POWELL Algorithm for Minimum Value Optimization

The POWELL algorithm is an efficient gradient-free optimization method specifically designed for solving multidimensional function minimization problems. Proposed by British mathematician Michael J. D. Powell in 1964, this algorithm has gained widespread application in scientific computing and engineering fields due to its exceptional convergence speed and computational accuracy. In code implementation, the algorithm typically requires defining an objective function and initial starting point as primary inputs.

The algorithm's core adopts the conjugate direction method concept, iteratively improving search directions without requiring gradient information of the objective function. This makes the POWELL algorithm particularly suitable for optimizing complex functions where derivatives are difficult to obtain or non-existent. From a programming perspective, the algorithm can be implemented using directional vectors and line search subroutines, often involving function evaluations along specified directions.

The working process primarily consists of three stages: First, establishing an initial set of search directions, typically selecting coordinate axis directions through identity matrix initialization. Then performing one-dimensional line searches to find current optimal solutions along each direction, which can be implemented using golden section search or Brent's method. Finally, generating new conjugate directions through direction update strategies. This approach effectively prevents "zigzagging phenomena" and ensures efficient algorithm convergence. Code implementation often includes direction replacement logic where the oldest direction is discarded when adding new conjugate directions.

The advantages of the POWELL algorithm lie in its strong robustness, insensitivity to initial point selection, and relatively small computational requirements. It can handle medium-scale optimization problems up to several dozen dimensions and has been successfully applied in mechanical design, economic modeling, and machine learning parameter tuning. In programming practice, developers should note that algorithm performance decreases as problem dimensionality increases, and ultra-high-dimensional problems may require integration with other techniques like dimensionality reduction methods. The algorithm typically terminates when direction improvements fall below a specified tolerance threshold.