Critical points via monodromy and local methods

TL;DR

This paper presents a numerical algebraic geometry approach using homotopy continuation and monodromy to compute all critical points of objective functions, aiding in optimization problems in applied mathematics.

Contribution

It introduces a novel method combining homotopy continuation and monodromy to efficiently find all complex critical points of objective functions.

Findings

Successfully computes critical points for Euclidean distance functions.

Determines maximum likelihood degrees in statistical models.

Demonstrates applicability to various optimization problems.

Abstract

In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function restricted to the model. We compile ideas arising from numerical algebraic geometry to compute the critical points of an objective function. Our method consists of using numerical homotopy continuation and a monodromy action on the total critical space to compute all of the complex critical points of an objective function. To illustrate the relevance of our method, we apply it to the Euclidean distance function to compute ED-degrees and the likelihood function to compute maximum likelihood degrees.