# An Opposition-Based Group Search Optimizer with Diversity Guidance

Authors: Dan Wang; Congcong Xiong; Xiankun  Zhang
Journal: Mathematical Problems in Engineering
 (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.


---

## Body

## 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. Section2 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 Algorithm1, where n is the dimension; Xik is in the ith member at the kth iteration; φik is a head angle; Dik is a unit vector [13].Algorithm 1:Procedure of the GSO algorithm.
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)EndIn the producing operator, the producer scans three points through the following expression [19]:(1)Xz=Xpk+r1lmaxDpkφk,Xr=Xpk+r1lmaxDpkφk+r2θmax2,Xl=Xpk+r1lmaxDpkφk-r2θmax2,where θmax is the maximum search angle; lmax is the maximum pursuit distance; r1 and r2 are parameters generated under the normally distribution; and r2 is also a real number in the interval [0,1].If the current point changed, the angle is(2)φk+1=φk+r2αmax,where αmax stands for the maximum turning angle [12].Otherwise the angle is fixed, and the angle is(3)φk+a=φk,where a 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(4)li=a·r1lmax.The new point is expressed:(5)Xik+1=Xik+liDikφk+1.


## 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 Algorithm2.Algorithm 2:Procedure of diversity-guided GSO (DGSO) with opposition-based learning (OBL).
/∗ 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 P=(X1,…,Xn) is a point, and the dimension is in n, where Xi is a real number, Xi∈[ai,bi]. The opposite point of P is P~=(X1~,…,Xn~), where Xi~=ai+bi-Xi.Opposition-based optimization (OBL) is as follows [26]: we assume f(·) 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 f(P~)≥f(P), then the point P replaces P~; otherwise the point is fixed and does not change. Hence, the point P and its opposite point P~ 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. Algorithm3 described the procedure of opposition-based learning operator, where P is the population, OPi,j is the opposite point according to Pi,j, NP is the population size, and aj=min(xjk), bj=max(xjk).Algorithm 3:Procedure of opposition-based learning operator.
Line# Procedure of opposition-based learning operator
(1)Begin
(2) Generate the populationP
(3)  for (inti=0; i < NP; i++)
(4)  for (intj=0; j<n; j++)
(5)OPi,j=aj+bj-Pi,j;
(6) Calculate the fitness value of theOPi,j
(7)Output the some optimal candidates from {P,OP};
(8)End


### 3.2. Diversity-Guided Operator

To alleviate the problem of premature convergence, we use a mechanism of diversity guidance as shown in Algorithm4. Here the diversity of group (population) is computed by the following formula [27]:(6)diversity=1S·L·∑i=1S∑j=1Npij-p-j2,where S is the group size, L is the length of the longest diagonal in the search space [26], N is the dimension, pij is the jth value in the ith member (individual), p- represents the average point, and p-j stands for the jth value of p-. Algorithm 4 described the procedure of diversity-based operator, where lmax is a given number used in formula (1); w1, w2 are set as 0.33 and 0.66, individually.Algorithm 4:Procedure of diversity-guided operator.
#Line Diversity-guided operator procedure
(1)Begin
(2)if(diversity < dLow)
(3)lmax⁡=w1∗lmax⁡;
(4)else if(diversity ≥ dHigh)
(5)lmax⁡=lmax⁡;
(6)else if ((diversity ≥ dLow) && (diversity < dHigh))
(7)lmax⁡=w2∗lmax⁡;
(8)End IF
(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 P=(X1,…,Xn) is a point, and the dimension is in n, where Xi is a real number, Xi∈[ai,bi]. The opposite point of P is P~=(X1~,…,Xn~), where Xi~=ai+bi-Xi.Opposition-based optimization (OBL) is as follows [26]: we assume f(·) 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 f(P~)≥f(P), then the point P replaces P~; otherwise the point is fixed and does not change. Hence, the point P and its opposite point P~ 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. Algorithm3 described the procedure of opposition-based learning operator, where P is the population, OPi,j is the opposite point according to Pi,j, NP is the population size, and aj=min(xjk), bj=max(xjk).Algorithm 3:Procedure of opposition-based learning operator.
Line# Procedure of opposition-based learning operator
(1)Begin
(2) Generate the populationP
(3)  for (inti=0; i < NP; i++)
(4)  for (intj=0; j<n; j++)
(5)OPi,j=aj+bj-Pi,j;
(6) Calculate the fitness value of theOPi,j
(7)Output the some optimal candidates from {P,OP};
(8)End


## 3.2. Diversity-Guided Operator

To alleviate the problem of premature convergence, we use a mechanism of diversity guidance as shown in Algorithm4. Here the diversity of group (population) is computed by the following formula [27]:(6)diversity=1S·L·∑i=1S∑j=1Npij-p-j2,where S is the group size, L is the length of the longest diagonal in the search space [26], N is the dimension, pij is the jth value in the ith member (individual), p- represents the average point, and p-j stands for the jth value of p-. Algorithm 4 described the procedure of diversity-based operator, where lmax is a given number used in formula (1); w1, w2 are set as 0.33 and 0.66, individually.Algorithm 4:Procedure of diversity-guided operator.
#Line Diversity-guided operator procedure
(1)Begin
(2)if(diversity < dLow)
(3)lmax⁡=w1∗lmax⁡;
(4)else if(diversity ≥ dHigh)
(5)lmax⁡=lmax⁡;
(6)else if ((diversity ≥ dLow) && (diversity < dHigh))
(7)lmax⁡=w2∗lmax⁡;
(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 f1(x)–f6(x) are selected from CEC 2008 Special Session [29], the functions f7(x)–f11(x) were provided from ISDA 2009 Workshop [30], and the functions f12(x)–f19(x) are the hybrid functions based on the previous functions f1(x)–f11(x) [28]. Test functions f1(x)–f11(x) are shown in Table 1; test functions f12(x)–f19(x) 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 D, we set D∈50,100,200. According to [31], MAX_FES is set to 5000∗D. All of the algorithms are terminated when the MAX_FES is reached.Table 1
Test functionsf1(x)–f11(x).
Function
Definition
Domain
Fitness optimum
f
1
(
x
)
∑
i
=
1
D
x
i
2
[−100, 100]D
0
f
2
(
x
)
max
i
⁡
x
i
,
1
≤
i
≤
D
[−100, 100]D
0
f
3
(
x
)
∑
i
=
1
D
100
x
i
2
+
x
i
+
1
2
+
x
i
-
1
2
[−100, 100]D
0
f
4
(
x
)
∑
i
=
1
D
x
i
2
-
10
c
o
s
2
π
x
i
+
10
[−5, 5]D
0
f
5
(
x
)
∑
i
=
1
D
x
i
2
4000
-
∏
i
=
1
D
cos
⁡
x
i
i
+
1
[−600, 600]D
0
f
6
(
x
)
-
20
exp
⁡
-
0.2
1
D
∑
i
=
1
D
x
i
2
-
e
x
p
1
D
∑
i
=
1
D
cos
⁡
2
π
x
i
+
20
+
e
[−32, 32]D
0
f
7
(
x
)
∑
i
=
1
D
x
i
+
∏
i
=
1
D
x
i
[−10, 10]D
0
f
8
(
x
)
∑
i
=
1
D
∑
j
=
1
i
x
j
2
[−65.536, 65.536]D
0
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
[−100, 100]D
0
f
10
(
x
)
∑
i
=
1
D
x
i
2
+
2
x
i
+
1
2
-
0.3
cos
⁡
3
π
x
i
-
0.4
cos
⁡
4
π
x
i
+
1
+
0.7
[−15, 15]D
0
f
11
x
∑
i
=
1
D
-
1
x
i
2
+
x
i
+
1
2
0.25
s
i
n
2
50
⋅
x
i
2
+
x
i
+
1
2
0.1
+
1
[−100, 100]D
0Table 2
Test functionsf12(x)–f19(x).
Function
m
n
s
Domain
FO
f
12
x
=
f
9
x
⊕
f
1
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.25
[−100, 100]D
0
f
1
(
x
)
∑
i
=
1
D
x
i
2
f
13
x
=
f
9
x
⊕
f
3
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.25
[−100, 100]D
0
f
3
(
x
)
∑
i
=
1
D
100
x
i
2
+
x
i
+
1
2
+
x
i
-
1
2
f
14
x
=
f
9
x
⊕
f
4
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.25
[−5, 5]D
0
f
4
(
x
)
∑
i
=
1
D
x
i
2
-
10
c
o
s
2
π
x
i
+
10
f
15
x
=
f
10
x
⊕
f
7
x
f
10
(
x
)
∑
i
=
1
D
x
i
2
+
2
x
i
+
1
2
-
0.3
c
o
s
3
π
x
i
-
0.4
c
o
s
4
π
x
i
+
1
+
0.7
0.25
[−10, 10]D
0
f
7
(
x
)
∑
i
=
1
D
x
i
+
∏
i
=
1
D
x
i
f
16
x
=
f
9
x
⊕
f
1
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.5
[−100, 100]D
0
f
1
(
x
)
∑
i
=
1
D
x
i
2
f
17
x
=
f
9
x
⊕
f
3
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.75
[−100, 100]D
0
f
3
(
x
)
∑
i
=
1
D
100
x
i
2
+
x
i
+
1
2
+
x
i
-
1
2
f
18
x
=
f
9
x
⊕
f
4
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.75
[−5, 5]D
0
f
4
(
x
)
∑
i
=
1
D
x
i
2
-
10
c
o
s
2
π
x
i
+
10
f
19
x
=
f
10
x
⊕
f
7
x
f
10
(
x
)
∑
i
=
1
D
x
i
2
+
2
x
i
+
1
2
-
0.3
c
o
s
3
π
x
i
-
0.4
c
o
s
4
π
x
i
+
1
+
0.7
0.75
[−10, 10]D
0
f
7
(
x
)
∑
i
=
1
D
x
i
+
∏
i
=
1
D
x
i
∗FO denotes fitness optimum.(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 φik to be π/4; the maximum search angle θmax is π/a2, where the value of a is a given number and θmax/2 is the maximum turning angle (see [13]).Table 3
Strategies used in the five GSO algorithms.
Algorithms
Manner of initialization
Manner of evolutionary operators
GSO
Random
Random
RGSO
Random
Opposition-based learning
OGSO
Opposition-based learning
Opposition-based learning
RDGSO
Random
Opposition-based learning and diversity-guided operator
DGSO
Opposition-based learning
Opposition-based learning and diversity-guided operator


### 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 D 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 4
Average fitness values tested on benchmark functions (D=30).
∗
F
GSO
OGSO
RDGSO
DGSO
Mean
Std.
Mean
Std.
Mean
Std.
Mean
Std.
F1
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F2
0.00
E
+
00
0.00
E
+
00
1.43
E
-
05
1.24
E
-
05
0.00
E
+
00
0.00
E
+
00
2.14
E
-
03
2.03
E
-
04
F3
7.99
E
+
01
5.25
E
-
01
6.97
E
+
02
2.86
E
+
01
6.56
E
+
01
6.03
E
-
01
4.55
E
+
01
3.94
E
+
01
F4
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F5
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F6
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F7
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F8
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F9
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F10
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F11
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F12
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F13
6.43
E
+
01
7.47
E
-
01
2.34
E
+
01
2.46
E
+
01
3.93
E
+
01
8.87
E
-
01
2.87
E
+
01
2.49
E
+
01
F14
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F15
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F16
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
1.39
E
-
158
0.00
E
+
00
0.00
E
+
00
F17
3.07
E
+
01
8.95
E
-
01
2.14
E
+
02
2.40
E
+
01
1.66
E
+
01
7.42
E
-
01
9.38
E
+
00
1.63
E
+
01
F18
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F19
2.15
E
+
01
2.98
E
-
01
3.45
E
+
01
2.64
E
+
01
1.14
E
-
02
4.29
E
-
03
6.96
E
-
03
4.98
E
-
03
∗
F: functions.Table5 summarizes the comparison of DGSO, GSO, and RDGSO with the different dimensions (D=50, 100, 200), respectively. It is evident that DGSO performs better than the other reported algorithms especially in case of the high dimension functions.Table 5
Average fitness values tested on benchmark functions with increasing dimension.
Functions
GSO
RDGSO
DGSO
dim = 50
100
200
dim = 50
100
200
dim = 50
100
200
F1
Mean
5.00
E
+
00
4.60
E
+
00
5.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.36
E
+
00
0.00
E
+
00
2.17
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F2
Mean
6.20
E
+
00
5.80
E
+
00
4.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
5.00
E
+
00
3.67
E
+
00
Std.
3.27
E
+
00
0.00
E
+
00
3.91
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
3.61
E
+
00
3.51
E
+
00
F3
Mean
6.60
E
+
00
3.20
E
+
00
4.60
E
+
00
2.40
E
+
00
2.33
E
+
00
2.33
E
+
00
4.80
E
+
00
2.33
E
+
00
4.00
E
+
00
Std.
1.30
E
+
00
4.76
E
-
02
1.52
E
+
00
3.36
E
+
00
2.08
E
+
00
1.53
E
+
00
0.00
E
+
00
2.52
E
+
00
2.65
E
+
00
F4
Mean
6.40
E
+
00
5.20
E
+
00
3.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
2.77
E
+
00
0.00
E
+
00
2.83
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F5
Mean
5.60
E
+
00
4.40
E
+
00
6.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.65
E
+
00
0.00
E
+
00
2.51
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
3.49
E
+
00
0.00
E
+
00
0.00
E
+
00
F6
Mean
4.80
E
+
00
2.40
E
+
00
7.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.91
E
+
00
0.00
E
+
00
1.82
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F7
Mean
3.00
E
+
00
3.00
E
+
00
7.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
1.00
E
+
00
0.00
E
+
00
2.95
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F8
Mean
4.00
E
+
00
5.80
E
+
00
3.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
6.00
E
+
00
Std.
2.17
E
+
00
0.00
E
+
00
3.42
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
1.00
E
+
00
F9
Mean
5.40
E
+
00
6.20
E
+
00
4.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
1.92
E
+
00
0.00
E
+
00
2.88
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F10
Mean
4.20
E
+
00
3.40
E
+
00
5.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.13
E
+
00
0.00
E
+
00
3.49
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F11
Mean
3.80
E
+
00
3.40
E
+
00
4.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
2.51
E
+
00
0.00
E
+
00
2.70
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F12
Mean
4.20
E
+
00
4.40
E
+
00
2.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
2.30
E
+
00
0.00
E
+
00
2.61
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F13
Mean
6.40
E
+
00
5.80
E
+
00
3.00
E
+
00
3.80
E
+
00
1.00
E
+
00
4.67
E
+
00
3.60
E
+
00
3.33
E
+
00
2.67
E
+
00
Std.
2.86
E
+
00
1.23
E
-
03
2.35
E
+
00
2.59
E
+
00
1.73
E
+
00
5.77
E
-
01
1.82
E
+
00
4.16
E
+
00
3.06
E
+
00
F14
Mean
6.40
E
+
00
4.60
E
+
00
5.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.65
E
+
00
0.00
E
+
00
2.83
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F15
Mean
3.20
E
+
00
3.60
E
+
00
4.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.78
E
+
00
0.00
E
+
00
3.36
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F16
Mean
6.40
E
+
00
6.40
E
+
00
4.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.58
E
+
00
0.00
E
+
00
2.07
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F17
Mean
4.80
E
+
00
4.80
E
+
00
4.40
E
+
00
2.60
E
+
00
5.67
E
+
00
2.33
E
+
00
5.80
E
+
00
5.33
E
+
00
4.00
E
+
00
Std.
4.09
E
+
00
1.71
E
-
06
2.97
E
+
00
2.88
E
+
00
1.15
E
+
00
3.21
E
+
00
3.11
E
+
00
2.31
E
+
00
2.00
E
+
00
F18
Mean
3.60
E
+
00
3.60
E
+
00
1.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.51
E
+
00
0.00
E
+
00
1.34
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F19
Mean
5.00
E
+
00
5.00
E
+
00
3.40
E
+
00
5.40
E
+
00
4.67
E
+
00
3.00
E
+
00
2.80
E
+
00
5.33
E
+
00
3.33
E
+
00
Std.
2.92
E
+
00
2.12
E
+
01
2.30
E
+
00
3.36
E
+
00
1.53
E
+
00
1.73
E
+
00
1.79
E
+
00
2.08
E
+
00
1.53
E
+
00(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 f1–f19. It shows that the convergence speed of DGSO is faster than the other four algorithms.Figure 1
Convergence results for test functionsf1–f19.
(a)
f
1
(b)
f
2
(c)
f
3
(d)
f
4
(e)
f
5
(f)
f
6
(g)
f
7
(h)
f
8
(i)
f
9
(j)
f
10
(k)
f
11
(l)
f
12
(m)
f
13
(n)
f
14
(o)
f
15
(p)
f
16
(q)
f
17
(r)
f
18
(s)
f
19(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.Table 6
Diversity values of the OGSO and DGSO.
Functions
OGSO
DGSO
DIV mean
DIV std.
DIV mean
DIV std.
F1
1.19
E
-
02
3.33
E
-
04
2.36
E
-
02
8.77
E
-
03
F2
1.42
E
-
02
3.74
E
-
04
1.198
E
-
01
1.85
E
-
01
F3
2.18
E
-
02
2.00
E
-
02
9.01
E
-
02
1.12
E
-
01
F4
1.11
E
+
00
6.88
E
-
01
1.40
E
+
00
1.93
E
+
00
F5
4.23
E
-
03
3.76
E
-
03
2.15
E
-
03
1.15
E
-
04
F6
1.66
E
-
01
1.35
E
-
01
3.89
E
-
01
6.01
E
-
01
F7
8.32
E
-
01
6.65
E
-
01
1.16
E
-
01
3.13
E
-
02
F8
5.76
E
-
02
2.12
E
-
02
2.87
E
-
02
1.26
E
-
02
F9
1.96
E
-
02
1.13
E
-
02
5.40
E
-
02
7.14
E
-
02
F10
1.15
E
+
00
1.12
E
+
00
1.05
E
-
01
3.56
E
-
02
F11
1.53
E
-
01
1.43
E
-
01
9.49
E
-
02
8.01
E
-
02
F12
1.23
E
-
01
1.19
E
-
01
1.37
E
-
01
2.09
E
-
01
F13
3.59
E
-
01
6.22
E
-
01
5.70
E
-
02
7.82
E
-
02
F14
2.12
E
-
01
2.01
E
-
01
3.39
E
-
01
1.10
E
-
01
F15
5.08
E
-
01
4.46
E
-
01
1.95
E
-
01
1.52
E
-
01
F16
1.31
E
-
01
1.01
E
-
01
4.17
E
-
02
3.35
E
-
02
F17
1.92
E
-
01
1.65
E
-
01
6.37
E
-
01
1.10
E
+
00
F18
4.49
E
-
02
3.44
E
-
02
5.97
E
-
02
6.92
E
-
02
F19
1.27
E
-
01
3.98
E
-
02
9.09
E
-
02
2.74
E
-
02(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.Table7 describes the comparison results of DGSO and some other algorithms when D=100. As shown in Table 7, the proposed DGSO obtains promising results in most of benchmark functions when compared with the other algorithms.Table 7
Average fitness values tested on benchmark functions.
∗
F
DGSO
PSO [10]
DE [5]
RGSO [21]
Mean
Std.
Mean
Std.
Mean
Std.
Mean
Std.
F1
0.00
E
+
00
0.00
E
+
00
4.69
E
+
05
3.22
E
+
04
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F2
5.00
E
+
00
3.61
E
+
00
7.57
E
+
01
6.37
E
+
00
5.21
E
+
01
4.12
E
+
00
3.46
E
-
02
1.02
E
-
03
F3
2.33
E
+
00
2.52
E
+
00
3.26
E
+
11
2.06
E
+
10
2.17
E
+
02
1.1
5
E
+
02
5.69
E
+
01
4.21
E
-
01
F4
0.00
E
+
00
0.00
E
+
00
7.41
E
+
02
4.96
E
+
0
1
6.38
E
+
02
1.01
E
+
02
4.3
E
-
06
2.31
E
-
05
F5
0.00
E
+
00
0.00
E
+
00
9.63
E
-
01
6.15
E
-
0
1
0.00
E
+
00
0.00
E
+
00
4.20
E
+
00
1.03
E
-
01
F6
0.00
E
+
00
0.00
E
+
00
1.96
E
+
01
1.83
E
-
0
1
2.28
E
-
10
1.33
E
-
10
3.72
E
+
00
0.48
E
+
00
F7
0.00
E
+
00
0.00
E
+
00
1.02
E
+
41
2.26
E
+
41
2.49
E
-
11
1.23
E
-
11
4.20
E
+
00
2.03
E
-
01
F8
0.00
E
+
00
0.00
E
+
00
2.60
E
+
05
6.45
E
+
04
3.94
E
+
03
7.69
E
+
02
4.78
E
+
00
4.03
E
-
01
F9
0.00
E
+
00
0.00
E
+
00
5.84
E
+
02
4.06
E
+
01
1.14
E
+
00
9.30
E
-
01
5.13
E
+
00
2.46
E
-
01
F10
0.00
E
+
00
0.00
E
+
00
5.35
E
+
01
3.82
E
+
00
5.25
E
+
00
1.29
E
+
00
3.40
E
+
00
1.02
E
+
00
F11
0.00
E
+
00
0.00
E
+
00
5.77
E
+
02
2.09
E
+
01
5.17
E
+
00
3.32
E
+
00
0.00
E
+
00
0.00
E
+
00
F12
0.00
E
+
00
0.00
E
+
00
3.05
E
+
05
2.71
E
+
04
6.44
E
-
03
9.60
E
-
03
0.00
E
+
00
0.00
E
+
00
F13
3.33
E
+
00
4.16
E
+
00
2.00
E
+
11
6.15
E
+
10
7.96
E
+
01
2.45
E
+
01
4.31
E
+
01
2.10
E
-
01
F14
0.00
E
+
00
0.00
E
+
00
5
.
3
5
E
+
02
5.24
E
+
01
4.67
E
+
02
6.71
E
+
01
3.27
E
+
00
1.04
E
+
00
F15
0.00
E
+
00
0.00
E
+
00
3.79
E
+
27
6.57
E
+
27
4.20
E
-
01
5.75
E
-
01
5.41
E
+
00
1.02
E
+
00
F16
0.00
E
+
00
0.00
E
+
00
3.41
E
+
05
1.89
E
+
04
5.45
E
+
02
2.32
E
-
07
7.02
E
+
00
1.34
E
+
00
F17
5.33
E
+
00
2.31
E
+
00
5.24
E
+
10
7.13
E
+
08
3.59
E
+
10
7.14
E
+
04
2.43
E
+
01
3.69
E
-
01
F18
0.00
E
+
00
0.00
E
+
00
2.05
E
+
05
4.85
E
+
03
8
.
25
E
-
02
6.47
E
-
02
3.21
E
+
00
2.68
E
-
01
F19
5.33
E
+
00
2.08
E
+
00
2.13
E
+
10
9.43
E
+
09
1.93
E
+
01
1.53
E
+
00
2.15
E
+
01
2.98
E
-
01
∗
F: functions.


## 4.1. Experimental Settings

(1)Test Functions. We select 19 classical benchmark functions described in [28], where the functions f1(x)–f6(x) are selected from CEC 2008 Special Session [29], the functions f7(x)–f11(x) were provided from ISDA 2009 Workshop [30], and the functions f12(x)–f19(x) are the hybrid functions based on the previous functions f1(x)–f11(x) [28]. Test functions f1(x)–f11(x) are shown in Table 1; test functions f12(x)–f19(x) 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 D, we set D∈50,100,200. According to [31], MAX_FES is set to 5000∗D. All of the algorithms are terminated when the MAX_FES is reached.Table 1
Test functionsf1(x)–f11(x).
Function
Definition
Domain
Fitness optimum
f
1
(
x
)
∑
i
=
1
D
x
i
2
[−100, 100]D
0
f
2
(
x
)
max
i
⁡
x
i
,
1
≤
i
≤
D
[−100, 100]D
0
f
3
(
x
)
∑
i
=
1
D
100
x
i
2
+
x
i
+
1
2
+
x
i
-
1
2
[−100, 100]D
0
f
4
(
x
)
∑
i
=
1
D
x
i
2
-
10
c
o
s
2
π
x
i
+
10
[−5, 5]D
0
f
5
(
x
)
∑
i
=
1
D
x
i
2
4000
-
∏
i
=
1
D
cos
⁡
x
i
i
+
1
[−600, 600]D
0
f
6
(
x
)
-
20
exp
⁡
-
0.2
1
D
∑
i
=
1
D
x
i
2
-
e
x
p
1
D
∑
i
=
1
D
cos
⁡
2
π
x
i
+
20
+
e
[−32, 32]D
0
f
7
(
x
)
∑
i
=
1
D
x
i
+
∏
i
=
1
D
x
i
[−10, 10]D
0
f
8
(
x
)
∑
i
=
1
D
∑
j
=
1
i
x
j
2
[−65.536, 65.536]D
0
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
[−100, 100]D
0
f
10
(
x
)
∑
i
=
1
D
x
i
2
+
2
x
i
+
1
2
-
0.3
cos
⁡
3
π
x
i
-
0.4
cos
⁡
4
π
x
i
+
1
+
0.7
[−15, 15]D
0
f
11
x
∑
i
=
1
D
-
1
x
i
2
+
x
i
+
1
2
0.25
s
i
n
2
50
⋅
x
i
2
+
x
i
+
1
2
0.1
+
1
[−100, 100]D
0Table 2
Test functionsf12(x)–f19(x).
Function
m
n
s
Domain
FO
f
12
x
=
f
9
x
⊕
f
1
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.25
[−100, 100]D
0
f
1
(
x
)
∑
i
=
1
D
x
i
2
f
13
x
=
f
9
x
⊕
f
3
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.25
[−100, 100]D
0
f
3
(
x
)
∑
i
=
1
D
100
x
i
2
+
x
i
+
1
2
+
x
i
-
1
2
f
14
x
=
f
9
x
⊕
f
4
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.25
[−5, 5]D
0
f
4
(
x
)
∑
i
=
1
D
x
i
2
-
10
c
o
s
2
π
x
i
+
10
f
15
x
=
f
10
x
⊕
f
7
x
f
10
(
x
)
∑
i
=
1
D
x
i
2
+
2
x
i
+
1
2
-
0.3
c
o
s
3
π
x
i
-
0.4
c
o
s
4
π
x
i
+
1
+
0.7
0.25
[−10, 10]D
0
f
7
(
x
)
∑
i
=
1
D
x
i
+
∏
i
=
1
D
x
i
f
16
x
=
f
9
x
⊕
f
1
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.5
[−100, 100]D
0
f
1
(
x
)
∑
i
=
1
D
x
i
2
f
17
x
=
f
9
x
⊕
f
3
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.75
[−100, 100]D
0
f
3
(
x
)
∑
i
=
1
D
100
x
i
2
+
x
i
+
1
2
+
x
i
-
1
2
f
18
x
=
f
9
x
⊕
f
4
x
f
9
(
x
)
∑
i
=
1
D
-
1
f
100
x
i
,
x
i
+
1
+
f
100
x
D
,
x
1 f100=x2+y20.25⋅sin250⋅x2+y20.1+1
0.75
[−5, 5]D
0
f
4
(
x
)
∑
i
=
1
D
x
i
2
-
10
c
o
s
2
π
x
i
+
10
f
19
x
=
f
10
x
⊕
f
7
x
f
10
(
x
)
∑
i
=
1
D
x
i
2
+
2
x
i
+
1
2
-
0.3
c
o
s
3
π
x
i
-
0.4
c
o
s
4
π
x
i
+
1
+
0.7
0.75
[−10, 10]D
0
f
7
(
x
)
∑
i
=
1
D
x
i
+
∏
i
=
1
D
x
i
∗FO denotes fitness optimum.(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 φik to be π/4; the maximum search angle θmax is π/a2, where the value of a is a given number and θmax/2 is the maximum turning angle (see [13]).Table 3
Strategies used in the five GSO algorithms.
Algorithms
Manner of initialization
Manner of evolutionary operators
GSO
Random
Random
RGSO
Random
Opposition-based learning
OGSO
Opposition-based learning
Opposition-based learning
RDGSO
Random
Opposition-based learning and diversity-guided operator
DGSO
Opposition-based learning
Opposition-based learning and diversity-guided operator


## 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 D 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 4
Average fitness values tested on benchmark functions (D=30).
∗
F
GSO
OGSO
RDGSO
DGSO
Mean
Std.
Mean
Std.
Mean
Std.
Mean
Std.
F1
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F2
0.00
E
+
00
0.00
E
+
00
1.43
E
-
05
1.24
E
-
05
0.00
E
+
00
0.00
E
+
00
2.14
E
-
03
2.03
E
-
04
F3
7.99
E
+
01
5.25
E
-
01
6.97
E
+
02
2.86
E
+
01
6.56
E
+
01
6.03
E
-
01
4.55
E
+
01
3.94
E
+
01
F4
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F5
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F6
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F7
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F8
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F9
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F10
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F11
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F12
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F13
6.43
E
+
01
7.47
E
-
01
2.34
E
+
01
2.46
E
+
01
3.93
E
+
01
8.87
E
-
01
2.87
E
+
01
2.49
E
+
01
F14
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F15
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F16
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
1.39
E
-
158
0.00
E
+
00
0.00
E
+
00
F17
3.07
E
+
01
8.95
E
-
01
2.14
E
+
02
2.40
E
+
01
1.66
E
+
01
7.42
E
-
01
9.38
E
+
00
1.63
E
+
01
F18
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F19
2.15
E
+
01
2.98
E
-
01
3.45
E
+
01
2.64
E
+
01
1.14
E
-
02
4.29
E
-
03
6.96
E
-
03
4.98
E
-
03
∗
F: functions.Table5 summarizes the comparison of DGSO, GSO, and RDGSO with the different dimensions (D=50, 100, 200), respectively. It is evident that DGSO performs better than the other reported algorithms especially in case of the high dimension functions.Table 5
Average fitness values tested on benchmark functions with increasing dimension.
Functions
GSO
RDGSO
DGSO
dim = 50
100
200
dim = 50
100
200
dim = 50
100
200
F1
Mean
5.00
E
+
00
4.60
E
+
00
5.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.36
E
+
00
0.00
E
+
00
2.17
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F2
Mean
6.20
E
+
00
5.80
E
+
00
4.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
5.00
E
+
00
3.67
E
+
00
Std.
3.27
E
+
00
0.00
E
+
00
3.91
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
3.61
E
+
00
3.51
E
+
00
F3
Mean
6.60
E
+
00
3.20
E
+
00
4.60
E
+
00
2.40
E
+
00
2.33
E
+
00
2.33
E
+
00
4.80
E
+
00
2.33
E
+
00
4.00
E
+
00
Std.
1.30
E
+
00
4.76
E
-
02
1.52
E
+
00
3.36
E
+
00
2.08
E
+
00
1.53
E
+
00
0.00
E
+
00
2.52
E
+
00
2.65
E
+
00
F4
Mean
6.40
E
+
00
5.20
E
+
00
3.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
2.77
E
+
00
0.00
E
+
00
2.83
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F5
Mean
5.60
E
+
00
4.40
E
+
00
6.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.65
E
+
00
0.00
E
+
00
2.51
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
3.49
E
+
00
0.00
E
+
00
0.00
E
+
00
F6
Mean
4.80
E
+
00
2.40
E
+
00
7.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.91
E
+
00
0.00
E
+
00
1.82
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F7
Mean
3.00
E
+
00
3.00
E
+
00
7.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
1.00
E
+
00
0.00
E
+
00
2.95
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F8
Mean
4.00
E
+
00
5.80
E
+
00
3.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
6.00
E
+
00
Std.
2.17
E
+
00
0.00
E
+
00
3.42
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
1.00
E
+
00
F9
Mean
5.40
E
+
00
6.20
E
+
00
4.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
1.92
E
+
00
0.00
E
+
00
2.88
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F10
Mean
4.20
E
+
00
3.40
E
+
00
5.20
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.13
E
+
00
0.00
E
+
00
3.49
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F11
Mean
3.80
E
+
00
3.40
E
+
00
4.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
2.51
E
+
00
0.00
E
+
00
2.70
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F12
Mean
4.20
E
+
00
4.40
E
+
00
2.40
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
2.30
E
+
00
0.00
E
+
00
2.61
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F13
Mean
6.40
E
+
00
5.80
E
+
00
3.00
E
+
00
3.80
E
+
00
1.00
E
+
00
4.67
E
+
00
3.60
E
+
00
3.33
E
+
00
2.67
E
+
00
Std.
2.86
E
+
00
1.23
E
-
03
2.35
E
+
00
2.59
E
+
00
1.73
E
+
00
5.77
E
-
01
1.82
E
+
00
4.16
E
+
00
3.06
E
+
00
F14
Mean
6.40
E
+
00
4.60
E
+
00
5.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.65
E
+
00
0.00
E
+
00
2.83
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F15
Mean
3.20
E
+
00
3.60
E
+
00
4.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.78
E
+
00
0.00
E
+
00
3.36
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F16
Mean
6.40
E
+
00
6.40
E
+
00
4.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.58
E
+
00
0.00
E
+
00
2.07
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F17
Mean
4.80
E
+
00
4.80
E
+
00
4.40
E
+
00
2.60
E
+
00
5.67
E
+
00
2.33
E
+
00
5.80
E
+
00
5.33
E
+
00
4.00
E
+
00
Std.
4.09
E
+
00
1.71
E
-
06
2.97
E
+
00
2.88
E
+
00
1.15
E
+
00
3.21
E
+
00
3.11
E
+
00
2.31
E
+
00
2.00
E
+
00
F18
Mean
3.60
E
+
00
3.60
E
+
00
1.60
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
Std.
3.51
E
+
00
0.00
E
+
00
1.34
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F19
Mean
5.00
E
+
00
5.00
E
+
00
3.40
E
+
00
5.40
E
+
00
4.67
E
+
00
3.00
E
+
00
2.80
E
+
00
5.33
E
+
00
3.33
E
+
00
Std.
2.92
E
+
00
2.12
E
+
01
2.30
E
+
00
3.36
E
+
00
1.53
E
+
00
1.73
E
+
00
1.79
E
+
00
2.08
E
+
00
1.53
E
+
00(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 f1–f19. It shows that the convergence speed of DGSO is faster than the other four algorithms.Figure 1
Convergence results for test functionsf1–f19.
(a)
f
1
(b)
f
2
(c)
f
3
(d)
f
4
(e)
f
5
(f)
f
6
(g)
f
7
(h)
f
8
(i)
f
9
(j)
f
10
(k)
f
11
(l)
f
12
(m)
f
13
(n)
f
14
(o)
f
15
(p)
f
16
(q)
f
17
(r)
f
18
(s)
f
19(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.Table 6
Diversity values of the OGSO and DGSO.
Functions
OGSO
DGSO
DIV mean
DIV std.
DIV mean
DIV std.
F1
1.19
E
-
02
3.33
E
-
04
2.36
E
-
02
8.77
E
-
03
F2
1.42
E
-
02
3.74
E
-
04
1.198
E
-
01
1.85
E
-
01
F3
2.18
E
-
02
2.00
E
-
02
9.01
E
-
02
1.12
E
-
01
F4
1.11
E
+
00
6.88
E
-
01
1.40
E
+
00
1.93
E
+
00
F5
4.23
E
-
03
3.76
E
-
03
2.15
E
-
03
1.15
E
-
04
F6
1.66
E
-
01
1.35
E
-
01
3.89
E
-
01
6.01
E
-
01
F7
8.32
E
-
01
6.65
E
-
01
1.16
E
-
01
3.13
E
-
02
F8
5.76
E
-
02
2.12
E
-
02
2.87
E
-
02
1.26
E
-
02
F9
1.96
E
-
02
1.13
E
-
02
5.40
E
-
02
7.14
E
-
02
F10
1.15
E
+
00
1.12
E
+
00
1.05
E
-
01
3.56
E
-
02
F11
1.53
E
-
01
1.43
E
-
01
9.49
E
-
02
8.01
E
-
02
F12
1.23
E
-
01
1.19
E
-
01
1.37
E
-
01
2.09
E
-
01
F13
3.59
E
-
01
6.22
E
-
01
5.70
E
-
02
7.82
E
-
02
F14
2.12
E
-
01
2.01
E
-
01
3.39
E
-
01
1.10
E
-
01
F15
5.08
E
-
01
4.46
E
-
01
1.95
E
-
01
1.52
E
-
01
F16
1.31
E
-
01
1.01
E
-
01
4.17
E
-
02
3.35
E
-
02
F17
1.92
E
-
01
1.65
E
-
01
6.37
E
-
01
1.10
E
+
00
F18
4.49
E
-
02
3.44
E
-
02
5.97
E
-
02
6.92
E
-
02
F19
1.27
E
-
01
3.98
E
-
02
9.09
E
-
02
2.74
E
-
02(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.Table7 describes the comparison results of DGSO and some other algorithms when D=100. As shown in Table 7, the proposed DGSO obtains promising results in most of benchmark functions when compared with the other algorithms.Table 7
Average fitness values tested on benchmark functions.
∗
F
DGSO
PSO [10]
DE [5]
RGSO [21]
Mean
Std.
Mean
Std.
Mean
Std.
Mean
Std.
F1
0.00
E
+
00
0.00
E
+
00
4.69
E
+
05
3.22
E
+
04
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
0.00
E
+
00
F2
5.00
E
+
00
3.61
E
+
00
7.57
E
+
01
6.37
E
+
00
5.21
E
+
01
4.12
E
+
00
3.46
E
-
02
1.02
E
-
03
F3
2.33
E
+
00
2.52
E
+
00
3.26
E
+
11
2.06
E
+
10
2.17
E
+
02
1.1
5
E
+
02
5.69
E
+
01
4.21
E
-
01
F4
0.00
E
+
00
0.00
E
+
00
7.41
E
+
02
4.96
E
+
0
1
6.38
E
+
02
1.01
E
+
02
4.3
E
-
06
2.31
E
-
05
F5
0.00
E
+
00
0.00
E
+
00
9.63
E
-
01
6.15
E
-
0
1
0.00
E
+
00
0.00
E
+
00
4.20
E
+
00
1.03
E
-
01
F6
0.00
E
+
00
0.00
E
+
00
1.96
E
+
01
1.83
E
-
0
1
2.28
E
-
10
1.33
E
-
10
3.72
E
+
00
0.48
E
+
00
F7
0.00
E
+
00
0.00
E
+
00
1.02
E
+
41
2.26
E
+
41
2.49
E
-
11
1.23
E
-
11
4.20
E
+
00
2.03
E
-
01
F8
0.00
E
+
00
0.00
E
+
00
2.60
E
+
05
6.45
E
+
04
3.94
E
+
03
7.69
E
+
02
4.78
E
+
00
4.03
E
-
01
F9
0.00
E
+
00
0.00
E
+
00
5.84
E
+
02
4.06
E
+
01
1.14
E
+
00
9.30
E
-
01
5.13
E
+
00
2.46
E
-
01
F10
0.00
E
+
00
0.00
E
+
00
5.35
E
+
01
3.82
E
+
00
5.25
E
+
00
1.29
E
+
00
3.40
E
+
00
1.02
E
+
00
F11
0.00
E
+
00
0.00
E
+
00
5.77
E
+
02
2.09
E
+
01
5.17
E
+
00
3.32
E
+
00
0.00
E
+
00
0.00
E
+
00
F12
0.00
E
+
00
0.00
E
+
00
3.05
E
+
05
2.71
E
+
04
6.44
E
-
03
9.60
E
-
03
0.00
E
+
00
0.00
E
+
00
F13
3.33
E
+
00
4.16
E
+
00
2.00
E
+
11
6.15
E
+
10
7.96
E
+
01
2.45
E
+
01
4.31
E
+
01
2.10
E
-
01
F14
0.00
E
+
00
0.00
E
+
00
5
.
3
5
E
+
02
5.24
E
+
01
4.67
E
+
02
6.71
E
+
01
3.27
E
+
00
1.04
E
+
00
F15
0.00
E
+
00
0.00
E
+
00
3.79
E
+
27
6.57
E
+
27
4.20
E
-
01
5.75
E
-
01
5.41
E
+
00
1.02
E
+
00
F16
0.00
E
+
00
0.00
E
+
00
3.41
E
+
05
1.89
E
+
04
5.45
E
+
02
2.32
E
-
07
7.02
E
+
00
1.34
E
+
00
F17
5.33
E
+
00
2.31
E
+
00
5.24
E
+
10
7.13
E
+
08
3.59
E
+
10
7.14
E
+
04
2.43
E
+
01
3.69
E
-
01
F18
0.00
E
+
00
0.00
E
+
00
2.05
E
+
05
4.85
E
+
03
8
.
25
E
-
02
6.47
E
-
02
3.21
E
+
00
2.68
E
-
01
F19
5.33
E
+
00
2.08
E
+
00
2.13
E
+
10
9.43
E
+
09
1.93
E
+
01
1.53
E
+
00
2.15
E
+
01
2.98
E
-
01
∗
F: functions.


## 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.



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*Source: 546181-2015-12-28.xml*