Second-order optimality conditions for SDCMPCC and application to rank optimization problems

This paper is concerned with second-order optimality conditions for the mathematical program with semidefinite cone complementarity constraints (SDCMPCC).To achieve this goal, we first provide an exact characterization on the second-order tangent set to the semidefinite cone complementarity (SDCC) set in terms of the second-order directional derivative of the projection operator onto the SDCC set, and then use the second-order tangent set to the SDCC set to derive second-order necessary and sufficient conditions for SDCMPCC under suitable subregularity constraint qualifications. An application is also illustrated in characterizing the second-order necessary optimality condition for the semidefinite rank regularized problem.