Scaled constraint qualifications and necessary optimality conditions for nonsmooth mathematical programs with second-order cone complementarity constraints

In this paper, a nonsmooth mathematical program with second-order cone complementarity constraints (SOCMPCC) is studied. First scaled and strong scaled necessary optimality results are derived for a general nonsmooth optimization problem. Then the SOCMPCC is formulated as a general optimization problem whose constraint system is described by the graph of the projection operator over the second-order cone. Next exact expressions for the tangents and normals to this graph are established. Finally, the scaled M- and S-stationary necessary optimality results are verified for nonsmooth SOCMPCC under new weak scaled constraint qualifications. A number of examples are provided to illustrate the results obtained.