Characterization of generic parameter families of constraint mappings in optimization

TL;DR

This paper investigates the generic behavior of constraint functions in optimization using singularity theory, classifying map-germs and establishing the prevalence of certain normal forms in parameterized families.

Contribution

It classifies constraint map-germs with small codimensions and demonstrates that most constraint mappings are generically equivalent to known normal forms in a residual set.

Findings

Classified map-germs with small stratum codimensions.

Calculated codimensions of orbits and their complements.

Proved that typical constraint mappings are generically equivalent to normal forms.

Abstract

The purpose of this paper is to understand generic behavior of constraint functions in optimization problems relying on singularity theory of smooth mappings. To this end, we will focus on the subgroup $\mathcal{K}[G]$ of the Mather's group $\mathcal{K}$, whose action to constraint map-germs preserves the corresponding feasible set-germs (i.e.~the set consisting of points satisfying the constraints). We will classify map-germs with small stratum $\mathcal{K}[G]_e$-codimensions, and calculate the codimensions of the $\mathcal{K}[G]$-orbits of jets represented by germs in the classification lists and those of the complements of these orbits. Applying these results and a variant of the transversality theorem, we will show that families of constraint mappings whose germ at any point in the corresponding feasible set is $\mathcal{K}[G]$-equivalent to one of the normal forms in the classification list compose a residual set in the entire space of constraint mappings with at most $4$-parameters. These results enable us to quantify genericity of given constraint mappings, and thus evaluate to what extent known test suites are generic.