Inexact Levenberg-Marquardt methods under H\"{o}lder metric subregularity

TL;DR

This paper develops and analyzes two inexact Levenberg-Marquardt algorithms for nonlinear systems, establishing convergence properties and demonstrating efficiency through numerical experiments on real-world problems.

Contribution

It introduces two novel inexact LM methods with convergence analysis under H"older metric subregularity and Lipschitz conditions, including complexity bounds and practical implementation strategies.

Findings

The first method achieves local superlinear convergence.

The second method guarantees global convergence and complexity bounds.

Numerical experiments confirm efficiency on real-world nonlinear systems.

Abstract

This paper investigates two inexact Levenberg-Marquardt (LM) methods for solving systems of nonlinear equations. Both approaches compute approximate search directions by solving the LM linear system inexactly, subject to specific residual-based conditions. The first method uses an adaptive scheme to update the LM parameter, and we establish its local superlinear convergence under H\"older metric subregularity and local H\"older continuity of the gradient. The second method combines an inexact LM step with a nonmonotone quadratic regularization strategy. For this variant, we prove global convergence under the assumption of Lipschitz continuous gradients and derive a worst-case global complexity bound, showing that an approximate stationary point can be found in $\mathcal{O}(\epsilon^{-2})$ function and gradient evaluations. Finally, we justify the use of the LSQR algorithm for efficiently solving the linear systems involved, which is used in our numerical experiment on several nonlinear systems, including those appearing in real-world biochemical reaction networks, monotone and nonlinear equations, and image deblurring problems.