Nonlinear Equations Solving with Intelligent Optimization Algorithms: A Survey

Abstract

Nonlinear Equations (NEs), which may usually have multiple roots, are ubiquitous in diverse fields. One of the main purposes of solving NEs is to locate as many roots as possible simultaneously in a single run, however, it is a difficult and challenging task in numerical computation. In recent years, Intelligent Optimization Algorithms (IOAs) have shown to be particularly effective in solving NEs. This paper provides a comprehensive survey on IOAs that have been exploited to locate multiple roots of NEs. This paper first revisits the fundamental definition of NEs and reviews the most recent development of the transformation techniques. Then, solving NEs with IOAs is reviewed, followed by the benchmark functions and the performance comparison of several state-of-the-art algorithms. Finally, this paper points out the challenges and some possible open issues for solving NEs.

Real-Time Laparoscopic Cholecystectomy Simulation Using a Particle-Based Physical System

“…The system was implemented on hardware comprising a workstation with an NVIDIA GeForce RTX 2060 GPU, an Intel i5-9500 CPU, 8 GB of RAM, and two haptic devices. The haptic devices were Geomagic Touch [26] models, enabling 6-DOF motion of the surgical tools. The average cluster radius of the virtual hepatobiliary system was 0.675 cm (Fig.…”

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

“…( 1) has also been referred to as a nonlinear equation system (NES) [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54]. The survey in article [55] uses the terminology Nonlinear Equations (NEs) to refer to a system of one or more nonlinear equations.…”

“…Other surveys discussing solution techniques for SNEs include [60] and [55]. We have decided to conduct this comprehensive literature review because many new solution techniques for SNEs have been introduced since the publication of [60] in 1994, and because the recent survey presented in [55] focuses mainly on methods which first convert a SNE into an optimization problem, and then search for multiple solutions to the optimization problem using Intelligent Optimization Algorithms (IOAs). The IOAs discussed in article [55] are primarily metaheuristics for global optimization.…”

“…We have decided to conduct this comprehensive literature review because many new solution techniques for SNEs have been introduced since the publication of [60] in 1994, and because the recent survey presented in [55] focuses mainly on methods which first convert a SNE into an optimization problem, and then search for multiple solutions to the optimization problem using Intelligent Optimization Algorithms (IOAs). The IOAs discussed in article [55] are primarily metaheuristics for global optimization. Although the survey in article [55] provides a very nice discussion of IOAs for solving SNEs reformulated as optimization problems, many of the IOAs they discuss are only introduced at a very high level, only eight IOAs were tested in their computational study, and the IOAs were evaluated on SNEs comprised of 20 equations or less.…”

“…The IOAs discussed in article [55] are primarily metaheuristics for global optimization. Although the survey in article [55] provides a very nice discussion of IOAs for solving SNEs reformulated as optimization problems, many of the IOAs they discuss are only introduced at a very high level, only eight IOAs were tested in their computational study, and the IOAs were evaluated on SNEs comprised of 20 equations or less. Also, article [55] only briefly mentions methods which can be used to search for solutions to SNEs without transforming them into optimization problems.…”