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

Edson Fernandez (), Victor Huilcapi, Isabela Birs and Ricardo Cajo
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Edson Fernandez: Laboratorio de Sistemas Automáticos de Control, Facultad de Ingeniería Mecánica Eléctrica, Universidad de Piura, UDEP, Av. Ramón Mugica 131, Piura 20009, Peru
Victor Huilcapi: Facultad de Ingenierías, Universidad Politécnica Salesiana, Robles 107 y Chambers, Guayaquil 090109, Ecuador
Isabela Birs: Automation Department, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Ricardo Cajo: Faculty of Electrical and Computer Engineering, Escuela Superior Politécnica del Litoral (ESPOL), Campus Gustavo Galindo Km 30.5 Vía Perimetral, Guayaquil 090902, Ecuador

Mathematics, 2025, vol. 13, issue 19, 1-34

Abstract: 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.

Keywords: fractional calculus; optimization algorithms; gradient-based optimization; fractional-order derivatives; memory effects; machine learning; artificial intelligence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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