IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 41, NO. 4, AUGUST 2011 1003 A Hybrid PSO-BFGS Strategy for Global Optimization of Multimodal Functions Shutao Li, Member, IEEE, Mingkui Tan, Ivor W. Tsang, and James Tin-Yau Kwok Abstract—Particle swarm optimizer (PSO) is a powerful opti- mization algorithm that has been applied to a variety of problems. It can, however, suffer from premature convergence and slow convergence rate. Motivated by these two problems, a hybrid global optimization strategy combining PSOs with a modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is presented in this paper. The modified BFGS method is integrated into the context of the PSOs to improve the particles’ local search ability. In addition, in conjunction with the territory technique, a reposition technique to maintain the diversity of particles is proposed to improve the global search ability of PSOs. One advantage of the hybrid strategy is that it can effectively find multiple local solutions or global solutions to the multimodal functions in a box- constrained space. Based on these local solutions, a reconstruction technique can be adopted to further estimate better solutions. The proposed method is compared with several recently devel- oped optimization algorithms on a set of 20 standard benchmark problems. Experimental results demonstrate that the proposed ap- proach can obtain high-quality solutions on multimodal function optimization problems. Index Terms—Local diversity, particle swarm optimizer (PSO), reconstruction technique, territory. I. I NTRODUCTION P ARTICLE swarm optimizer (PSO), which was proposed by Kennedy and Eberhart in 1995 [1], is a population- based stochastic optimization technique inspired by the social behavior of bird flocking or fish schooling for finding an opti- mal solution in complex search spaces. Due to its effectiveness and simple implementation in solving multidimensional prob- lems, PSO and its variants have been applied in many areas. However, one drawback of the canonical PSO is that it suffers from premature convergence and slow convergence rate [2], [3]. To address this problem, many improvements of the PSO Manuscript received January 9, 2010; revised August 9, 2010 and November 21, 2010; accepted December 15, 2010. Date of publication January 28, 2011; date of current version July 20, 2011. This work was supported in part by the National Natural Science Foundation of China under Grant 60871096 and Grant 60835004 and in part by the Ministry of Education of China through the Ph.D. Programs Foundation under Grant 200805320006. This paper was recommended by Associate Editor Q. Zhang. S. Li is with the College of Electrical and Information Engineering, Hunan University, Changsha 410082, China (e-mail: shutao_li@yahoo.com.cn). M. Tan was with the College of Electrical and Information Engineering, Hunan University, Changsha 410082, China. He is now with the School of Computer Engineering, Nanyang Technological University, Singapore 639798 (e-mail: tanmingkui@gmail.com). I. W. Tsang is with the School of Computer Engineering, Nanyang Techno- logical University, Singapore 639798. J. T.-Y. Kwok is with the Department of Computer Science and Engineer- ing, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2010.2103055 algorithms have been proposed. Traditional improved variants can be generally categorized into three groups [3]. The first category adjusts parameters to trade off the global and local search abilities of PSO [4], [5]. The second category designs efficient population utilization strategy or dynamic multiple swarms to improve the global search ability [6]–[8]. In the third category, a hybrid mechanism combining PSO with other evolutionary algorithms is explored to keep the population diversity and improve the local convergence rate [9]–[11]. Another drawback of the canonical PSO and the traditional variants is that it is difficult for them to find multiple optima due to an intrinsic restriction that all particles must converge to only one point at the final step [12]. To address this problem, a multigrouped particle swarm optimization technique was proposed in [12]. It allows particles to converge to multiple points rather than to only one point, and thus, it can find multiple local optima. However, it has the limitation that each local optimum needs to be supported by an independent swarm [12]. Parsopoulos and Vrahatis [13] introduced a repulsion technique as well as deflection and stretching techniques into PSO to compute all the global optima. This is an efficient algorithm that has the ability to detect all global minimizers of a function, under the assumption that the global optimum was known a priori. However, this assumption does not hold for most problems in real problems. Recently, improving the performance of evolutionary algo- rithms by introducing the local search method into the evo- lutionary algorithms has attracted much attention [14]–[16]. Based on the estimation of distribution algorithm, Zhang et al. [17] introduced a hybrid evolutionary algorithm for continuous global optimization problems where the simplex method was introduced to implement the local search. To improve the local search ability of genetic algorithm (GA), a large collection of methods, named as memetic algorithm (MA), has been thoroughly studied in recent years [18]–[20]. In particular, in [19], a dynamical approach is proposed to start the local search and determine the local search intensity. However, this strategy may lead to too many local searches. As for PSO, Liang and Suganthan developed a hybrid strategy combining a dynamic multiswarm (DMS) PSO with a local search technique to main- tain the particles’ diversity as well as local search ability [2]. In addition, Fan and Zahara [21] also proposed to integrate the simplex search method into the PSO iterations for uncon- strained optimizations. There are also some other combination strategies [22], [23]. For example, Coelho and Mariani [23] recently developed a novel chaotic PSO combined with an im- plicit filtering local search method to solve economic dispatch problems. 1083-4419/$26.00 © 2011 IEEE