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We consider nonlinear optimization problems with cardinality constraints.Based on a continuous reformulation we introduce second order necessary andsufficient optimality conditions. Under such a second order condition, we canguarantee local uniqueness of M-stationary points. Finally, we use thisobservation to provide extended local convergence theory for a Scholtes-typerelaxation method for cardinality constrained optimization problems, whichguarantees the existence and convergence of the iterates under suitableassumptions.

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