# Finding Zeros of H\"{o}lder Metrically Subregular Mappings via Globally Convergent Levenberg-Marquardt Methods

## TL;DR

This paper introduces two globally convergent Levenberg-Marquardt algorithms for finding zeros of H"{o}lder metrically subregular mappings, with proven convergence, complexity bounds, and successful application to biological data.

## Contribution

The paper develops two novel Levenberg-Marquardt methods with global convergence and complexity analysis for non-isolated zeros of H"{o}lder metrically subregular mappings.

## Key findings

- Methods achieve global convergence to stationary points.
- Worst-case complexity is O(ε^{-2}) evaluations.
- Numerical results successfully applied to biological data.

## Abstract

We present two globally convergent Levenberg-Marquardt methods for finding zeros of H\"{o}lder metrically subregular mappings that may have non-isolated zeros. The first method unifies the Levenberg- Marquardt direction and an Armijo-type line search, while the second incorporates this direction with a nonmonotone trust-region technique. For both methods, we prove the global convergence to a first-order stationary point of the associated merit function. Furthermore, the worst-case global complexity of these methods are provided, indicating that an approximate stationary point can be computed in at most $\mathcal{O}(\varepsilon^{-2})$ function and gradient evaluations, for an accuracy parameter $\varepsilon>0$. We also study the conditions for the proposed methods to converge to a zero of the associated mappings. Computing a moiety conserved steady state for biochemical reaction networks can be cast as the problem of finding a zero of a H\"{o}lder metrically subregular mapping. We report encouraging numerical results for finding a zero of such mappings derived from real-world biological data, which supports our theoretical foundations.