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

TL;DR

This paper develops second-order optimality conditions for semidefinite cone complementarity constrained problems, providing precise tangent set characterizations and applying these results to rank optimization problems.

Contribution

It offers an exact characterization of the second-order tangent set for SDCC and derives new second-order necessary and sufficient optimality conditions for SDCMPCC.

Findings

Characterization of the second-order tangent set for SDCC.

Derivation of second-order optimality conditions for SDCMPCC.

Application to rank regularized semidefinite problems.

Abstract

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.