  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.
