A Morse theoretic approach to non-isolated singularities and applications to optimization

Abstract

Let $X$ be a complex affine variety in ${\mathbb{C}}^N$, and let $f:{\mathbb{C}}^N\to \mathbb{C}$ be a polynomial function whose restriction to $X$ is nonconstant. For $g:{\mathbb{C}}^N\to \mathbb{C}$ a general linear function, we study the limiting behavior of the critical points of the one-parameter family of $f_t:=f-tg$ as $t\to 0$. Our main result gives an expression of this limit in terms of critical sets of the restrictions of $g$ to the singular strata of $(X,f)$. We apply this result in the context of optimization problems. For example, we consider nearest point problems (e.g., Euclidean distance degrees) for affine varieties and a possibly nongeneric data point.

24 pages

References in corpus (1)

• Global Euler obstruction and polar invariants