Critical points via monodromy and local methods

Abstract

A fundamental problem in many areas of applied mathematics and statistics is to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the function restricted to the model.We compile ideas arising from numerical algebraic geometry to compute these critical points. 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. 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.