Newton's Method, Steepest Descent, Quasi-Newton Methods, Golden Section Search, and One-Dimensional Search Techniques

In this article, we will explore MATLAB programming techniques for implementing key numerical optimization algorithms: Newton's Method, Steepest Descent, Quasi-Newton Methods, Golden Section Search, and One-Dimensional Search methods. These algorithms form crucial components in numerical computing, requiring careful attention to implementation details for optimal performance.

First, let's examine Newton's Method. This iterative approach finds roots or extremum points of functions by using local first and second derivatives to create quadratic approximations. The algorithm solves for the roots of these approximations to converge toward the true solution. In MATLAB implementation, the fminunc function is commonly employed for unconstrained optimization using Newton's method, where developers can specify gradient and Hessian information through function handles for improved accuracy.

Next, we consider Steepest Descent (also known as Gradient Descent). This optimization technique minimizes functions by iteratively moving in the direction opposite to the gradient vector. The algorithm uses first-order derivative information to determine the optimal descent direction at each point. MATLAB's fminsearch function implements a derivative-free version of this approach using the Nelder-Mead simplex method, while custom implementations often involve calculating gradients numerically or analytically for better convergence control.

Beyond Newton's Method and Steepest Descent, Quasi-Newton Methods represent another important class of numerical optimization algorithms. These methods approximate the Hessian matrix using gradient information, avoiding expensive second-derivative calculations while maintaining faster convergence than basic gradient descent. The BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm is a popular Quasi-Newton implementation available in MATLAB's fminunc function when configured with appropriate options, providing efficient memory usage through matrix updates rather than full recomputation.

Additionally, Golden Section Search and One-Dimensional Search methods serve as fundamental optimization techniques. Golden Section Search is an interval reduction method for unimodal function minimization that divides the search space using the golden ratio (approximately 1.618). One-Dimensional Search techniques operate within constrained intervals to locate function minima. MATLAB's fminbnd function implements these approaches, combining golden section search with parabolic interpolation for enhanced convergence in single-variable optimization problems, requiring only function evaluations without derivative information.

In summary, this article has demonstrated MATLAB implementation strategies for Newton's Method, Steepest Descent, Quasi-Newton Methods, Golden Section Search, and One-Dimensional Search techniques. These algorithms play vital roles in numerical computing, and mastering their implementation approaches significantly enhances our ability to write efficient and effective MATLAB code for optimization problems.