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

TL;DR

This paper demonstrates that the local topological structure of stationary point sets in nonlinear optimization can be universally represented by convex quadratic problems, revealing a normal form for their analysis.

Contribution

It establishes that convex quadratic problems capture all possible local topological structures of stationary point sets in parametric nonlinear optimization.

Findings

Convex quadratic problems produce a normal form for stationary point set structures.

The closure of stationary point sets forms a manifold with boundary, defined by constraint qualification violations.

Stationary point sets and violation sets share the same local stratified space structures.

Abstract

We investigate the local topological structure, stationary point sets in parametric optimization genericly 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.