# Local convergence of the Levenberg-Marquardt method under H\"{o}lder metric subregularity

## TL;DR

This paper investigates the local convergence of the Levenberg-Marquardt method for nonlinear equations under H"older metric subregularity, proposing an adaptive parameter and analyzing convergence with additional ojasiewicz inequality, supported by biochemical network applications.

## Contribution

It introduces an adaptive Levenberg-Marquardt parameter formula and analyzes convergence under H"older metric subregularity and ojasiewicz inequality, extending existing convergence theory.

## Key findings

- Convergence is established under H"older metric subregularity.
- Numerical results confirm theoretical convergence for biochemical problems.
- The method effectively computes steady states in biochemical networks.

## Abstract

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under H\"{o}lder metric subregularity of the function defining the equation and H\"older continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the \L{}ojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the \L{}ojasiewicz gradient inequality assumption.