Interior Point Penalty Function Method: Algorithm Description and MATLAB Implementation

In the following text, we will describe the fundamental concepts and principles of the interior point penalty function method algorithm, and provide a detailed guide for implementing this algorithm using MATLAB. The interior point penalty function method is an optimization algorithm designed to solve specific types of constrained problems. It employs penalty functions to penalize constraint violations and incorporates a penalty parameter within the function. The algorithm progressively approaches the optimal solution by continuously reducing the penalty parameter value during iterative processes. We will first introduce the algorithm's basic concepts and mathematical principles, followed by detailed MATLAB implementation steps and sample code. The implementation will include key programming aspects such as: - Initialization of penalty parameters and tolerance thresholds - Implementation of the barrier function for interior point maintenance - Gradient calculation for objective and constraint functions - Iterative update mechanisms for penalty parameter reduction - Convergence criteria implementation The MATLAB code examples will demonstrate practical implementation techniques, including function handle usage for objective functions, constraint handling through logarithmic barrier functions, and step-size control strategies. This comprehensive approach will help readers better understand the concrete implementation process of the algorithm, with particular emphasis on numerical stability considerations and convergence optimization techniques.