On the directional asymptotic approach in optimization theory Part A: approximate, M-, and mixed-order stationarity

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.