Hooke-Jeeves Pattern Search Algorithm for Multivariable Optimization

The Hooke-Jeeves pattern search algorithm provides an effective approach for multivariable optimization in direct search methods. This algorithm employs a two-phase process consisting of exploratory moves and pattern moves to efficiently navigate the search space. During implementation, the algorithm typically starts with an initial point and step size, performing coordinate-wise explorations to determine promising directions. The pattern move then accelerates convergence by leveraging successful search directions from previous iterations.

A key feature of this algorithm is its dynamic step size adjustment mechanism, which allows it to refine the search granularity based on performance. When pattern moves successfully reduce the objective function value, the algorithm expands the step size to explore larger regions. Conversely, if pattern moves fail, it contracts the step size for more localized search. This adaptive behavior enables better exploration of the solution space while maintaining computational efficiency.

The algorithm's implementation typically involves maintaining a current base point, testing points in different coordinate directions, and establishing pattern directions when improvements are found. Common termination criteria include reaching maximum iterations, achieving satisfactory convergence tolerance, or exhausting minimum step size limits. The Hooke-Jeeves method demonstrates particular strength in handling nonlinear optimization problems where derivative information is unavailable or computationally expensive.

With its rapid convergence characteristics and straightforward implementation, the Hooke-Jeeves pattern search algorithm serves as an efficient tool for solving multivariable optimization problems across various engineering and scientific applications.