Mathematical Programs with Semidefinite Cone Complementarity Constraints: Constraint Qualifications and Optimality Conditions

The tangent cone of gph $N_{S^n_+}$ plays an important role in developing necessary conditions for mathematical programs with semidefinite cone complementarity constraints. We demonstrate an elegant formula for the tangent cone of gph $N_{S^n_+}$ , based on which the Bouligand stationary point is characterized explicitly. The relationships among different stationary points under certain constraint qualifications are discussed. Then we propose a second order sufficient condition which can be weakened under the strict complementarity condition. Importantly, for the sake of algorithm design, under the assumption of strict complementarity condition, we give a nonsmooth equation reformulation of the stationary point, whose smoothing system is verified to be nonsingular at the stationary point under the proposed second order sufficient condition.