In this study, a framework for processing gradient algorithms is proposed in accordance with nabla fractional-order system theory. Unlike most of the literature, the gradient algorithm is initially recast into a nabla fractional-order dynamic system. To be specific, the algorithm is designed using control theory and analyzed using the Lyapunov theory, which is a more general and effective way to render high-performance algorithm. Three types of algorithms are built in this study, i.e., the asymptotic convergence case, the finite-time convergence case, and the fixed-time convergence case. Finally, a comprehensive simulation study is conducted verifying the correctness, usefulness, and practicality of the framework.

Lévy flights is a random walk where the step-lengths have a probability distribution that is heavy-tailed. It has been shown that Lévy flights can maximize the efficiency of resource searching in uncertain environments and also the movements of many foragers and wandering animals have been shown to follow a Lévy distribution. The reason mainly comes from the fact that the Lévy distribution has an infinite second moment and hence is more likely to generate an offspring that is farther away from its parent. However, the investigation into the efficiency of other different heavy-tailed probability distributions in swarm-based searches is still insufficient up to now. For swarm-based search algorithms, randomness plays a significant role in both exploration and exploitation or diversification and intensification. Therefore, it is necessary to discuss the optimal randomness in swarm-based search algorithms. In this study, cuckoo search (CS) is taken as a representative method of swarm-based optimization algorithms, and the results can be generalized to other swarm-based search algorithms. In this paper, four different types of commonly used heavy-tailed distributions, including Mittag-Leffler distribution, Pareto distribution, Cauchy distribution, and Weibull distribution, are considered to enhance the searching ability of CS. Then four novel CS algorithms are proposed and experiments are carried out on 20 benchmark functions to compare their searching performances. Finally, the proposed methods are used to system identification to demonstrate the effectiveness.