  We consider nonlinear optimization problems with cardinality constraints.
Based on a continuous reformulation we introduce second order necessary and
sufficient optimality conditions. Under such a second order condition, we can
guarantee local uniqueness of M-stationary points. Finally, we use this
observation to provide extended local convergence theory for a Scholtes-type
relaxation method for cardinality constrained optimization problems, which
guarantees the existence and convergence of the iterates under suitable
assumptions.
