Computational Procedures of Different Search Methods (Steepest Descent, Conjugate Gradient, Newton's, and Quasi-Newton Methods)

This article introduces the computational procedures of different search algorithms including Steepest Descent, Conjugate Gradient, Newton's, and Quasi-Newton methods, along with a comparison of their respective advantages and disadvantages. The Steepest Descent method employs a simple gradient-based approach where each iteration moves in the direction opposite to the gradient, making it easy to implement but suffering from slow convergence rates. The Conjugate Gradient method utilizes orthogonal direction vectors to accelerate convergence, requiring specific conditions like quadratic functions for optimal performance. Newton's method achieves rapid quadratic convergence by leveraging second-order derivative information through Hessian matrix calculations, though this involves computational complexity. Quasi-Newton methods approximate the Hessian matrix using update formulas (like BFGS or DFP), eliminating direct Hessian computation while requiring careful selection and updating of the approximation matrix. Consequently, choosing appropriate search algorithms based on different application scenarios can better meet optimization requirements, with implementations typically involving gradient calculations, direction updates, and step size determination in iterative loops.