Adaptive Modified Levenberg–Marquardt Algorithm for Nonlinear Least Squares: Analysis and Applications

Abstract

Nonlinear least squares (NLS) methods such as Levenberg–Marquardt (LM) often suffer from sensitivity to initialization and instability in ill-conditioned problems. This paper proposes an Adaptive Modified Levenberg–Marquardt (AMLM) algorithm with curvature-aware diagonal damping and a reduction-ratio-based update strategy. Experiments on exponential, power-law and logistic models (3 initialization levels, σ = 0–0.1, 100 runs per setting) demonstrate that AMLM achieves improved robustness and convergence stability compared with the traditional LM algorithm under challenging conditions. In exponential models, AMLM achieves win rates above 55% under high noise and poor initialization and successfully recovers correct nonlinear structures where LM fails. In queueing-inspired near-singular scenarios, AMLM maintains stable convergence while LM becomes unstable. These gains come at the cost of increased computation, with runtime and evaluation cost rising by approximately 10–30%. Overall, AMLM provides a simple and effective enhancement to LM for ill-conditioned and strongly nonlinear optimization problems.