A Comprehensive Review of Numerical Methods for Solving Nonlinear Equations and Optimization Problems

Abstract

Nonlinear equations and optimization problems lie at the heart of computational science, appearing in fields as diverse as engineering design, economics, machine learning, power systems, and the physical and biological sciences. Because closed-form analytical solutions rarely exist, numerical methods provide the practical means of obtaining accurate approximate solutions. This review presents a structured and comprehensive survey of the principal numerical methods used to solve nonlinear equations and optimization problems, tracing their development from classical iterative schemes such as the Newton-Raphson and quasi-Newton families to modern higher-order multi-step methods, derivative-free techniques, gradient-based optimizers, and population-based metaheuristics. We examine the theoretical foundations governing convergence order, computational efficiency, and stability, and we discuss the efficiency index as a unifying measure for comparing methods that differ in the number of function evaluations per iteration. The review synthesizes recent advances reported between 2015 and 2021, highlighting trends toward high-order convergence with reduced functional evaluations, the avoidance of second derivatives through finite-difference approximations, and the hybridization of deterministic and stochastic search strategies. Particular attention is given to the parallel evolution of optimization algorithms in machine learning, where adaptive gradient methods such as AdaGrad, RMSProp, and Adam have reshaped large-scale training, and to the emergence of Newton-inspired metaheuristics that balance exploration and exploitation in non-convex landscapes. Across these developments, recurring challenges include guaranteeing global convergence, handling ill-conditioning and high dimensionality, and escaping local optima. The review concludes by identifying open problems and promising directions, including the integration of higher-order curvature information, robustness under limited data, and the convergence of classical numerical analysis with data-driven optimization. By consolidating these strands, this paper aims to serve as a coherent reference for researchers and practitioners navigating the rich landscape of nonlinear solvers and optimizers.