An Opposition-Based Group Search Optimizer with Diversity Guidance

Wang, Dan; Xiong, Congcong; Zhang, Xiankun

doi:https://doi.org/10.1155/2015/546181

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusion Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 546181 | https://doi.org/10.1155/2015/546181

An Opposition-Based Group Search Optimizer with Diversity Guidance

Dan Wang,¹Congcong Xiong,¹and Xiankun Zhang¹

Academic Editor: Miguel A. Salido

Received13 Jul 2015

Revised01 Nov 2015

Accepted08 Nov 2015

Published28 Dec 2015

Abstract

Group search optimizer (GSO) which is an effective evolutionary algorithm has been successfully applied in many applications. However, the purely stochastic resampling or selection mechanism in GSO leads to long computing time and premature convergence. In this paper, we propose a diversity-guided group search optimizer (DGSO) with opposition-based learning (OBL) to overcome these limitations. Opposition-based learning is utilized to accelerate the convergence rate of GSO, while the diversity guidance (DG) is used to increase the diversity of population. When compared with the standard GSO, a novel operator using OBL and DG is developed for the population initialization as well as the generation jumping. A comprehensive set of 19 complex benchmark functions is used for experiment verification and is compared to the original GSO algorithm. Numerical experiments show that the proposed DGSO leads to better performance in comparison with the standard GSO in the convergence rate and the solution accuracy.

1. Introduction

During the past few decades, evolutionary algorithms (EAs) have been developed rapidly and even improved the social development increasingly [1–3]. Diversity of EAs such as Genetic Algorithm (GA) [4], Differential Evolutionary (DE) algorithm [5–7], Ant Colony Optimization (ACO) [8, 9], and Particle Swarm Optimizer (PSO) [10, 11] have been proposed, and they also have been successfully applied in lots of applications in practice. It is noted that most of EAs are developed based on the swarm intelligence coming from the nature world.

Along with this viewpoint, a novel swarm intelligence algorithm, namely, group search optimizer (GSO), inspired by animal behavior [12], has been proven to get more competitive performance in comparison with some typical swarm intelligence algorithms such as PSO when dealing with some complex optimization problems [13]. The GSO algorithm is composed by one initialization operator and three evolutionary operators. Like the other EAs, the initialization operator of GSO is realized based on purely random searching. In spite of the guidance of animal behavior mechanism, the three operators are also based on random searching to some extent. It is evident that sometimes such random searching mechanism does not come with the optimal solution in the given time. In other words, the defect of the random mechanism leads to long computing time and premature convergence [14–16]. On the one hand, the GSO algorithm is a population-based optimization algorithm, which is prone to suffer from long computing times. On the other hand, random initialization and selection arrives at the best optimization solution only in good luck without any prior information. These two limitations prevent the development of the GSO and restrict the application of GSO in practice.

In this paper, we propose a new evolutionary algorithm, namely, diversity-guided group search optimizer (DGSO), to alleviate the two drawbacks of GSO. The proposed DGSO is realized based on diversity guidance (DG) and a so-called opposition-based learning (OBL) originally coming from neural networks field [17–20]. Diversity guidance is used to increase the diversity of the population of GSO, while opposition-based learning is utilized to accelerate the convergence speed. Numerical experiments are carried out on several benchmark functions to evaluate the performance of the proposed DGSO.

The structure of the paper is arranged as follows. Section 2 recalls the GSO algorithm. Section 3 presents diversity-guided group search optimizer (DGSO) with opposition-based learning. Section 4 reports the experimental results. At last, some conclusions are drawn in Section 5.

2. Group Search Optimizer Algorithm

Let us recall the original GSO. In the GSO, there are mainly three evolutionary operators: producing, scrounging, and dispersion. The detailed procedure of the GSO is described in Algorithm 1, where is the dimension; is in the th member at the th iteration; is a head angle; is a unit vector [13].

Line# Procedure of GSO algorithm
(1) Begin
(2) Initialization
(3) Evaluate the fitness of the current member;
(4) Find a producer;
(5) Producing
(6) Randomly choose three points based on (1);
(7) Search the best resource, otherwise stay the current position based on (2);
(8) If the producer can’t search a better area, then change the angle by using (3);
(9) Scrounging
(10) Randomly choose some group members as the scrounger
(11) Dispersion
(12) Generate a head angle using (2)
(13) Obtain a random distance based on (4) and then move to a new point using (5)
(14) Search a better solution, report the final optimal solution
(15) End

In the producing operator, the producer scans three points through the following expression [19]:where is the maximum search angle; is the maximum pursuit distance; and are parameters generated under the normally distribution; and is also a real number in the interval .

If the current point changed, the angle iswhere stands for the maximum turning angle [12].

Otherwise the angle is fixed, and the angle iswhere is a constant.

In the scrounging operator, we select 80% candidates from the remainder of population.

In the dispersion operator, we used the random walks [22–25]. And the random distance is

The new point is expressed:

3. Diversity-Guided Group Search Optimizer (DGSO) with Opposition-Based Learning

As one of the evolutionary algorithms, GSO is the population-based algorithm, which has the common default; that is, the individuals are selected from a given population randomly generated. To resolve this problem, we used a priori information such as the opposition-based learning, which is applied for making the opposition individuals, and diversity-guided strategy, which is used for improving the diversity of the original GSO. More details of the DGSO are summarized as shown in Algorithm 2.

/ Diversity-Guided GSO (DGSO) with Opposition-Based Learning (OBL) /
(1) BEGIN
(2) Initialization (Generate uniformly distributed random population);
(3) Opposition-Based Operator (for Group Initialization);
(4) While the termination condition isn’t satisfied
(5) Diversity-Guided Operator;
(6) GSO algorithms;
(7) Opposition-Based Operator (for Generation Jumping)
(8) End While
(9) END

3.1. Opposition-Based Search Operator

To speed up the convergence of the DGSO, we use a method of opposition-based learning (OBL) [17, 18, 26]. The essence of OBL is using opposite numbers, which would accelerate the convergence of the DGSO and prevent DGSO from suffering expensive computational times which always happens as one weakness in evolutionary algorithm. Let us memorize the basic concepts.

Opposition-based point is as follows [24]: assume is a point, and the dimension is in , where is a real number, . The opposite point of is , where .

Opposition-based optimization (OBL) is as follows [26]: we assume is the fitness function which is to measure the candidate’s fitness value for finding the best candidate solution. According to the above definition, if , then the point replaces ; otherwise the point is fixed and does not change. Hence, the point and its opposite point are evaluated simultaneously to continue the fitter point.

Based on the upper concepts, we embedded the opposition-based operators in the DGSO’s initialized group and generation jumping. Algorithm 3 described the procedure of opposition-based learning operator, where is the population, is the opposite point according to , NP is the population size, and , .

Line# Procedure of opposition-based learning operator
(1) Begin
(2) Generate the population
(3) for (int ; < NP; ++)
(4) for (int ; ; ++)
(5) ;
(6) Calculate the fitness value of the
(7) Output the some optimal candidates from ;
(8) End

3.2. Diversity-Guided Operator

To alleviate the problem of premature convergence, we use a mechanism of diversity guidance as shown in Algorithm 4. Here the diversity of group (population) is computed by the following formula [27]:where is the group size, is the length of the longest diagonal in the search space [26], is the dimension, is the th value in the th member (individual), represents the average point, and stands for the th value of . Algorithm 4 described the procedure of diversity-based operator, where is a given number used in formula (1); , are set as 0.33 and 0.66, individually.

#Line Diversity-guided operator procedure
(1) Begin
(2) if (diversity < dLow)
(3) ;
(4) else if (diversity ≥ dHigh)
(5) ;
(6) else if ((diversity ≥ dLow) && (diversity < dHigh))
(7) ;
(8) End IF
(9) End

4. Experimental Studies

In this section, we demonstrate the detailed experimental studies of the proposed DGSO. First, we describe test functions and many experimental settings. Second, we compared the DGSO to other algorithms using four comparative analyses, mean and standard deviation, convergence speed, contribution of diversity-guided operator, and contribution of opposite points.

4.1. Experimental Settings

(1) Test Functions. We select 19 classical benchmark functions described in [28], where the functions are selected from CEC 2008 Special Session [29], the functions were provided from ISDA 2009 Workshop [30], and the functions are the hybrid functions based on the previous functions [28]. Test functions are shown in Table 1; test functions are shown in Table 2. In the experimental process, we set the population size to 100. For each algorithm, we run 25 times. In the problem size , we set . According to [31], MAX_FES is set to . All of the algorithms are terminated when the MAX_FES is reached.

(2) Experimental Settings. We used five different algorithms for comparing: GSO [13], RGSO, OGSO, RDGSO, and DGSO. All of them are based on the GSO algorithm. The only difference is whether they are applied in the manner of initialization and evolutionary operators. More details are described in Table 3. Through all experiments, we set the head angle to be ; the maximum search angle is , where the value of is a given number and is the maximum turning angle (see [13]).

4.2. Experimental Results

In this section, we report and analyze the results of the proposed algorithm with some popular algorithms reported in the literatures.

(1) Mean and Standard Deviation. Two performance indexes, mean and standard deviation, are utilized to measure the performance of different algorithms whose strategies are described in Table 3. Table 4 provides the average values when is set as 30. As shown in Table 4, the proposed algorithm DGSO performs better than the other algorithms (GSO, OGSO, and RDGSO) in most benchmark functions.

Table 5 summarizes the comparison of DGSO, GSO, and RDGSO with the different dimensions (, 100, 200), respectively. It is evident that DGSO performs better than the other reported algorithms especially in case of the high dimension functions.

(2) Convergence Speed. Next we compare the convergence speed between the DGSO and the other four algorithms including GSO, OGSO, RGSO, and RDGSO. Figure 1 depicts the values of fitness with the increasing of FES [32–34] for . It shows that the convergence speed of DGSO is faster than the other four algorithms.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(3) Contribution of Diversity-Guided Operator. Now let us analyze the contribution of diversity-guided operator. For convenience, two algorithms DGSO that used the diversity-guided operator and OGSO that is the opposition-based algorithm without diversity-guided operator are compared. The experiment results are summarized in Table 6. It is clear that the group (population) of DGSO obtains more diversity in comparison with OGSO in most of benchmark functions.

(4) Comparison Results of Other Optimization Algorithms. The proposed DGSO is also compared with some other well-known evolutionary algorithms [21] based on these benchmark functions.

Table 7 describes the comparison results of DGSO and some other algorithms when . As shown in Table 7, the proposed DGSO obtains promising results in most of benchmark functions when compared with the other algorithms.

5. Conclusion

In this paper, we proposed a diversity-guided group search optimizer (DGSO) that is realized based on an opposition-based learning method and a diversity-guided operator mechanism. Experiments are carried out on 19 benchmark functions to demonstrate that the proposed DGSO performs much better results than the other four GSOs including RDGSO, OGSO, RGSO, and GSO.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Foundation of Educational Commission of Tianjin City, China (Grant no. 20140803), supported by the Innovation Foundation for Young Teachers of Tianjin University of Science and Technology, China (Grant no. 2014CXLG30), supported by the National Natural Science Foundation of China (Grants nos. 61272509, 61402332, and 61402331), and supported by National High Technology Research and Development Program of China (Grant no. SQ2011SF11801036).

References

F. J. Rodriguez, C. García-Martinez, and M. Lozano, “Hybrid metaheuristics based on evolutionary algorithms and simulated annealing: taxonomy, comparison, and synergy test,” IEEE Transactions on Evolutionary Computation, vol. 16, no. 6, pp. 787–800, 2012.
View at: Publisher Site | Google Scholar
W. Huang and L. X. Ding, “Project-scheduling problem with random time-dependent activity duration times,” IEEE Transactions on Engineering Management, vol. 58, no. 2, pp. 377–387, 2011.
View at: Publisher Site | Google Scholar
W. Huang and L. X. Ding, “The shortest path problem on a fuzzy time-dependent network,” IEEE Transactions on Communications, vol. 60, no. 11, pp. 3376–3385, 2012.
View at: Publisher Site | Google Scholar
J.-N. Choi, S.-K. Oh, and W. Pedrycz, “Structural and parametric design of fuzzy inference systems using hierarchical fair competition-based parallel genetic algorithms and information granulation,” International Journal of Approximate Reasoning, vol. 49, no. 3, pp. 631–648, 2008.
View at: Publisher Site | Google Scholar
R. Storn and K. Price, “Differential evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces,” Tech. Rep. TR-95-012, Berkeley, Calif, USA, 1995.
View at: Google Scholar
K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Springer, Berlin, Germany, 2005.
View at: MathSciNet
M. Vasile, E. Minisci, and M. Locatelli, “An inflationary differential evolution algorithm for space trajectory optimization,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 2, pp. 267–281, 2011.
View at: Publisher Site | Google Scholar
A. M. Mora, P. García-Sánchez, J. J. Merelo, and P. A. Castillo, “Pareto-based multi-colony multi-objective ant colony optimization algorithms: an island model proposal,” Soft Computing, vol. 17, no. 7, pp. 1175–1207, 2013.
View at: Publisher Site | Google Scholar
C.-F. Lu, C.-H. Hsu, and C.-F. Juang, “Coordinated control of flexible AC transmission system devices using an evolutionary fuzzy lead-lag controller with advanced continuous ant colony optimization,” IEEE Transactions on Power Systems, vol. 28, no. 1, pp. 385–392, 2013.
View at: Publisher Site | Google Scholar
M. R. Khouadjia, B. Sarasola, E. Alba, L. Jourdan, and E.-G. Talbi, “A comparative study between dynamic adapted PSO and VNS for the vehicle routing problem with dynamic requests,” Applied Soft Computing Journal, vol. 12, no. 4, pp. 1426–1439, 2012.
View at: Publisher Site | Google Scholar
Y. Marinakis and M. Marinaki, “Particle swarm optimization with expanding neighborhood topology for the permutation flowshop scheduling problem,” Soft Computing, vol. 17, no. 7, pp. 1159–1173, 2013.
View at: Publisher Site | Google Scholar
S. He, Q. H. Wu, and J. R. Saunders, “A novel group search optimizer inspired by animal behavioural ecology,” in Proceedings of the IEEE Congress on Evolutionary Computation (CEC '06), pp. 1272–1278, Vancouver, Canada, July 2006.
View at: Google Scholar
S. He, Q. H. Wu, and J. R. Saunders, “Group search optimizer: an optimization algorithm inspired by animal searching behavior,” IEEE Transactions on Evolutionary Computation, vol. 13, no. 5, pp. 973–990, 2009.
View at: Publisher Site | Google Scholar
K. A. de Jong, Evolutionary Computation: a Unified Approach, MIT Press, Cambridge, Mass, USA, 2006.
View at: MathSciNet
S. Nesmachnow, H. Cancela, and E. Alba, “A parallel micro evolutionary algorithm for heterogeneous computing and grid scheduling,” Applied Soft Computing, vol. 12, no. 2, pp. 626–639, 2012.
View at: Publisher Site | Google Scholar
Y. Mei, K. Tang, and X. Yao, “A memetic algorithm for periodic capacitated arc routing problem,” IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics, vol. 41, no. 6, pp. 1654–1667, 2011.
View at: Publisher Site | Google Scholar
H. R. Tizhoosh, “Opposition-based learning: a new scheme for machine intelligence,” in Proceedings of the International Conference on Computational Intelligence for Modelling, Control and Automation, vol. 1, pp. 695–701, Vienna, Austria, November 2005.
View at: Google Scholar
H. R. Tizhoosh, “Reinforcement learning based on actions and opposite actions,” in Proceedings of the ICGST International Conference on Artificial Intelligence and Machine Learning, Cairo, Egypt, 2005.
View at: Google Scholar
W. Huang, S.-K. Oh, Z. Guo, and W. Pedrycz, “A space search optimization algorithm with accelerated convergence strategies,” Applied Soft Computing, vol. 13, no. 12, pp. 4659–4675, 2013.
View at: Publisher Site | Google Scholar
W. Huang, S.-K. Oh, and W. Pedrycz, “Design of hybrid radial basis function neural networks (HRBFNNs) realized with the aid of hybridization of fuzzy clustering method (FCM) and polynomial neural networks (PNNs),” Neural Networks, vol. 60, pp. 166–181, 2014.
View at: Publisher Site | Google Scholar
C. Xiong, L. Hao, and D. Wang, “Group search optimizer with ranking-based differential operator,” in Proceedings of the International Symposium on Information Technology Convergence (ISITC '15), vol. 4, pp. 3279–3285, Tianjin, China, October 2015.
View at: Google Scholar
D. Mustard, “Numerical integration over the n-dimensional spherical shell,” Mathematics of Computation, vol. 18, pp. 578–589, 1964.
View at: Google Scholar | MathSciNet
G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. Da Luz, E. P. Raposo, and H. E. Stanley, “Optimizing the success of random searches,” Nature, vol. 401, no. 6756, pp. 911–914, 1999.
View at: Publisher Site | Google Scholar
K. Li, “Performance analysis and evaluation of random walk algorithms on wireless networks,” in Proceedings of the IEEE International Symposium on Parallel and Distributed Processing, Workshops and Phd Forum (IPDPSW '10), pp. 1–8, Atlanta, Ga, USA, April 2010.
View at: Publisher Site | Google Scholar
H. Ono, “Fast random walks on finite graphs and graph topological information,” in Proceedings of the 2nd International Conference on Networking and Computing (ICNC '11), pp. 360–363, IEEE, Osaka, Japan, December 2011.
View at: Publisher Site | Google Scholar
R. S. Rahnamayan, H. R. Tizhoosh, and M. M. A. Salama, “Opposition-based differential evolution,” IEEE Transactions on Evolutionary Computation, vol. 12, no. 1, pp. 64–79, 2008.
View at: Publisher Site | Google Scholar
J. Riget and J. S. Vesterstrøm, “A diversity-guided particle swarm optimizer—the ARPSO,” Tech. Rep. 2002-02, Department of Computer Science, University of Aarhus, EVALife, 2002.
View at: Google Scholar
F. Herrera, M. Lozano, and D. Molina, “Test suite for the special issue of soft computing on scalability of evolutionary algorithms and other metaheuristics for large scale continuous optimization problems,” Tech. Rep., University of Granada, Granada, Spain, 2010.
View at: Google Scholar
K. Tang, X. Yao, P. N. Suganthan et al., “Benchmark functions for the CEC '2008 special session and competition on high-dimensional real-parameter optimization,” Tech. Rep., Nature Inspired Computation and Applications Laboratory, USTC, Hefei, China, 2008.
View at: Google Scholar
F. Herrera and M. Lozano, “Workshop for evolutionary algorithms and other metaheuristics for continuous optimization problems—a scalability test,” in Proceedings of the International Conference on Intelligent System Design and Applications (ISDA '09), Pisa, Italy, November-December 2009.
View at: Google Scholar
N. Q. Huy, O. Y. Soon, L. M. Hiot, and N. Krasnogor, “Adaptive cellular memetic algorithms,” Evolutionary Computation, vol. 17, no. 2, pp. 231–256, 2009.
View at: Publisher Site | Google Scholar
M. Vasile, E. Minisci, and M. Locatelli, “An inflationary differential evolution algorithm for space trajectory optimization,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 2, pp. 267–281, 2011.
View at: Publisher Site | Google Scholar
S. K. Goudos, K. A. Gotsis, K. Siakavara, E. E. Vafiadis, and J. N. Sahalos, “A multi-objective approach to subarrayed linear antenna arrays design based on memetic differential evolution,” IEEE Transactions on Antennas and Propagation, vol. 61, no. 6, pp. 3042–3052, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
U. Halder, S. Das, and D. Maity, “A cluster-based differential evolution algorithm with external archive for optimization in dynamic environments,” IEEE Transactions on Cybernetics, vol. 43, no. 3, pp. 881–897, 2013.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2015 Dan Wang 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.

PDF Download Citation

Download other formats

Order printed copies

Views

786

Downloads

816

Citations