  Approximate necessary optimality conditions in terms of Fr\'echet
subgradients and normals for a rather general optimization problem with a
potentially non-Lipschitzian objective function are established with the aid of
Ekeland's variational principle, the fuzzy Fr\'echet subdifferential sum rule,
and a novel notion of lower semicontinuity relative to a set-valued mapping or
set. Feasible points satisfying these optimality conditions are referred to as
approximately stationary. As applications, we derive a new general version of
the extremal principle. Furthermore, we study approximate stationarity
conditions for an optimization problem with a composite objective function and
geometric constraints, a qualification condition guaranteeing that
approximately stationary points of such a problem are M-stationary, and a
multiplier-penalty-method which naturally computes approximately stationary
points of the underlying problem. Finally, necessary optimality conditions for
an optimal control problem with a non-Lipschitzian sparsity-promoting term in
the objective function are established.
