Performance Analysis of the Optimization Method for Escaping Saddle Points in FEMS Energy Management

Non-convex optimization problems in FEMS frequently encounter saddle points, which can significantly hinder the efficiency of traditional optimization methods such as Newton's method. This paper introduces the Saddle Free Newton method, an innovative approach designed to overcome the limitations of traditional methods by modifying the eigenvalues of the Hessian matrix. By adjusting negative eigenvalues to positive ones, this method enables the algorithm to escape saddle points effectively. We implemented this method in Python and applied it to a non-convex function, visualizing the optimization path through both 2D and 3D graphs. These visualizations clearly demonstrate the method's ability to navigate complex landscapes and efficiently converge to optimal solutions, showcasing significant improvements over traditional approaches.