  Asymptotic stationarity and regularity conditions turned out to be quite
useful to study the qualitative properties of numerical solution methods for
standard nonlinear and complementarity-constrained programs. In this paper, we
first extend these notions to nonlinear optimization problems with nonsmooth
but Lipschitzian data functions in order to find reasonable notions of
asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's
subdifferential construction. Particularly, we compare the associated novel
asymptotic constraint qualifications with already existing ones. The second
part of the paper presents two applications of the obtained theory. On the one
hand, we specify our findings for complementarity-constrained optimization
problems and recover recent results from the literature which demonstrates the
power of the approach. Furthermore, we hint at potential extensions to or- and
vanishing-constrained optimization. On the other hand, we demonstrate the
usefulness of asymptotic regularity in the context of bilevel optimization.
More precisely, we justify a well-known stationarity system for affinely
constrained bilevel optimization problems in a novel way. Afterwards, we
suggest a solution algorithm for this class of bilevel optimization problems
which combines a penalty method with ideas from DC-programming. After a brief
convergence analysis, we present results of some numerical experiments.
