On stability of the Scholtes regularization for mathematical programs with complementarity constraints

TL;DR

This paper investigates the stability of Scholtes regularization in MPCCs, establishing conditions under which the topological type of solutions is preserved and addressing the index shift phenomenon.

Contribution

It introduces a generic condition to maintain the topological type of solutions and relates first- and second-order information in the stability analysis.

Findings

Identifies conditions to preserve the topological type of solutions.

Shows possible index shifts between MPCC solutions and regularization approximations.

Provides a method to uniquely trace solutions maintaining their topological properties.

Abstract

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.