  The main goal of this paper is to relate the topologically relevant
stationary points of a cardinality-constrained optimization problem and its
continuous reformulation up to their type. For that, we focus on the
nondegenerate M- and T-stationary points, respectively. Their so-called M- and
T-indices, which uniquely determine the global and local structure of
optimization problems under consideration in algebraic terms, are traced. As
novelty, we suggest to regularize the continuous reformulation for this
purpose. The main consequence of our analysis is that the number of saddle
points of the regularized continuous reformulation grows exponentially as
compared to that of the initial cardinality-constrained optimization problem.
Additionally, we obtain the Morse theory for the regularized continuous
reformulation by using the corresponding results on mathematical programs with
orthogonality type constraints.
