  As a starting point of our research, we show that, for a fixed order
$\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization
problem in Euclidean spaces is either M-stationary in the classical sense
(corresponding to stationarity of order $1$), satisfies stationarity conditions
in terms of a coderivative construction of order $\gamma$, or is asymptotically
stationary with respect to a critical direction as well as order $\gamma$ in a
certain sense. By ruling out the latter case with a constraint qualification
not stronger than directional metric subregularity, we end up with new
necessary optimality conditions comprising a mixture of limiting variational
tools of orders $1$ and $\gamma$. These abstract findings are carved out for
the broad class of geometric constraints and $\gamma:=2$, and visualized by
examples from complementarity-constrained and nonlinear semidefinite
optimization. As a byproduct of the particular setting $\gamma:=1$, our general
approach yields new so-called directional asymptotic regularity conditions
which serve as constraint qualifications guaranteeing M-stationarity of local
minimizers. We compare these new regularity conditions with standard constraint
qualifications from nonsmooth optimization. Further, we extend directional
concepts of pseudo- and quasi-normality to arbitrary 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 used to construct sufficient conditions for 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.
