Critical points via monodromy and local methods

Abstract

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.

Solving polynomial systems via homotopy continuation and monodromy

“…A 2B Applying the laws of mass-action kinetics to the reaction network above, we obtain the polynomial system (7) consisting of the corresponding steadystate and conservation equations. Here the k i 's represent the reaction rates, x i 's represent species concentrations, and the c i 's are parameters.…”

“…Although we may obtain large systems, they typically have very low root counts compared to the sparse case. The polynomial system (7) has four solutions. A larger example is the wnt signaling pathway from Systems Biology [14] consisting of 19 polynomial equations with 9 solutions.…”

Tensor decomposition and homotopy continuation

“…In practice, we may compute a pseudowitness point set U by starting with one sufficiently general point in the image and performing monodromy loops. Such an approach has been used in various applications, e.g., [44,70], and will be used in many of the examples in Section 5. Since Y = π(X), we can compute a general point y ∈ Y given a general point x ∈ X.…”

“…The completeness of the set is verified via a trace test. More information about this procedure can be found in, e.g., [44,70] and [41, § 2.4.2].…”

What is numerical algebraic geometry?

“…The fourth article [27] considers exploiting the structure of polynomial systems arising in the computation of critical points.…”