Optimization Methods: Hooke-Jeeves and Powell Algorithms

This article explores two optimization methods: the Hooke-Jeeves and Powell algorithms, demonstrating their implementation through MATLAB programs. To facilitate understanding, we begin with a simple example: finding the minimum value of a function. The implementation involves key MATLAB functions like fminsearch alternatives and custom pattern search routines. We'll detail how to select initial points using rand or fixed coordinates, set step sizes through parameters like alpha, and define convergence criteria using tolerance thresholds. The algorithm workflow includes pattern moves and exploratory steps for Hooke-Jeeves, while Powell's method implements conjugate direction updates through directional minimization. Through this example, we'll examine the underlying mechanics of these algorithms, including how they handle objective functions defined via @(x) anonymous functions, and demonstrate their application to solve various optimization problems in practical scenarios.