Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions

Ioannis K Argyros,Roman Iakymchuk,Michael I Argyros,Halyna Yarmola,Stepan Shakhno

doi:10.3390/axioms10030158

Abstract

We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort.

Highlights

Nonlinear least squares problems often arise while solving overdetermined systems of nonlinear equations, estimating parameters of physical processes by measurement results, constructing nonlinear regression models for solving engineering problems, etc
We study the local convergence of the Gauss–Newton–Secant method under the classical and generalized Lipschitz conditions for first-order Fréchet derivative and divided differences
We developed an improved local convergence analysis of the Gauss–Newton–Secant method for solving nonlinear least squares problems with nondifferentiable operator

Summary

Introduction

Nonlinear least squares problems often arise while solving overdetermined systems of nonlinear equations, estimating parameters of physical processes by measurement results, constructing nonlinear regression models for solving engineering problems, etc. Some nonlinear functions have a differentiable and a nondifferentiable part In this case, a good idea is to use a sum of the derivative of the differentiable part of the operator and the divided difference of the nondifferentiable part instead of the Jacobian [4,5,6]. We study the local convergence of the Gauss–Newton–Secant method under the classical and generalized Lipschitz conditions for first-order Fréchet derivative and divided differences. The convergence domain is small (in general), and error estimates are pessimistic These problems restrict the applicability of these methods. Our idea is to use a center and restricted radius Lipschitz conditions Such an approach to the study of the convergence of methods allows for extending the convergence ball of the method and improving error estimates.

Local Convergence Analysis

Numerical Examples

Conclusions