Anomalies of the Scholtes regularization for mathematical programs with complementarity constraints

Abstract: For mathematical programs with complementarity constraints (MPCC), we refine the convergence analysis of the Scholtes regularization. Our goal is to relate nondegenerate C-stationary points of MPCC with nondegenerate Karush-Kuhn-Tucker points of its Scholtes regularization. We detected the following anomalies: (i) in a neighborhood of a nondegenerate C-stationary point there could be degenerate Karush-Kuhn-Tucker points of the Scholtes regularization; (ii) even if nondegenerate, they might be locally non-unique; (iii) if nevertheless unique, their quadratic index potentially differs from the C-index of the C-stationary point under consideration. Thus, a change of the topological type for Karush-Kuhn-Tucker points of the Scholtes regularization is possible. In particular, a nondegenerate minimizer of MPCC might be approximated by saddle points. In order to bypass the mentioned anomalies, an additional generic condition for nondegenerate C-stationary points of MPCC is identified. Then, we uniquely trace nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization and successively maintain their topological type.