A Newton-Kantorovich Inverse Function Theorem in Quasi-Metric Spaces

TL;DR

This paper extends the Newton-Kantorovich inverse function theorem to quasi-metric spaces, enabling superlinear convergence analysis of Newton-type methods for root finding in complex spaces relevant to biology and phylogenetics.

Contribution

It introduces a Newton differentiability concept and proves a Newton-Kantorovich theorem in quasi-metric spaces, broadening the theoretical foundation for Newton-type methods.

Findings

Superlinear convergence of Newton-type methods established.

A Newton-Kantorovich inverse function theorem applicable to quasi-metric spaces.

Framework applicable to biological and phylogenetic optimization problems.

Abstract

The purpose of this work is to investigate root finding problems defined on (quasi-)metric spaces, and ranging in Euclidean spaces. The motivation for this line of inquiry stems from recent models in biology and phylogenetics, where problems of great practical significance are cast as optimization problems on (quasi-)metric spaces. We investigate a minimal algebraic setup that allows us to study a notion of differentiability suitable for Newton-type methods, called Newton differentiability. This notion of differentiability benefits from calculus rules and is sufficient to prove superlinear convergence of a Newton-type method. Finally, a Newton-Kantorovich-type theorem provides an inverse function result, applicable on (quasi-)metric spaces.