Extended Levenberg-Marquardt Method for Composite Function Minimization

Abstract

In this paper, we propose an extended Levenberg-Marquardt (ELM) framework that generalizes the classic Levenberg-Marquardt (LM) method to solve the unconstrained minimization problem min $ρ(r(x))$, where $r$ : $\mathbb{R}^n$ → $\mathbb{R}^m$ and $ρ$ : $\mathbb{R}^m$ → $\mathbb{R}$. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jacobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.