A Parameter-Self-Adjusting Levenberg-Marquardt Method for Solving Nonsmooth Equations

Abstract

A parameter-self-adjusting Levenberg-Marquardt method (PSA-LMM) is proposed for solving a nonlinear system of equations $F(x) = 0$, where $F : \mathbb{R}^n$ →$\mathbb{R}^n$ is a semismooth mapping. At each iteration, the LM parameter $μ_k$ is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA-LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.