MATLAB Implementation of Optimization Algorithms: Steepest Descent, Newton's Method, and Conjugate Gradient Methods

This article presents MATLAB implementations of three essential optimization algorithms: the Steepest Descent Method, Newton's Method, and the Conjugate Gradient Method. These numerical methods are widely used in optimization problems, including solving nonlinear equations, least squares problems, and function minimization. The implementation typically involves defining objective functions, calculating gradients (for Steepest Descent), computing Hessian matrices (for Newton's Method), and implementing iterative convergence criteria. Key MATLAB functions employed include fminunc for comparison, gradient computation using finite differences or symbolic differentiation, and matrix operations for Hessian calculations. The Conjugate Gradient implementation features the Fletcher-Reeves or Polak-Ribière formula for beta parameter computation and efficient line search techniques. We will examine the fundamental principles and implementation details of these algorithms, providing MATLAB code examples for reference. The code structure generally includes initialization parameters, iteration loops with stopping criteria (tolerance or maximum iterations), and performance monitoring through residual norms or function value reduction. Additionally, we will discuss practical applications of these algorithms in real-world problems and provide guidance on selecting the most appropriate method based on problem characteristics such as problem dimension, function smoothness, and computational resource constraints. Through studying this material, readers will gain a deeper understanding of these crucial numerical methods and be able to apply them effectively in their scientific research work.