Local convergence of the Levenberg–Marquardt method under Hölder metric subregularity

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 Holder metric subregularity of the function defining the equation and Holder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Łojasiewicz 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 Łojasiewicz gradient inequality assumption.