  We extend the convergence analysis of the Scholtes-type regularization method
for cardinality-constrained optimization problems. Its behavior is clarified in
the vicinity of saddle points, and not just of minimizers as it has been done
in the literature before. This becomes possible by using as an intermediate
step the recently introduced regularized continuous reformulation of a
cardinality-constrained optimization problem. We show that the Scholtes-type
regularization method is well-defined locally around a nondegenerate
T-stationary point of this regularized continuous reformulation. Moreover, the
nondegenerate Karush-Kuhn-Tucker points of the corresponding Scholtes-type
regularization converge to a T-stationary point having the same index, i.e. its
topological type persists. Overall, we conclude that the global structure of
the regularized continuous reformulation and its Scholtes-type regularization
essentially coincide.
