Sufficient optimality conditions hold for almost all nonlinear semidefinite programs

Abstract

We derive a new genericity result for nonlinear semidefinite programming (NLSDP). Namely, almost all linear perturbations of a given NLSDP are shown to be nondegenerate. Here, nondegeneracy for NLSDP refers to the transversality constraint qualification, strict complementarity and second-order sufficient condition. Due to the presence of the second-order sufficient condition, our result is a nontrivial extension of the corresponding results for linear semidefinite programs (SDP) from Alizadeh et al. (Math Program 77(2, Ser. B):111---128, 1997). The proof of the genericity result makes use of Forsgren's derivation of optimality conditions for NLSDP in Forsgren (Math Program 88(1, Ser. A):105---128, 2000). Due to the latter approach, the positive semidefiniteness of a symmetric matrix G(x), depending continuously on x, is locally equivalent to the fact that a certain Schur complement S(x) of G(x) is positive semidefinite. This yields a reduced NLSDP by considering the new semidefinite constraint $$S(x) \succeq 0$$S(x)ź0, instead of $$G(x) \succeq 0$$G(x)ź0. While deriving optimality conditions for the reduced NLSDP, the well-known and often mentioned "H-term" in the second-order sufficient condition vanishes. This allows us to access the proof of the genericity result for NLSDP.