Computational Intelligence and Neuroscience

Volume 2015 (2015), Article ID 326431, 12 pages

http://dx.doi.org/10.1155/2015/326431

## An Improved Quantum-Behaved Particle Swarm Optimization Algorithm with Elitist Breeding for Unconstrained Optimization

^{1}School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China^{2}School of Information Engineering, Guangzhou Panyu Polytechnic, Guangzhou 511483, China^{3}Department of Electronic Engineering, City University of Hong Kong, Tat Chee Ave, Hong Kong

Received 12 December 2014; Accepted 15 April 2015

Academic Editor: Thomas DeMarse

Copyright © 2015 Zhen-Lun Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An improved quantum-behaved particle swarm optimization with elitist breeding (EB-QPSO) for unconstrained optimization is presented and empirically studied in this paper. In EB-QPSO, the novel elitist breeding strategy acts on the elitists of the swarm to escape from the likely local optima and guide the swarm to perform more efficient search. During the iterative optimization process of EB-QPSO, when criteria met, the personal best of each particle and the global best of the swarm are used to generate new diverse individuals through the transposon operators. The new generated individuals with better fitness are selected to be the new personal best particles and global best particle to guide the swarm for further solution exploration. A comprehensive simulation study is conducted on a set of twelve benchmark functions. Compared with five state-of-the-art quantum-behaved particle swarm optimization algorithms, the proposed EB-QPSO performs more competitively in all of the benchmark functions in terms of better global search capability and faster convergence rate.

#### 1. Introduction

Particle swarm optimization (PSO), inspired by the social behavior of bird flocks [1], is an important and widely used population-based stochastic algorithm. Unlike evolutionary algorithms, PSO is computationally inexpensive and its implementation is straightforward. Each potential solution in PSO, represented by a particle, flies in a multidimensional search space with a velocity dynamically adjusted by the particle’s own former information and the experience of the other particles. For its superiority, PSO has rapidly developed with applications in solving real-world optimization problems in recent years [2–5].

However, as demonstrated by van den Bergh [6], PSO is not a guaranteed global convergence algorithm according to the convergence criteria in [7]. Based on quantum mechanics and trajectory analysis of PSO [8], Sun et al. [9] proposed a variant of PSO, quantum-behaved PSO (QPSO) algorithm, which is theoretically proved to be global convergent using Markov process [10, 11]. The global convergence of QPSO guarantees to find the global optimal solution upon unlimited number of search iterations. Nevertheless, such condition is unrealistic when it comes to the real-world problems as only a finite number of iterations are allowed for the search of optimal solution on using any optimization algorithm. Thus, QPSO is also likely to be trapped in local optima or with slow convergence speed when it is used to solve complex problems. So far, many researchers developed various strategies to improve performance of QPSO in terms of convergence speed and global optimality [12–21]. However, it is rather difficult to improve the global search capability and accelerate the rate of convergence simultaneously. If any attempt focuses on avoiding being stuck at local optima, it is likely to have a slower convergence rate.

In PSO, the personal best (*pbest*) of each particle and global best (*gbest*) of the swarm found so far in the search process can be considered as the elitists of the whole swarm at any search iteration. In most of the current QPSOs, the information of elitists is either used directly or with some simple extra processing to guide the flying behavior of each particle in the search space; to the best of our knowledge, deep exploration with the elitist is not taken into account to assist the search of solutions in any work of QPSO reported in literatures. The exploration on the elitists will produce some extra information that may be beneficial for the search of the optimal solution.

In this study, a novel variant of the QPSO algorithm, called the quantum-behaved particle swarm optimization with elitist breeding (EB-QPSO), is proposed for the desirable aims to achieve better global search capability and convergence rate by employing a breeding scheme through transposon operators on elitists of the swarm. Elitist breeding is a kind of advanced elitist exploration method treating the elitist members as parents to create new diverse individuals with transposon operators. On one hand, elitist breeding helps to diversify the particle swarm during the search and thus enhance the global search capability of the algorithm. On the other hand, the new bred individuals with better fitness are selected as the new members of elitists and used to guide the swarm to perform exploration and exploitation more efficiently. Experiment results on twelve benchmark functions show that EB-QPSO outperforms the original QPSO and other four state-of-the-art QPSO variants.

The rest of this paper is organized as follows. A brief introduction of QPSO is presented in Section 2. In Section 3, an overview of related work is given. The proposed EB-QPSO algorithm is elaborated and compared with various existing QPSO algorithms over twelve benchmark functions in Sections 4 and 5, respectively. Finally, the general conclusions of the paper are given in Section 6.

#### 2. Quantum-Behaved Particle Swarm Optimization

In the original PSO, each particle is defined by a position vector which signifies a solution in the search space and associated with a velocity vector responsible for the exploration of the search space. Let denote the swarm size and the dimensionality of the search space, during the evolutionary process, the velocity and the position of each particle are updated with the following rules:where and , and are the th dimension component of velocity and position of particle in search iteration , respectively, and are the th dimension of the personal best of particle and the global best of the swarm in search iteration , respectively, is the inertia weight, and are two positive constant acceleration coefficients, and and are two random numbers uniformly distributed in the interval (0, 1).

According to the trajectory analysis given by Clerc and Kennedy [8], the convergence of the PSO algorithm may be achieved if each particle converges to its local attractor , of which the coordinates are defined as where .

The concept of the QPSO was developed based on the analysis above. Each single particle in QPSO is treated as a spin-less one moving in quantum space and the probability of the particle’s appearing at position in the search iteration is determined from a probability density function [22]. Employing the Monte Carlo method, each particle flies with the following rules:where is a parameter called contraction-expansion coefficient; both and are random numbers uniformly distributed on ; is a global virtual point called mainstream or mean best defined as

A time-varying decreasing method [23] usually is adapted to control the contraction-expansion coefficient defined as follows:where and are the initial and final values of , respectively; is the maximum number of iterations; is the current search iteration number.

The QPSO algorithm has simpler evolutional equation forms and fewer parameters than classical PSO, substantially facilitating the control and convergence in the search space. Without loss of generality, let be the objective function to be minimized; the procedure for implementing the QPSO is given in Algorithm 1.