The Role of Fractional Calculus in Modern Optimization: A Survey of Algorithms, Applications, and Open Challenges

This paper provides a comprehensive overview of the application of fractional calculus in modern optimization methods, with a focus on its impact in artificial intelligence (AI) and computational science. We examine how fractional-order derivatives have been integrated into traditional methodologies, including gradient descent, least mean squares algorithms, particle swarm optimization, and evolutionary methods. These modifications leverage the intrinsic memory and nonlocal features of fractional operators to enhance convergence, increase resilience in high-dimensional and non-linear environments, and achieve a better trade-off between exploration and exploitation. A systematic and chronological analysis of algorithmic developments from 2017 to 2025 is presented, together with representative pseudocode formulations and application cases spanning neural networks, adaptive filtering, control, and computer vision. Special attention is given to advances in variable- and adaptive-order formulations, hybrid models, and distributed optimization frameworks, which highlight the versatility of fractional-order methods in addressing complex optimization challenges in AI-driven and computational settings. Despite these benefits, persistent issues remain regarding computational overhead, parameter selection, and rigorous convergence analysis. This review aims to establish both a conceptual foundation and a practical reference for researchers seeking to apply fractional calculus in the development of next-generation optimization algorithms.