importantly, we
show that the modulus of the convergence rate is inversely proportional to the
penalty parameter. That is, an asymptotically superlinear convergence is
obtained if the penalty parameter used in the algorithm is increasing to
infinity, or an arbitrarily Q-linear rate of convergence can be guaranteed if
the penalty parameter is fixed but it is sufficiently large. Besides, as a
byproduct, the convergence, as well as the convergence rate, of the distance
from the primal sequence to the solution set of the problem is obtained.
