Linear optimization on varieties and Chern-Mather classes

TL;DR

This paper explores the algebraic and geometric aspects of linear optimization on algebraic varieties, revealing how conormal varieties and bidegrees determine Chern-Mather classes and relate to optimization degrees.

Contribution

It establishes a novel connection between the geometry of conormal varieties and the algebraic complexity of linear optimization on varieties.

Findings

Conormal variety bidegrees determine Chern-Mather classes.

Bidegrees coincide with linear optimization degrees of generic sections.

Provides a geometric interpretation of optimization complexity.

Abstract

The linear optimization degree gives an algebraic measure of complexity of optimizing a linear objective function over an algebraic model. Geometrically, it can be interpreted as the degree of a projection map on the {affine} conormal variety. Fixing an affine variety, our first result shows that the geometry of {this} conormal variety, expressed in terms of bidegrees, completely determines the Chern-Mather classes of the given variety. We also show that these bidegrees coincide with the linear optimization degrees of generic affine sections.