Sufficient optimality conditions hold for almost all nonlinear semidefinite programs

• 成果类型: Article
• 署名作者: Dorsch, Dominik; Gomez, Walter; Shikhman, Vladimir
• 署名单位: RWTH Aachen University; Universidad de La Frontera; Universite Catholique Louvain
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-015-0915-0
• 发表日期: 2016
• 页码: 77-97
• 关键词: Optimization; CONVERGENCE

摘要

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 , instead of . 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.