Stationary Point Sets: Convex Quadratic Optimization Is Universal in Nonlinear Optimization

Abstract

We investigate the local topological structure that stationary point sets in parametric optimization generically may have. Our main result states that up to stratified isomorphism, any such structure is already present in the small subclass of parametric problems with convex quadratic objective function and affine-linear constraints. In other words, the convex quadratic problems produce a normal form for the local topological structure of stationary point sets. As a consequence we see, as far as no equality constraints are involved, that the closure of the stationary point set constitutes a manifold with boundary. The boundary is exactly the violation set of the Mangasarian--Fromovitz constraint qualification. A side result states that stationary point sets and violation sets of Mangasarian--Fromovitz constraint qualification carry the same set of possible local structures as stratified spaces.