Adaptive fuzzy particle swarm optimization for global optimization of multimodal functions
Introduction
The particle swarm optimization (PSO) algorithm was first introduced by Kennedy and Eberhart [5] and described in detail in [6]. It is an evolutionary computational technique motivated by the social cooperative and competitive behavior of bird flocking or fish schooling. In the PSO algorithm, a swarm is defined as a population of interacting elements and a particle is a member in the swarm, representing a potential solution to the optimization process. Each member of the swarm adjusts search patterns according to its experience and that of the other members.
Recently, several variants of PSO have been developed. To balance local exploitation and global exploration, Shi and Eberhart [20] introduced a parameter referred to as inertia weight, and this modification is often referred to as the standard PSO (SPSO) algorithm. Parsopoulos and Vrahatis [18] integrated exploration and exploitation to form a unified particle swarm optimization (UPSO). A number of variants of the PSO algorithm involve hybridization with other search techniques to improve performance. A PSO incorporating evolutionary operators such as mutation, selection, and crossover was presented in [1]. The evolutionary self-adapting PSO [11], [12], [13] is a hybrid method combining evolution strategies with the PSO technique. The quadratic interpolation PSO (QIPSO) algorithm [16] introduced a quadratic crossover operator to improve numerical results. Liang and Suganthan [8] presented the dynamic multi-swarm optimizer (DMSPSO) algorithm by dividing the entire population into many small swarms, and then regrouping them. In [2], Du and Li divided all of the particles into two parts and proposed a multi-strategy ensemble PSO algorithm for dynamic optimization. Mendes et al. [10] introduced a fully informed particle swarm (FIPS) algorithm, using the information from neighboring particles to update velocity. The comprehensive learning particle swarm optimizer (CLPSO) algorithm was proposed in [7] in which the historical best information of all other particles is compiled to update the velocity of particles. The CLPSO significantly improves the performance of many recent variants of the PSO algorithms with regard to multimodal problems. To deal with discrete events, an algorithm based on discrete-particle-swarm-optimization was developed in [24]. This approach solves the overlapping coalition formation problem in multiple virtual organizations.
The standard PSO algorithm is easy to implement and effective in solving optimization problems. However, due to its tendency to terminate prematurely when solving complex multimodal problems, it often gets trapped in a local optimum [3], [7], [8]. To improve the performance of the PSO in complex multimodal problems, this paper presents a PSO method demonstrating a significant performance improvement over the SPSO, QIPSO, UPSO, FIPS, DMSPSO, and CLPSO algorithms. Because the proposed method utilizes fuzzy set theory for the adaptation of parameters, it is referred to as the adaptive fuzzy PSO (AFPSO).
The proposed AFPSO has two advantages. One is the ability to adjust the acceleration coefficients of the standard PSO adaptively, thereby improving the accuracy and efficiency of searches. The other advantage is its flexibility in integrating with other PSO techniques to further enhance search performance. We will illustrate this by including the quadratic crossover operator [16] in the proposed AFPSO algorithm, naming the AFPSO-QI algorithm. The results of the experiments show that the proposed algorithms outperform the SPSO, QIPSO, UPSO, FIPS, the DMSPSO, and CLPSO algorithms in solving most of the multimodal function problems considered in this study.
In Section 2, we introduce the standard PSO algorithm. In Section 3, we present the proposed AFPSO algorithm and its variant, AFPSO-QI. In Section 4, we present the experimental results. Conclusions are made in Section 5.
Section snippets
Standard PSO algorithm
In the standard PSO (SPSO) algorithm, each particle searches for an optimal solution to the objective function in the search space. Each particle dynamically updates its position based on its previous position and new information regarding velocity. Its best location found in the search space so far is called pbest and the best location found for all the particles in the population is called gbest. The procedure of the SPSO is shown as follows [15], [20].
- (1)
Initialize position and velocity vectors.
Proposed AFPSO algorithm
In this section, we propose an adaptive fuzzy PSO algorithm (AFPSO). The acceleration coefficients c1 and c2 in Eq. (1) are adaptively adjusted so as to improve the searching accuracy and efficiency.
Experimental results
In this section, the proposed AFPSO and AFPSO-QI algorithms are compared with the SPSO, QIPSO, UPSO, FIPS, DMSPSO, and CLPSO algorithms. The main aim of this paper was to improve the performance of the PSO when dealing with multimodal problems; therefore, we tested the proposed algorithms with various multimodal functions. The sixteen multimodal benchmark functions with respective dimensions 10 and 30 are for comparison. Functions 1–4 and their rotated functions 5–8 were selected from [4], [7],
Conclusions
We have been presented an adaptive fuzzy PSO algorithm in this paper. In conventional PSO algorithms, when updating the search velocity of each particle in the swarm, the acceleration coefficients are usually kept at particular fixed empirical values. To extend the search capability and improve convergent efficiency, this paper suggests three simple fuzzy inference rules to adaptively tune these acceleration coefficients. A variant AFPSO-QI algorithm was also proposed incorporating the
Acknowledgements
This work was partially supported by the National Science Council of the Republic of China under the contract NSC 98-2221-E-008-092. The authors would like to thank the anonymous Reviewers for providing valuable comments and Prof. Suganthan for providing the CEC-05 benchmarks code on the website.
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