  The cardinality constrained optimization problem (CCOP) is an optimization
problem where the maximum number of nonzero components of any feasible point is
bounded. In this paper, we consider CCOP as a mathematical program with
disjunctive subspaces constraints (MPDSC). Since a subspace is a special case
of a convex polyhedral set, MPDSC is a special case of the mathematical program
with disjunctive constraints (MPDC). Using the special structure of subspaces,
we are able to obtain more precise formulas for the tangent and (directional)
normal cones for the disjunctive set of subspaces. We then obtain first and
second order optimality conditions by using the corresponding results from
MPDC. Thanks to the special structure of the subspace, we are able to obtain
some results for MPDSC that do not hold in general for MPDC. In particular we
show that the relaxed constant positive linear dependence (RCPLD) is a
sufficient condition for the metric subregularity/error bound property for
MPDSC which is not true for MPDC in general. Finally we show that under all
constraint qualifications presented in this paper, certain exact penalization
holds for CCOP.
