On the Topology of the Karush–Kuhn–Tucker Set under Mangasarian–Fromovitz Constraint Qualification

Abstract

This paper deals with smooth optimization problems $\cal p$ in $\R^n$ depending on parameter $y\in\R^p$. The problem ${\cal p} (y)$ is defined by means of a finite number of equality and inequality constraints. We study the set ${\bf \sum}_{KKT}$ of pairs $(x,y)$ such that $x$ is a Karush-- Kuhn--Tucker point of the problem ${\cal p} (y)$. Let ${\bf \sum}$ denote the subset of ${\bf \sum}_{KKT}$ at which the Mangasarian--Fromovitz constraint qualification is fulfilled. For problem data in general position we prove that ${\bf \sum}$ is a topological manifold of dimension $p$.