Survey of Methods for Solving Systems of Nonlinear Equations, Part II: Optimization Based Approaches

TL;DR

This survey comprehensively reviews optimization-based methods for solving systems of nonlinear equations, detailing transformations, algorithms, and future research directions to aid practitioners in selecting suitable techniques.

Contribution

It introduces a detailed classification and analysis of optimization approaches for SNEs, including transformations and promising future research avenues.

Findings

Transformations enable solving SNEs via optimization algorithms.

Analysis of method characteristics guides method selection.

Future research directions identified for improving solution techniques.

Abstract

This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting a thorough description and analysis of the known methods capable of finding one or many solutions to SNEs, and to assist interested readers seeking to identify solution techniques which are well suited for solving the various classes of SNEs which one may encounter in real world applications. To accomplish these objectives, we present a multi-part survey. In part one, we focused on root-finding approaches which can be used to search for solutions to a SNE without transforming it into an optimization problem. In part two, we introduce the various transformations which have been utilized to transform a SNE into an optimization problem, and we discuss optimization algorithms which can then be used to search for solutions. We emphasize the important characteristics of each method, and we discuss promising directions for future research. In part three, we will present a robust quantitative comparative analysis of methods capable of searching for solutions to SNEs.