  In this paper, we characterize Lipschitzian properties of different
multiplier-free and multiplier-dependent perturbation mappings associated with
the stationarity system of a so-called generalized nonlinear program
popularized by Rockafellar. Special emphasis is put on the investigation of the
isolated calmness property at and around a point. The latter is decisive for
the locally fast convergence of the so-called semismooth* Newton-type method by
Gfrerer and Outrata. Our central result is the characterization of the isolated
calmness at a point of a multiplier-free perturbation mapping via a combination
of an explicit condition and a rather mild assumption, automatically satisfied
e.g. for standard nonlinear programs. Isolated calmness around a point is
characterized analogously by a combination of two stronger conditions. These
findings are then related to so-called criticality of Lagrange multipliers, as
introduced by Izmailov and extended to generalized nonlinear programming by
Mordukhovich and Sarabi. We derive a new sufficient condition (a
characterization for some problem classes) of nonexistence of critical
multipliers, which has been also used in the literature as an assumption to
guarantee local fast convergence of Newton-, SQP-, or multiplier-penalty-type
methods. The obtained insights about critical multipliers seem to complement
the vast literature on the topic.
