Date Published: June 20, 2019
Publisher: Public Library of Science
Author(s): Lai Xu, Aamir Muhammad, Yifei Pu, Jiliu Zhou, Yi Zhang, Nicholas Chancellor.
Motivated by the concepts of quantum mechanics and particle swarm optimization (PSO), quantum-behaved particle swarm optimization (QPSO) was developed to achieve better global search ability. This paper proposes a new method to improve the global search ability of QPSO with fractional calculus (FC). Based on one of the most frequently used fractional differential definitions, the Grünwald-Letnikov definition, we introduce its discrete expression into the position updating of QPSO. Extensive experiments on well-known benchmark functions were performed to evaluate the performance of the proposed fractional-order quantum particle swarm optimization (FQPSO). The experimental results demonstrate its superior ability in achieving optimal solutions for several different optimizations.
Particle swarm optimization (PSO) , which is inspired by animal social behaviors, such as birds, was first proposed by Kennedy and Eberhart as a population-based optimization technique. In PSO, the potential solutions, which are called particles, go through the solution space by relying on their own experiences and current best particle. PSO has a competitive performance with the classical Genetic Algorithm (GA) , evolutionary programming (EP) , evolution strategies (ES) , genetic programming (GP) and other classic algorithms. It has attracted increasing attention during recent years thanks to its effectiveness in different optimization problems .
Grünwald-Letnikov (GL) , Riemann-Liouville (RL) , and Caputo  definitions are three different definitions for fractional calculus in Euclidean space. Due to its convenient computational form, GL definition for the fractional derivative is commonly used for engineering problems.
Inspired by the properties of fractional calculus, we presented a novel QPSO algorithm incorporated with fractional calculus strategy, which is based on the properties of long time memory and non-locality of fractional calculus. The goal is to employ the proposed method to accelerate not only the convergence speed but also avoid the local optimums. Since the property of fractional calculus enables quantum-particles in FQPSO to appear anywhere during iterations, it significantly improves the global searching ability. Furthermore, FQPSO also increases the convergence rate for the quantum particles. As a result, the proposed FQPSO method achieves more favorable results than all the other algorithms.