  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 approximately stationary with respect to a critical
direction as well as $\gamma$ 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 order $1$ and $\gamma$. These abstract
findings are carved out for the broad class of geometric constraints. As a
byproduct, we obtain new constraint qualifications ensuring M-stationarity of
local minimizers. The paper closes by illustrating these results in the context
of standard nonlinear, complementarity-constrained, and nonlinear semidefinite
programming.
