On Stability of the Scholtes Regularization for Mathematical Programs with Complementarity Constraints

• 成果类型: Article; Early Access
• 署名作者: Lammel, Sebastian; Shikhman, Vladimir
• 署名单位: Technische Universitat Chemnitz
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2025.0895
• 发表日期: 2025-11-27
• 关键词: mathematical programs with complementarity constraints; Scholtes regularization; STABILITY; nondegeneracy; index shift; topological type; convergence properties

摘要

For mathematical programs with complementarity constraints (MPCC), we study the stability properties of their Scholtes regularization. Our goal is to relate nondegenerate C-stationary points of MPCC with nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization up to their topological type. As it is standard in the framework of Morse theory, the topological types are captured by the C-index and the quadratic index, respectively. It turns out that a change of the topological type for the approximating Karush-Kuhn-Tucker points of the Scholtes regularization and their limiting C-stationary point is possible. In particular, a minimizer of MPCC with zero C-index might be approximated by saddle points of the Scholtes regularization with nonzero quadratic index. In order to bypass this index shift phenomenon, an additional generic condition for nondegenerate C-stationary points of MPCC is identified. It says that nonbiactive multipliers under consideration should not vanish. Then, we uniquely trace nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization and successively maintain the topological type of their limiting C-stationary point. The main technical issue here is to relate the first-order information of the defining functions, which enters the biactive part of the C-index, with the secondorder information, which enters the quadratic index of the Karush-Kuhn-Tucker points.