  During the last years, asymptotic (or sequential) constraint qualifications,
which postulate upper semicontinuity of certain set-valued mappings and provide
a natural companion of asymptotic stationarity conditions, have been shown to
be comparatively mild, on the one hand, while possessing inherent practical
relevance from the viewpoint of numerical solution methods, on the other one.
Based on recent developments, the theory in this paper enriches asymptotic
constraint qualifications for very general nonsmooth optimization problems over
inverse images of set-valued mappings by incorporating directional data. We
compare these new directional asymptotic regularity conditions with standard
constraint qualifications from nonsmooth optimization. Further, we introduce
directional concepts of pseudo- and quasi-normality which apply to set-valued
mappings. It is shown that these properties provide sufficient conditions for
the validity of directional asymptotic regularity. Finally, a novel
coderivative-like variational tool is introduced which allows to study the
presence of directional asymptotic regularity. For geometric constraints, it is
illustrated that all appearing objects can be calculated in terms of initial
problem data.
