On the derivative-free quasi-Newton-type algorithm for separable systems of nonlinear equations

Hassan Mohammad,Aliyu Muhammed Awwal,Ahmad Salihu Ben Musa,Auwal Bala Abubakar

doi:10.1051/ro/2021154

Abstract

A derivative-free quasi-Newton-type algorithm in which its search direction is a product of a positive definite diagonal matrix and a residual vector is presented. The algorithm is simple to implement and has the ability to solve large-scale nonlinear systems of equations with separable functions. The diagonal matrix is simply obtained in a quasi-Newton manner at each iteration. Under some suitable conditions, the global and R-linear convergence result of the algorithm are presented. Numerical test on some benchmark separable nonlinear equations problems reveal the robustness and efficiency of the algorithm.

Highlights

Consider the problem of finding a solution of nonlinear system of equations g(x) = 0, (1.1)where g = (g1, g2, . . . gn) : Rn → Rn is a separable function
We incorporate the diagonal Hessian approximation approach studied by Deng and Wan [2] and the spectral residual approach presented in [13] to propose, analyze and implement a derivative-free algorithm for separable problems, which can be seen as an improved version of the dfsane algorithm that used a positive definite diagonal matrix as the approximation of the Jacobian of the function g
We have presented, analyzed, and implemented a derivative-free quasi-Newton-type algorithm for solving nonlinear systems of equations with separable functions

Summary

Introduction

Derivative-free methods, quasi-Newton-type methods, convergence, numerical experiments. For finding the solution of general nonlinear equations, quasi-Newton methods are famous and commonly used algorithms because of their derivative-free nature [17, 21] Among these methods, some are not suitable for large-scale problems due to matrix storage requirements. We incorporate the diagonal Hessian approximation approach studied by Deng and Wan [2] and the spectral residual approach presented in [13] to propose, analyze and implement a derivative-free algorithm for separable problems, which can be seen as an improved version of the dfsane algorithm that used a positive definite diagonal matrix as the approximation of the Jacobian of the function g. ‖ · ‖ stands for the Euclidean norm of vectors and the induced 2-norm of matrices

Preliminaries and algorithm

Convergence results

Numerical experiments

Findings

Conclusions