Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 628357, 10 pages

http://dx.doi.org/10.1155/2014/628357

Research Article

Particle Swarm Optimization Based on Local Attractors of Ordinary Differential Equation System

Wenyu Yang,¹ Wei Wu,² Yetian Fan,² and Zhengxue Li²

¹College of Science, Huazhong Agricultural University, Wuhan 430070, China

²School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 24 April 2014; Revised 8 August 2014; Accepted 15 August 2014; Published 26 August 2014

Academic Editor: Manuel De la Sen

Copyright © 2014 Wenyu 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.

Particle swarm optimization (PSO) is inspired by sociological behavior. In this paper, we interpret PSO as a finite difference scheme for solving a system of stochastic ordinary differential equations (SODE). In this framework, the position points of the swarm converge to an equilibrium point of the SODE and the local attractors, which are easily defined by the present position points, also converge to the global attractor. Inspired by this observation, we propose a class of modified PSO iteration methods (MPSO) based on local attractors of the SODE. The idea of MPSO is to choose the next update state near the present local attractor, rather than the present position point as in the original PSO, according to a given probability density function. In particular, the quantum-behaved particle swarm optimization method turns out to be a special case of MPSO by taking a special probability density function. The MPSO methods with six different probability density functions are tested on a few benchmark problems. These MPSO methods behave differently for different problems. Thus, our framework not only gives an interpretation for the ordinary PSO but also, more importantly, provides a warehouse of PSO-like methods to choose from for solving different practical problems.

1. Introduction

Inspired by sociological behavior associated with bird flock, particle swarm optimization (PSO) was first introduced by Kennedy and Eberhart [1]. In a PSO, the individual particles of a swarm fly stochastically toward the positions of their own previous best performance and the best previous performance of the swarm. Researchers have been trying to excogitate new framework to interpret PSO in order to analyze the property of PSO and to construct new PSO-like methods. For instance, in [2] PSO is interpreted as a difference scheme for a second-order ordinary differential equation. Fernández-Martínez et al. interpret PSO algorithm as a stochastic damped mass-spring system: the so-called PSO continuous model and present a theoretical analysis of PSO trajectories in [3]. Based on a continuous vesion of PSO, Fernández Martínez and García Gonzalo propose Generalized PSO(GPSO) in [4] and introduce a delayed version of the PSO continuous model in [5]. Furthermore, Fernández-Martínez and García-Gonzalo give stochastic stability analysis of PSO models in [6] and propose two novel algorithms: PP-GPSO and RR-GPSO in [7]. PSO algorithms have been applied successfully for practical applications [8–10].

In [11], a so-called quantum-behaved particle swarm optimization (QPSO) is proposed based on an assumption that the individual particles in a PSO system have quantum behavior. A wide range of continuous optimization problems have been solved successfully by QPSO and many efficient strategies have been proposed to improve the algorithm [11–20]. A global convergence analysis of QPSO is given by Sun et al. in [21].

In this paper, we interpret PSO as a finite difference scheme for solving a system of stochastic ordinary differential equations (SODE in short). In this framework, the convergent point of the position points in the PSO iteration process corresponds to an equilibrium point (a global attractor) of the SODE. We observe that the local attractors, which are easily computed by using the present position points, also converge to the global attractor in the PSO iteration process. Inspired by this observation, we propose a class of modified PSO iteration methods (MPSO in short) based on local attractors of the SODE. The idea of MPSO is to choose the next update state in a neighbourhood of the present local attractor, rather than the present position point as in the original PSO, according to a given probability density function. We will test the MPSO methods with six different probability density functions for solving a few benchmark problems. These MPSO methods behave differently for different problems. Thus, our framework not only gives an interpretation for the ordinary PSO but also, more importantly, provides a warehouse of PSO-like methods to choose from for solving different practical problems.

Our work is partly inspired by the second-order ordinary differential equation framework for PSO in [2, 3]. But the solution of a second-order ordinary differential equation is somehow more difficult to describe and it seems more difficult to construct new PSO-like methods through this framework. Our framework of ordinary differential equation makes the job easier.

Our work is also inspired by the quantum-behaved particle swarm optimization (QPSO) method in [11]. It turns out that QPSO becomes a special case of our MPSO by choosing a special probability density function. And fortunately, the convergence analysis for QPSO given in [21] is still valid for our MPSO methods. In fact, what matters in the convergence analysis is a suitable choice of the probability density function, and the particular quantum behavior does nothing in the convergence analysis.

The rest of the paper is organized as follows. In Section 2 we interpret PSO as a system of ODE and propose a class of MPSO methods. Then, in Section 3 we test, on some nonlinear benchmark functions, our MPSO methods with different probability density functions. Finally, some conclusions are gathered in Section 4.

2. Particle Swarm Optimization Algorithms

2.1. The Original PSO

Particle swarm optimization algorithm (PSO) was first introduced in Kennedy and Eberhart [1] inspired by sociological behavior associated with bird flock. In a PSO with population size , the velocity vector and the position vector of each particle are iteratively adjusted to minimize an objective function with as input value. At the th iteration step, the particle updates its velocity vector and position vector according to

for and , where and are random numbers uniformly distributed on (0,1). The values of and are scaled by the constants and which are called acceleration coefficients. The vector denotes the personal best position (pbest in short), which is the position of particle giving the best objective function value so far, and the vector is the global best (gbest) position which is the position of the best particle among all particles. They are updated according to

When the PSO system is convergent, all particles converge, as tends to infinity, to a global attractor ; that is,

and their velocity vectors converge to zero.

2.2. Interpretation of PSO as a Finite Difference Scheme for an SODE

Let us rewrite the iteration formula (1) for PSO as a system of difference equations:

for , where

Generally but not necessarily, the constants and are set to be equal. For the sake of simple representation, we rewrite

where . We regard (4) as a finite difference scheme with time step length for solving the following system of stochastic ordinary differential equations (SODE):

where , , , and for all real numbers . Let ,  , , and , where , and so forth. Then, the SODE (7) is written as

where is a nonlinear function. The theoretical analysis of this SODE and its difference schemes is difficult, but it is not our concern. For our purpose, we recall that usually, or at least very often, the solution of a difference scheme for an ODE will converge to an equilibrium point satisfying when the iteration time tends to infinity. Thus, the convergent point of the PSO iteration process corresponds to an equilibrium point of the SODE.

Usually, for an equilibrium point of the SODE (7) we have

for some constant vector . Let us call the local attractor and the global attractor for the SODE at time .

2.3. MPSO for Iteratively Decreasing

Now, let us forget the original PSO and concentrate on the task of finding a global attractor (an equilibrium point) of the SODE (7). Based on (10), we see that finding an equilibrium point of the ODE system (7) is equivalent to making the position vector closer and closer to the local attractor . Thus, we choose the next update state in a neighbourhood of the present local attractor, rather than the present position point as in the original PSO, according to a given probability density function (PDF).

Define

where is an adjustable parameter and is the mbest position defined by the average of the pbest positions of all particles; that is, . We choose a PDF using as a parameter indicating the “width" of the support of the function. Choose a random number , according to the PDF . Then, the update formula of MPSO is as follows:

where the signs + and − are taken randomly in equal probability and . , a discrete approximation of the local attractor , is called discrete local attractor at th step. Therefore, is a point randomly chosen from a neighborhood of the discrete local attractor .

Our MPSO algorithm is described in detail in Algorithm 1.

Choose a PDF .

Initialize population position vector

Do

For to population size

    If Then

End For

For to population size

For to dimension

    Choose and compute .

    Choose a random number ,

    according to the PDF .

    Choose , .

    If Then

    If Then

End For

End For

Repeat the iteration until all the increments are small enough

or some other termination criterion is satisfied.

Algorithm 1

2.3.1. MPSO with Different PDFs

As examples, we give six different PDFs and depict them when in Figures 1, 2, 3, 4, 5, and 6.

Figure 1: Figure of the probability density function .

Figure 2: Figure of the probability density function .

Figure 3: Figure of the probability density function .

Figure 4: Figure of the probability density function .

Figure 5: Figure of the probability density function .

Figure 6: Figure of the probability density function .

Now, let us demonstrate how to choose a random number according to the probability density function . First, we obtain the corresponding probability distribution function by

Then, for a randomly given number , we solve the equation

to get

Therefore, the update formulas corresponding to can be written as follows:

where , , and . Here the superscripts and subscripts of and have been removed for the sake of clear representation. For instance, in the case of , the precise formula is

2.4. Quantum-Behaved Particle Swarm Optimization

In QPSO proposed by Sun et al. [11], the quantum state of a particle is depicted by a wave function , which is the solution of the Schrödinger equation

where Hamiltonian operator is

Note that the Schrödinger equation is a second-order partial differential equation. is chosen to be the PDF for the present position of the particle. In particular, and defined above are two such PDFs mentioned in [11]. Thus, QPSO can be regarded as a special case of MPSO. But the other four PDFs to are not related to the Schrödinger equation.

Let us elaborate a little bit on QPSO. In [11], Sun et al. assume that, at the th iteration, on the th dimension () of the search space, particle moves in a potential well centered at which is the th dimension coordinate of its local attractor.

Let , then they obtained the PDF

in Jun Sun et al. [17] is determined by

or

where is the mbest position. Finally,

Sun et al. gave a global convergence analysis of QPSO in [20], which employed certain properties of the PDF but had nothing to do with the particular Schördinger equation. Actually, all our six PDFs given Section 2.3.1 possess these properties, and hence the global convergence analysis applies.

3. Numerical Simulation

In this section, we use MPSO with six different PDFs mentioned in Section 2 to test five nonlinear benchmark functions used in [11]. The first function is the Sphere function described by

The second function is Rosenbrock function described by

The third function is the generalized Rastrigrin function described by

The fourth function is the generalized Griewank function described by

The last function is Shcaffer function described by

where is an -dimension real-valued vector. The initialization and search ranges of the four functions used in [11] are listed in Table 1.

Table 1: Asymmetric initialization and search ranges.

Function
Asymmetric initialization range
Search range

Sphere

Rosenbrock

Genralized Rastrigrin

Genralized Griewank

Schaffer

As in [11], different population sizes (20, 40, and 80) are used for each function to investigate the scalability. The maximum number of the generations is set to 1000, 1500, and 2000, corresponding to the dimensions 10, 20, and 30 for the four functions, respectively. The mean best fitness values and standard deviations, out of total 50 runs of MPSO with different PDFs on to , are shown in Tables 2, 3, 4, 5, and 6. In the tables, is the parameter in (12), and means that decreases linearly from 1 to 0.5. The boldface results are obtained by performing the QPSO algorithm in [11], which is equivalent to MPSO with the PDF . An efficiency comparison of MPSOs with different PDFs on to is presented in Table 7.

Table 2: Mean best fitness values and standard deviation for the sphere function.

PDF
Dim.

Mean best
Standard deviation
Mean best
Standard deviation
Mean best
Standard deviation

()
10
1000

20
1500

30
2000

()
10
1000

20
1500

30
2000

()
10
1000

20
1500

30
2000

()
10
1000

20
1500

30
2000

()
10
1000
1.18E  −  27
6.19E  −  27
8.47E  −  40
5.82E  −  39
8.09E  −  53
5.23E  −  52

20  
1500  
1.16E  −  14
4.48E  −  14
1.47E  −  23
7.46E  −  23
1.04E  −  31
3.46E  −  31

30  
2000  
7.16E  −  09
4.01E  −  08
2.73E  −  16
6.02E  −  16
9.68E  −  22
3.07E  −  21

()  
  10  
  1000  

  20  
  1500  

  30  
  2000  

Table 3: Mean best fitness values and standard deviation for the Rosenbrock function.

PDF
Dim.

Mean best
Standard deviation
Mean best
Standard deviation
Mean best
St andard deviation

()
10
1000
17.9685
29.8452
12.2590
21.7814
11.7597
26.2940

20
1500
111.6378
188.7111
60.2500
50.6469
46.2081
41.0317

30
2000
416.5972
1436.0
112.4362
145.4795
72.8064
99.0574

()
10
1000
33.6155
49.8280
26.2646
43.4507
8.2692
9.3772

20
1500
82.0246
67.5004
60.3972
51.3739
44.3745
42.1392

30
2000
138.9158
138.7761
83.9015
80.3806
68.5438
42.1898

()
10
1000
17.2888
36.1287
16.5371
31.0729
11.9782
26.6798

20
1500
89.6570
162.4040
54.0191
56.6993
49.7047
50.1423

30
2000
221.5118
539.5228
86.1085
152.8715
67.2254
103.1385

()
10
1000
71.5892
150.1641
29.8696
46.9684
17.7129
18.4078

20
1500
86.1848
107.0572
78.7056
95.2644
52.9101
47.3649

30
2000
154.0989
203.9596
104.0223
155.8188
80.6434
88.6188

()
10
1000
45.6945
93.5950
14.9285
18.8043
9.2944
12.6001

20
1500
120.3651
166.2435
77.0110
96.0561
57.0820
45.8037

30
2000
242.7943
408.3627
135.5536
204.6100
56.0855
45.8168

()
10
1000
20.5409
30.0800
14.8070
21.2420
10.4618
20.9702

20
1500
93.9863
127.0970
86.2331
129.4297
49.4717
51.3767

30
2000
504.8399
1090.4
145.6577
198.6213
80.3452
53.4474

Table 4: Mean best fitness values and standard deviation for the generalized Rastrigrin function.

PDF
Dim.

Mean best
Standard deviation
Mean best
Standard deviation
Mean best
Standard deviation

()
10
1000
5.8080
4.0212
4.1037
3.3882
3.1426
3.0072

20
1500
14.9072
4.9084
10.1125
3.1486
7.5353
2.4052

30
2000
31.5566
7.7407
20.4983
4.8202
14.4129
3.1085

()
10
1000
9.9946
5.2287
5.9174
3.0855
4.5410
2.3456

20
1500
25.7827
10.8980
19.7675
8.6651
15.2177
5.7206

30
2000
44.8329
14.6816
28.6026
8.0842
23.8493
10.6089

()
10
1000
4.9402
3.6175
3.0414
3.2103
1.8484
1.1974

20
1500
15.7036
4.8358
9.6034
2.6184
7.8317
2.4439

30
2000
31.3558
7.7029
18.3678
4.8567
14.8082
3.2635

()
10
1000
4.5903
2.4440
2.6680
1.5771
2.1146
1.3227

20
1500
15.2589
5.9192
11.4040
4.5605
9.1622
4.9490

30
2000
31.5361
12.5203
22.5729
10.0456
17.8487
8.0797

()
10
1000
6.4976
3.9073
3.0799
1.6581
2.6277
1.5756

20
1500
16.3392
6.5859
10.7055
3.4636
9.8460
3.6358

30
2000
32.8285
11.4557
23.2707
7.5592
16.1005
3.5702

()
10
1000
5.4773
4.4511
3.4689
1.8336
2.4370
1.3997

20
1500
16.2287
5.2804
10.9497
3.1922
8.7061
2.8914

30
2000
35.3445
9.7132
22.2326
5.6518
18.6051
5.6434

Table 5: Mean best fitness values and standard deviation for the generalized Griewank function.

PDF
Dim.

Mean best
Standard deviation
Mean best
Standard deviation
Mean best
Standard deviation

()
10
1000
0.0846
0.1467

20
1500
0.0049
0.0240

30
2000
0.0024
0.0094

()
10
1000
0.0531
0.0951

20
1500

0
0

30
2000

0
0

()
10
1000
0.0548
0.1063

20
1500
0.0013
0.0072

30
2000
0.0054
0.0096

()
10
1000
0.0748
0.1122

20
1500
0.0051
0.0174

30
2000

()
10
1000
0.0362
0.0553
0.0010
0.0072
4.36E  −  08
2.40E  −  07

20
1500
0.0022
0.0097
9.19E  −  08
6.36E  −  07
3.40E  −  09
2.24E  −  08

30
2000
5.61E  −  05
1.16E  −  04
5.80E  −  08
1.33E  −  07
6.33E  −  07
4.40E  −  06

()
10
1000
0.0452
0.0837
0.0092
0.0647

20
1500
0.0055
0.0190

30
2000
0.0012
0.0056

Table 6: Mean best fitness values and standard deviation for the Schaffer function.

PDF
Dim.

Mean best
Standard deviation
Mean best
Standard deviation
Mean best
Standard deviation

()
2
1000

0.0023

2
1500

0.0026

2
2000

0.0026

()
2
1000
0.0068
0.0045
0.0047
0.0049

0.0029

2
1500
0.0064
0.0046
0.0033
0.0046
0.0014
0.0034

2
2000
0.0075
0.0016
0.0033
0.0046
0.0019
0.0039

()
2
1000

0.0029

2
1500

2
2000

()
2
1000

0.0023

2
1500

0.0026

2
2000

0.0014

()
2
1000
0.0023
0.0041
2.05E  −  04
0.0014
5.85E  −  07
1.62E  −  06

2
1500
9.73E  −  04
0.0029
3.89E  −  04
0.0019
2.15E  −  08
5.06E  −  08

2
2000
7.78E  −  04
0.0026
1.94E  −  04
0.0014
2.05E  −  08
7.71E  −  08

()
2
1000

0.0019

2
1500

0.0019

2
2000

0.0014

Table 7: Descending order of MPSOs efficiency with different PDFs for testing benchmark functions.

Function
Descending order of MPSOs efficiency with different PDFs

Sphere

Rosenbrock

Generalized Rastrigrin

Generalized Griewank

Schaffer

4. Conclusion

Particle swarm optimization (PSO) algorithm is interpreted as a finite difference scheme for solving a system of stochastic ordinary differential equations (SODE in short). It is illustrated that the position points of the swarm and the local attractors, which are easily defined by the present position points, all converge to a global attractor of the SODE. A class of modified PSO iteration methods (MPSO in short) based on local attractors of the SODE is proposed such that the next update state is chosen near the present local attractor, rather than the present position point as in the original PSO, according to a given probability density function. In particular, the quantum-behaved particle swarm optimization method turns out to be a special case of MPSO by taking a special probability density function. The MPSO methods with six different probability density functions are tested on a few benchmark problems. These MPSO methods behave differently for different problems. Thus, our framework not only gives an interpretation for the ordinary PSO but also, more importantly, provides a warehouse of PSO-like methods to choose from for solving different practical problems.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11171367) and the Fundamental Research Funds for the Central Universities of China (2662013BQ049, 2662014QC011).

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