2025/01/13 by Vladimir Shikhman, Shikhman, Vladimir, Sebastian Lämmel +1
Computer Science · Mathematics · #90C26 #90C33 #Advanced Banach Space Theory #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2501.07383
openalex publication_date 2025/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 non-biactive 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 second-order information, which enters the quadratic index of the Karush-Kuhn-Tucker points.