Constrained Optimization Method – Steepest Descent Method (Also Known as Gradient Method)

In optimization problems, the Steepest Descent Method (also referred to as the Gradient Method) stands as one of the earliest techniques for solving extremum problems involving multivariable functions. The method operates through iterative updates along the negative gradient direction to search for function minima. A typical implementation involves calculating partial derivatives to determine the gradient vector, followed by parameter updates using a learning rate or step size. While this method is conceptually straightforward and easy to implement, its convergence rate may become slow in high-dimensional spaces due to zigzagging behavior near optimal points. In recent decades, numerous advanced optimization techniques have been developed to enhance efficiency and precision, including Conjugate Gradient Methods that overcome orthogonal direction limitations, and Quasi-Newton Methods that approximate Hessian matrices without costly computations. Beyond theoretical optimization, the Steepest Descent Method finds practical applications in signal processing for adaptive filtering, image processing for edge detection, and machine learning for parameter tuning, demonstrating its versatility in solving real-world engineering challenges.