Second Order Optimality Conditions and Improved Convergence Results for a Scholtes-type Regularization for a Continuous Reformulation of Cardinality Constrained Optimization Problems

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.