Elsevier

Neurocomputing

Volume 124, 26 January 2014, Pages 218-227
Neurocomputing

Enhanced particle swarm optimizer incorporating a weighted particle

https://doi.org/10.1016/j.neucom.2013.07.005Get rights and content

Abstract

This study proposes an enhanced particle swarm optimizer incorporating a weighted particle (EPSOWP) to improve the evolutionary performance for a set of benchmark functions. In conventional particle swarm optimizer (PSO), there are two principal forces to guide the moving direction of each particle. However, if the current particle lies too close to either the personal best particle or the global best particle, the velocity is mainly updated by only one term. As a result, search step becomes smaller and the optimization of the swarm is likely to be trapped into a local optimum. To address this problem, we define a weighted particle for incorporation into the particle swarm optimization. Because the weighted particle has a better opportunity getting closer to the optimal solution than the global best particle during the evolution, the EPSOWP is capable of guiding the swarm to a better direction to search the optimal solution. Simulation results show the effectiveness of the EPSOWP, which outperforms various evolutionary algorithms on a selected set of benchmark functions. Furthermore, the proposed EPSOWP is applied to controller design and parameter identification for an inverted pendulum system as well as parameter learning of neural network for function approximation to show its viability to solve practical design problems.

Introduction

Particle swarm optimization (PSO) was introduced by Kennedy and Eberhart [1] in 1995. Based on simulation of simplified animal social behaviors such as fish schooling, bird flocking, [2], [3], PSO has the advantages of simple implementation and quick convergence to a reasonably good solution [4] for finding the optimum of an objective function [5]. Because of its robustness, PSO has been widely adopted in various engineering applications [6], [7], [8]. There are, however, problems on conventional PSOs as far as a desired optimization method is concerned. For example, the convergence of optimization becomes staggered, especially in later stage during the evolution process [9]. Other issues include proper control of global exploration and local exploitation [2], [10] as well as sensitivity of the control parameters [11]. To address the aforementioned issues, many PSO variants were presented in recent years [1], [12], [13], [14], [15], [16], [17], [18], which solved the problems to some extent, including the selection of control parameters for optimality and convergence [19], [20], [21]. In particular, the convergence analysis of PSO algorithms was investigated by Clerc and Kennedy [19] and Trelea [20]. Ruben and Kamran [22] and Van den Bergh [23] provided criteria for determining the parameter bounds [24] to ensure the convergence of PSOs. Based on the convergence analysis, suitable acceleration parameters can be chosen for the PSO.

In order to improve the optimization performances, different variants of PSO have been developed, emphasizing particularly the velocity-updating rules and adjustment of evolution parameters. For example, a hybrid optimization approach was proposed by incorporating PSO with an extra particle which is the center position in a swarm [25]. An improved particle swarm optimization algorithm utilized particles from a sub-population [26] to provide alterative searching direction. In [27], an improved PSO algorithm adopting multi-particle information sharing strategy and a mutation operator was proposed so that particles can jump into other search areas of the solution space to improve the global search ability. The study of PSO in [28] suggested the use of an adaptive population size, which is periodically increased or decreased according to a ladder function. Also, an enhanced PSO, called cooperative random learning particle swarm optimization (CRPSO) [32] which adopted several sub-swarms and modified velocity updating equation, was proposed to search the solution space. As far as a global optimizer is concerned, there is still room for further improvements over the existing PSO approaches in terms of robustness and accuracy of the optimization algorithms. In this paper, an enhanced particle swarm optimizer incorporating a weighted particle (EPSOWP) is proposed to improve the evolutionary performance for a set of benchmark functions. Because the weighted particle has a better opportunity getting closer to the optimal solution than the global best particle during the evolution, the EPSOWP is capable of guiding the swarm to a better direction to search the optimal solution.

The paper is organized as follows. Section 2 introduces the background of PSO and discusses the convergence of PSO. In Section 3, the weighted particle is defined based on which the EPSOWP is proposed. The simulation results are presented in Section 4, and conclusions are drawn in Section 5.

Section snippets

Preliminaries of the particle swarm optimization

The particle swarm optimization (PSO) is a stochastic global optimization technique which has been shown to successfully optimize a wide range of continuous functions [1], [2]. The algorithm is based on a metaphor of social interaction like the movement of a flock of birds and a school of fish to look for foods [3]. The PSO searches potential optimal positions by adjusting the trajectories of individual vectors which are conceptualized as moving points, called particles, in multidimensional

Weighted particle

In a conventional PSO, a set of particles are initialized to form a population, each associated with a movement toward xiP(t) and xG(t) for searching the optimal solution. For convenience, we omit argument (t) in the subsequent presentations. With reference to (1), ϕ1i(xiPxi) and ϕ2i(xiGxi) are the two principal terms to guide the movement direction of each particle. If a particle xi lies close to xiP(t) or xG(t), one of the two terms vanishes and only the other one guides the xi to search

The effect of the weighted particle

To show that the weighted particle does provide a better search direction for PSO as evolution continues, six benchmark functions are evaluated to show the effect of the weighted particle.

  • 1.

    The Sphere function:F1(x)=k=1Dxk2.

  • 2.

    The Schwefel function:F2(x)=k=1D|xk|+k=1D|xk|.

  • 3.

    The Rosenbrock function:F3(x)=k=1D1100×(xk+1xk2)2+(1xk)2.

  • 4.

    The Ackley function:F4(x)=20e0.2(1/D)k=1Dxk2e(1/D)k=1Dcos(2πxk)+20+e.

  • 5.

    The Grewank function:F5(x)=k=1Dxk24000k=1Dcos(xkk).

  • 6.

    The Rastrigin function:F6(x)=k=1D(xk2

Conclusions

In this paper, an enhanced evolutionary algorithm, EPSOWP, based on particle swarm optimization incorporating a weighted particle is proposed. The convergence of the EPSOWP has been guaranteed by stability conditions for inertia weight w and acceleration constants, c1, c2, c3 and c4. Thanks to the weighted particle, the proposed evolutionary framework based on an attraction value has demonstrated itself as an effective approach to provide a more promising search direction for all particles

Acknowledgments

This work was supported by the National Science Council, Taiwan, under Grant NSC 99–2221-E-008–093-MY3, and the “Aim for the Top University Plan” from National Taiwan Normal University and the Ministry of Education, Taiwan.

Nai-Jen Li was born in Taipei, Taiwan, in 1984. He received the B.S. and M.S. degree in electrical engineering from Tamkang University, Tamsui, Taiwan, in 2007 and 2009, respectively. He is currently working toward his Ph.D. degree in electrical engineering at National Central University, Jhungli, Taiwan. His currently research interests are in the areas of evolutionary algorithms, neural networks, image processing and fuzzy control.

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    Nai-Jen Li was born in Taipei, Taiwan, in 1984. He received the B.S. and M.S. degree in electrical engineering from Tamkang University, Tamsui, Taiwan, in 2007 and 2009, respectively. He is currently working toward his Ph.D. degree in electrical engineering at National Central University, Jhungli, Taiwan. His currently research interests are in the areas of evolutionary algorithms, neural networks, image processing and fuzzy control.

    Wen-June Wang received the B.S. degree in the Department of Control Engineering from National Chiao-Tung University, Taiwan, in 1980; and the M.S. degree in the Department of Electrical Engineering from Tatung University, Taiwan, in 1984. Moreover, he received the Ph.D. degree in the Institute of Electronics from National Chiao-Tung University of Taiwan in 1987. Dr. Wang is presently a chair professor in the Department of Electrical Engineering, National Central University, Taiwan. He was a visiting scholar for one year in Department of Mechanical Engineering, Georgia Institute of Technology, USA, in 1994. Furthermore, he was the Dean of the College of Science and Technology, National Chi Nan University, Puli, Taiwan from 2005 to 2007 and served as the Dean of Research and Development Office, National Taipei University of Technology, Taiwan from 2007 to 2009. Until today, Dr. Wang has published more than 145 journal papers and 136 conference papers. He also received three times of Distinguished Research Award from the National Science Council of Taiwan. From 2003 to 2006, he served as the Convener of the Control Engineering Group of National Science Council in Taiwan. Dr. Wang was elected as an IEEE Fellow of 2008. He serves as a member of editorial board of numerous journals including IEEE Transactions on Systems, Man, and Cybernetics Part-B, IEEE Transactions on Fuzzy Systems, and the International Journal of Electrical Engineers, etc. He is now the Editor-in-Chief of the International Journal of Fuzzy Systems. Furthermore, Dr. Wang served as the Chairman of Taipei Chapter, IEEE Control Systems Society, from 1999 to 2001, and serves as the Chairman of Taipei Chapter, IEEE Systems, Man, and Cybernetics from 2006 to 2008. His research interests include the areas of fuzzy systems, robust control, neural networks, robotics, and pattern recognition.

    Chen-Chien James Hsu (M’06) was born in Hsinchu, Taiwan. He received the B.S. degree in electronic engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan, in 1987, the M.S. degree in control engineering from the National Chiao-Tung University, Hsinchu, in 1989, and the Ph.D. degree from the School of Microelectronic Engineering, Griffith University, Brisbane, Australia, in 1997. Before commencing his Ph.D. study, he was a Systems Engineer with IBM Corporation for three years, where he was responsible for information systems planning and application development. He joined the Department of Electronic Engineering, St. Johns University, Taipei, as an Assistant Professor in 1997 and was appointed as an Associate Professor in August 2004. From August 2006 to July 2009, he was with the Department of Electrical Engineering, Tamkang University, Taipei. Dr. Hsu is currently a Professor with the Department of Applied Electronics Technology, National Taiwan Normal University, Taipei, Taiwan. He is the author or a coauthor of more than 100 refereed journal and conference papers. His research interests include digital control systems, neural-fuzzy control systems, evolutionary computation, vision-based measuring systems, sensor applications, and mobile robot navigation.

    Wei Chang received his B.S. and M.S. degrees from the Department of Marine Engineering of the National Taiwan Ocean University, Taiwan, in 2003 and 2005. He is currently working toward his Ph.D. degree in the Department of Electrical Engineering at the National Central University, Jhongli, Taiwan. His research interests focus on fuzzy control and dynamic system analysis.

    Hao-Gong Chou was born in Taipei, Taiwan, in 1984. He received the B.S. and M.S. degree from Chien Hsin University of Science and Technology, Jhongli, Taiwan, in 2007 and 2010, respectively. He is currently working toward the Ph.D. degree with the Department of Electrical Engineering, National Central University, Jhongli, Taiwan. His current research interests include fuzzy control, and field-programmable gate array chip design.

    Jun-Wei Chang was born in Taipei, Taiwan, in 1985. He received the B.S. and M.S. degree from National Formosa University, Yunlin, Taiwan, in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree with the Department of Electrical Engineering, National Central University, Jhongli, Taiwan. His current research interests include fuzzy control, image processing, and intelligent robots.

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