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 : R n → 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.Mathematics subject classification: 65K05, 90C30.which arises from many important applications in optimization field. Pang and Qi [15] reviewed eight problems in the study of complementarity problems, variational inequality problems and optimization problems, which can be reformulated as systems of nonsmooth equations. From the early 1990s, a number of generalized Newton methods for solving nonsmooth equations were proposed and analyzed. For a comprehensive discussion on this topic, readers are referred to the monographs [4,5] and references therein. In this paper, we will discuss the Levenberg-Marquardt-type methods for solving nonsmooth equations.The Levenberg-Marquardt (LM) method proposed by [9,11] is a classical and popular approach for solving nonlinear equations H(x) = 0, (1.2) *

Slope-Steering Motion Planning for Unmanned Tracked Vehicles Based on SSTP-RRT

“…Therefore, PSO-LM uses PSO to calculate the initial value range before applying LM to find the numerical solution for nonlinear minimization. LM is an optimization method for least-squares estimation of regression parameters in nonlinear regression that provides a numerical solution for nonlinear minimization and can improve the problem of nonexistent inverse matrices of the Gauss-Newton algorithm [17] [18]. LM combines the Gauss-Newton algorithm and the gradient descent method by modifying the parameters during iteration, ensuring a velocity of convergence while searching along the descent direction overall.…”

An improved Levenberg–Marquardt method for nonsmooth equations with application to multi-stream heat exchangers

“…The formulation of the nonsmooth MHEX model is given by Eqs. ( 7) and (8), which represent the energy balance and the second law requirement that heat flows from hot to cold, respectively.…”

“…As a result, solving a system of nonsmooth equations has become one of most active research directions in mathematical programming. There exist many methods for solving systems of nonsmooth equations, such as Newton-type methods [3,4], Trust-region-type methods [5][6][7], Levenberg-Marquardt-type methods [8,9]. However, there are some drawbacks in the first and second type of methods, as pointed out in [6,10].…”

Applying Bayesian Regularization for Acceleration of Levenberg Marquardt based Neural Network Training

“…Qi proposed LM-method with self adjusting parameters for solving a nonlinear system of equations [3]. By their approach, at each step, the LM parameter µk is automatically adjusted on the basis of the correlation between actual reduction and predicted reduction.…”