Optimization Methods: Hooke-Jeeves and Powell Algorithms

The Hooke-Jeeves and Powell algorithms are two important direct search methods in optimization, particularly suitable for derivative-free optimization problems. Implementing these algorithms in MATLAB can help us efficiently solve various optimization problems.

### Hooke-Jeeves Algorithm The Hooke-Jeeves algorithm is a pattern search method that progressively approaches the optimal solution through exploration and pattern move phases. During the exploration phase, the algorithm conducts exploratory searches along coordinate directions to find local improvement points. In the pattern move phase, it accelerates convergence by making pattern moves based on exploration results. MATLAB implementation typically requires defining initial points, step sizes, and convergence criteria, making it suitable for low-dimensional optimization problems. The algorithm structure involves initializing parameters, implementing coordinate-wise search loops, and pattern move calculations using vector operations.

### Powell Algorithm The Powell algorithm belongs to conjugate direction methods, improving search efficiency by iteratively constructing conjugate directions. Unlike Hooke-Jeeves, it doesn't rely on coordinate axis directions but dynamically adjusts search directions, making it particularly suitable for high-dimensional optimization problems. MATLAB implementation requires careful attention to direction set update strategies and line search precision to ensure algorithm efficiency. Key implementation aspects include maintaining direction matrices, implementing accurate line search subroutines, and updating conjugate directions using vector transformations.

### MATLAB Implementation Examples In MATLAB, simple scripts can be written to implement both algorithms. For quadratic function optimization: Hooke-Jeeves: Set initial point, initial step size, and tolerance, then iteratively update search directions and step sizes through loop structures. The implementation involves nested loops for coordinate directions and pattern moves with condition checks for successful explorations. Powell: Initialize direction matrices (typically identity matrices), cyclically perform line searches and update conjugate directions until convergence conditions are met. The implementation requires functions for direction normalization, line search optimization, and direction set recombination using matrix operations.

Both methods are suitable for black-box optimization problems without analytical derivatives, but step size selection significantly impacts convergence speed. In practical applications, parameters can be adjusted according to specific problems or combined with other optimization strategies to improve efficiency. Implementation considerations include adaptive step size control, convergence monitoring, and handling of constraint boundaries through penalty functions.