Enhancing stochastic optimization: investigating fixed points of chaotic maps for global optimization

Abstract

Chaotic maps, despite their deterministic nature, can introduce controlled randomness into optimization algorithms. This chaotic map behaviour helps overcome the lack of mathematical validation in traditional stochastic methods. The chaotic optimization algorithm (COA) uses chaotic maps that help it achieve faster convergence and escape local optima. The effective use of these maps to find the global optimum would be possible only with a complete understanding of them, especially their fixed points. In chaotic maps, fixed points repeat indefinitely, disrupting the map's characteristic unpredictability. While using chaotic maps for global optimization, it is crucial to avoid starting the search at fixed points and implement corrective measures if they arise in between the sequence. This paper outlines strategies for addressing fixed points and provides a numerical evaluation (using Newton's method) of the fixed points for 20 widely used chaotic maps. By appropriately handling fixed points, researchers and practitioners across diverse fields can avoid costly failures, improve accuracy, and enhance the reliability of their systems.