  We study cardinality-constrained optimization problems (CCOP) in general
position, i. e. those optimization-related properties that are fulfilled for a
dense and open subset of their defining functions. We show that the well-known
cardinality-constrained linear independence constraint qualification (CC-LICQ)
is generic in this sense. For M-stationary points we define nondegeneracy and
show that it is a generic property too. In particular, the sparsity constraint
turns out to be active at all minimizers of a generic CCOP. Moreover, we
describe the global structure of CCOP in the sense of Morse theory, emphasizing
the strength of the generic approach. Here, we prove that multiple cells need
to be attached, each of dimension coinciding with the proposed M-index of
nondegenerate M-stationary points. Beyond this generic viewpoint, we study
singularities of CCOP. For that, the relation between nondegeneracy and strong
stability in the sense of Kojima (1980) is examined. We show that nondegeneracy
implies the latter, while the reverse implication is in general not true. To
fill the gap, we fully characterize the strong stability of M-stationary points
under CC-LICQ by first- and second-order information of CCOP defining
functions. Finally, we compare nondegeneracy and strong stability of
M-stationary points with second-order sufficient conditions recently introduced
in the literature.
