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

Abstract

Let X be a complex affine variety in C N , and let f : C N → C be a polynomial function whose restriction to X is nonconstant. For g : C N → 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 → 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.