Optimizing Complex Functions: A Numerical and ML Comparison

Abstract

Optimization plays a vital role across disciplines such as engineering, economics, and artificial intelligence. Complex functions, which map complex numbers to complex outputs, often cannot be solved analytically, necessitating numerical or machine learning-based approaches. This study presents a comparative analysis of numerical optimization methods—specifically Gradient Descent and Newton’s Method—against machine learning-based techniques, including Genetic Algorithms, Particle Swarm Optimization, and Deep Q-Learning. These methods are evaluated using standard benchmark functions: Ackley, Rastrigin, and Rosenbrock. The comparison focuses on convergence rate and runtime performance. Results show that numerical methods offer faster runtimes but lower convergence rates, while machine learning approaches achieve higher convergence at the cost of increased computational time. This analysis underscores the trade-offs between efficiency and robustness in optimization techniques, offering practical insights for selecting appropriate methods based on specific application needs, especially in scenarios involving complex, non-linear, or high-dimensional functions.