Steepest Descent Method in Nonlinear Programming with MATLAB Implementation

In this article, we explore various methods in nonlinear programming, including the Steepest Descent Method, BFGS method, and Conjugate Gradient Method. We provide corresponding MATLAB source codes for these algorithms, featuring a dedicated implementation for the Quasi-Newton BFGS method named "3-Quasi-Newton-BFGS-method.m". These implementations demonstrate key optimization techniques such as gradient computation, Hessian approximation updates, and line search procedures. The MATLAB codes include essential functions for: - Gradient calculation using numerical differentiation or analytical derivatives - Step size determination through Armijo line search or Wolfe conditions - BFGS Hessian matrix updating formula: H_{k+1} = (I - ρ_k s_k y_k^T)H_k(I - ρ_k y_k s_k^T) + ρ_k s_k s_k^T - Convergence criteria checking with tolerance thresholds These source programs serve as practical tools for studying the working mechanisms of optimization algorithms and their application to real-world nonlinear programming problems. By delving into these methods, researchers can better understand the challenges and opportunities in nonlinear optimization, contributing to future research developments and practical applications. The code structure follows modular programming principles, allowing easy modification and extension for specific problem requirements.