Hybrid Fractional-Order Optimization in High-Dimensional Engineering Systems: A Rigorous Convergence-Guaranteed Mathematical Framework

Abstract

Classical integer-order gradient methods update each iterate using only the instantaneous local gradient, discarding the trajectory's history and, in high-dimensional, multimodal engineering optimization landscapes, this local-memoryless behavior frequently produces premature convergence to poor local optima. Fractional-order calculus, particularly the Caputo derivative, offers a mathematically principled mechanism for injecting long-range memory into the optimization trajectory by replacing the integer-order derivative with a non-local, history-weighted convolution operator. This paper reviews the convergence theory and hybrid algorithmic designs that have emerged around fractional-order optimization for high-dimensional engineering systems, synthesizing formal convergence-rate results for Caputo fractional gradient descent (CFGD) under convex, strongly convex, and non-convex smoothness assumptions with the metaheuristic hybridization literature that embeds fractional-order memory into population-based optimizers such as differential evolution and the Young's Double-Slit Experimental optimizer. Particular attention is given to the mathematical mechanism connecting the fractional order α to convergence behavior: theoretical and empirical evidence converges on the finding that fractional orders above unity accelerate convergence while orders below unity improve steady-state accuracy, with a domain-dependent optimum typically located near α ≈ 0.65 for highly multimodal engineering benchmarks. Comparative tables map the reviewed convergence theorems to their underlying smoothness assumptions and guarantee type, cross-reference hybrid fractional-metaheuristic algorithms against the specific engineering optimization problem each was validated on, and set reported statistical performance, convergence-speed improvement, error reduction, and significance testing, against the baseline algorithms each hybrid method outperformed. The review concludes that while rigorous convergence guarantees for fractional gradient methods are now well established for smooth convex and non-convex settings, extending these formal guarantees to the population-based hybrid metaheuristics that currently dominate high-dimensional engineering practice remains the field's principal open mathematical problem.