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

Abstract

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we first propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under H\"{o}lder metric subregularity. We then introduce a bounded version of the Levenberg-Marquardt parameter and analyse the local convergence of the modified method under the \L ojasiewicz gradient inequality. 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 H\"{o}lder metric subregularity assumption.