Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 769245, 16 pages
http://dx.doi.org/10.1155/2015/769245

Research Article

A Hybrid Backtracking Search Optimization Algorithm with Differential Evolution

Lijin Wang,1,2 Yiwen Zhong,¹ Yilong Yin,2,3 Wenting Zhao,² Binqing Wang,² and Yulong Xu²

¹College of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou 350002, China
²School of Computer Science and Technology, Shandong University, Jinan 250101, China
³School of Computer Science and Technology, Shandong University of Finance and Economics, Jinan 250014, China

Received 14 November 2014; Revised 10 February 2015; Accepted 19 February 2015

Academic Editor: Antonio Ruiz-Cortes

Copyright © 2015 Lijin 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.

Abstract.

The backtracking search optimization algorithm (BSA) is a new nature-inspired method which possesses a memory to take advantage of experiences gained from previous generation to guide the population to the global optimum. BSA is capable of solving multimodal problems, but it slowly converges and poorly exploits solution. The differential evolution (DE) algorithm is a robust evolutionary algorithm and has a fast convergence speed in the case of exploitive mutation strategies that utilize the information of the best solution found so far. In this paper, we propose a hybrid backtracking search optimization algorithm with differential evolution, called HBD. In HBD, DE with exploitive strategy is used to accelerate the convergence by optimizing one worse individual according to its probability at each iteration process. A suit of 28 benchmark functions are employed to verify the performance of HBD, and the results show the improvement in effectiveness and efficiency of hybridization of BSA and DE.

1. Introduction

Optimization plays an important role in many fields, for example, decision science and physical system, and can be abstracted as the minimization or maximization of objective functions subject to constraints on their variables mathematically. Generally speaking, the optimization algorithms can be employed to find their solutions. The stochastic relaxation optimization algorithms, such as genetic algorithm (GA) [1], particle swarm optimization algorithm (PSO) [2, 3], ant colony algorithm (ACO) [4], and differential evolution (DE) [5], are one of the methods for solving solutions effectively and almost nature-inspired optimization techniques. For instance, DE, one of the most powerful stochastic optimization methods, employs the mutation, crossover, and selection operators at each generation to drive the population to global optimum. In DE, the mutation operator is one of core components and includes many differential mutation strategies which reveal different characteristics. For example, the strategies, which utilize the information of the best solution found so far, have fast convergence speed and favor exploitation. These strategies are classified as the exploitative strategies [6].

Inspired by the success of GA, PSO, ACO, and DE for solving optimization problems, new nature-inspired algorithms have been a hot topic in the development of the stochastic relaxation optimization techniques, such as artificial bee colony [7], cuckoo search [8], bat algorithm [9], firefly algorithm [10], social emotional optimization [11–13], harmony search [14], and biogeography based optimization [15]. A survey has pointed out that there are about 40 different nature-inspired algorithms [16].

The backtracking search optimization algorithm (BSA) [17] is a new stochastic method for solving real-valued numerical optimization problems. Similar to other evolutionary algorithms, BSA uses the mutation, crossover, and selection operators to generate trial solutions. When generating trial solutions, BSA employs a memory to store experiences gained from previous generation solutions. Taking advantage of historical information to guide the population to global optimum, BSA focuses on exploration and is capable of solving multimodal optimization problems. However, utilizing experiences may make BSA converge slowly and prejudice exploitation on later iteration stage.

On the other hand, researches have paid more and more attention to combine different search optimization algorithms or machine learning methods to improve the performance for real-world optimization problems. Some good surveys about hybrid metaheuristics or machine learning methods can be found in the literatures [18–20]. In this paper, we also concentrate on a hybrid metaheuristic algorithm, called HBD, which combines BSA and DE. HBD employs DE with exploitative mutation strategy to improve convergence speed and to favor exploitation. Furthermore, in HBD, DE is invoked to optimize only one worse individual selected with the help of its probability at each iteration process. We use 28 benchmark functions to verify the performance of HBD, and the results show the improvement in effectiveness and efficiency of hybridization of BSA and DE. The major advantages of our approach are as follows. (i) DE with exploitive strategies helps HBD converge fast and favor exploitation. (ii) Since DE optimizes one individual, HBD expends only one more function evaluation at each iteration and will not increase the overall complexity of BSA. (iii) DE is embedded behind BSA, and therefore HBD does not destroy the structure of BSA, and it is still very simple.

The remainder of this paper is organized as follows. Section 2 describes BSA and DE. Section 3 presents the HBD algorithm. Section 4 reports the experimental results. Section 5 concludes this paper.

2. Preliminary

2.1. BSA

The backtracking search optimization algorithm is a new stochastic search technique developed recently [17]. BSA has a single control parameter and a simple structure that is effective and capable of solving different optimization problems. Furthermore, BSA is a population-based method and possesses a memory in which it stores a population from a randomly chosen previous generation for generating the search-direction matrix. In addition, BSA is a nature-inspired method employing three basic genetic operators: mutation, crossover, and selection.

BSA employs a random mutation strategy that used only one direction individual for each target individual, formulated as follows:

where is the current population, is the historical population, and is a coefficient which controls the amplitude of the search-direction matrix .

BSA also uses a nonuniform and more complex crossover strategy. There are two steps in the crossover process. Firstly, a binary integer-values matrix (map) of size ( and are the population size and the problem dimensions) is generated to indicate the mutant individual to be manipulated by using the relevant individual. Secondly, the relevant dimensions of mutant individual are updated by using the relevant individual. This crossover process can be summarized as shown in Algorithm 1.

Input:   and

output:  

Step 1.

Initiate

  Generate drawn from uniformly distribution with the range between 0 and 1.

If then

     For from 1 to do

            Generate a vector containing a random permutation of the integers from 1 to

            Generate drawn from uniformly distribution with the range between 0 and 1.

     End For

Else

       For from 1 to do

            Generate a random integer from 1 to

     End For

End If

Step 2.

For from 1 to do

     For from 1 to do

            If then

            Else

            End If

     End For

End For

Algorithm 1: Crossover operator.

BSA has two types of selection operators. The first type selection operator is employed to select the historical population for calculating search direction. The rule is that the historical population should be replaced with the current population when the random number is smaller than the other one. The second type of selection operator is greedy to determine the better individuals to go into the next generation.

According to the above descriptions, the pseudocode of BSA is summarized as shown in Algorithm 2.

Initiate the population and the historical population randomly sampled from search space.

While (Stop Condition doesn’t meet)

    Perform the first type selection: in the case of , where   and are drawn from uniformly distribution with the

    range between 0 and 1.

    Permute arbitrary changes in position of oldP.

    Generate the mutant according to (1).

    Generate the population based on Algorithm 1.

    Perform the second type selection: select the population with better fitness from and .

    Update the best solution.

End While

Output the best solution.

Algorithm 2: BSA.

2.2. DE

DE is a powerful evolutionary algorithm for global optimization over continuous space. When being used to solve optimization problems, it evolves a population of candidate solutions with -dimensional parameter vectors, noted as . In DE, the population is initiated by uniform sampling within the prescribed minimum and maximum bounds.

After initialization, DE steps into the iteration process where the evolutionary operators, namely, mutation, crossover, and selection, are invoked in turn, respectively.

DE employs the mutation strategy to generate a mutant vector . So far, there are several mutant strategies, and the most well-known and widely used strategies are listed as follows [21, 22]:

 “DE/best/1”:

 “DE/current-to-best/1”:

 “DE/best/2”:

 “DE/rand/1”:

 “DE/current-to-rand/1”:

 “DE/rand/2”:

 where the indices , , , , and are uniformly random mutually different integers from 1 to , denotes the best individual obtained so far, and and are the th vector of and , respectively.

The crossover operator is performed to generate a trial vector according to each pair of and after the mutant vector is generated. The most popular strategy is the binomial crossover described as follows:

where is called the crossover rate, is randomly sampled from 1 to , and , , and are the th element of , , and , respectively.

Finally, DE uses a greedy mechanism to select the better vector from each pair of and . This can be described as follows:

3. HBD

In this section, we describe the HBD algorithm in detail. First, the motivations of this paper are given. Second, the framework of HBD is shown.

3.1. Motivations

BSA uses an external archive to store experiences gained from previous generation solutions and makes use of them to guide the population to global optimum. According to BSA, permuting arbitrary changes in position of historical population makes the individuals be chosen randomly in the mutation operator; therefore, the algorithm focuses on exploration and is capable of solving multimodal optimization problems. However, just due to random selection, by utilizing experiences, BSA may be led to converge slowly and to prejudice exploitation on later iteration stage. This motivates our approach which aims to accelerate the convergence speed and to enhance the exploitation of the search space to keep the balance between the exploration and exploitation capabilities of BSA.

On the other hand, some studies have investigated the exploration and exploitation ability of different DE mutation strategies and pointed out the mutation operators that incorporate the best individual (e.g., (2), (3), and (4)) favor exploitation because the mutant individuals are strongly attracted around the best individual [6, 23]. This motivates us to hybridize these exploitative mutation strategies to enhance the exploitation capability of BSA. In addition, this paper is also in light of some studies which have shown that it is an effective way to combine other optimization methods to improve the performance for real-world optimization problems [24–27].

3.2. Framework of HBD

Generally speaking, there are many ways to hybridize BSA with DE. In this study, we propose another hybrid schema between BSA with DE. In this schema, HBD employs DE with exploitive strategy behind BSA at each iteration process to share the information between BSA and DE. However, more individuals are optimized by DE, and more function evaluations will be spent. In this case, HBD would gain the premature convergence, resulting in prejudicing exploration. Thus, to keep the exploration capability of HBD, DE is used to optimize only one worse individual according to its probability. In addition, (2) is used as default mutation strategy in HBD because (3) and (4) have stronger exploration capabilities by introducing more perturbation with the random individual [6] or a modification combining “DE/best/1” and “DE/rand/1” [28]. The performance influenced by different exploitative strategies will be discussed in Section 4.3.

In order to select one individual for DE, in this work, we assign a probability model for each individual according to its fitness. It can be formulated as follows:

where is the population size and is the ranking value of each individual when the population is sorted from the worst fitness to the best one.

Note that the probability equation is similar to the selection probability in DE with ranking-based mutation operators [29]. In general, the worse individuals are more far away from the best individual than the better ones; thus, they will have higher probabilities to get around the best one. This selection strategy can be defined as follows:

where is selected individual and optimized by DE.

It is worth pointing out that our previous work [30], called BSADE, splits the whole iteration process into two parts: the previous two-third and the latter one-third stages. BSA is used in the first stage, and DE is employed in the second stage. In this case, DE does not share the population information with BSA. Moreover, it is difficult to split the whole iteration process into two parts. Thus, the difference between HBD and BSADE is that HBD shares the population information between BSA and DE, while BSADE does not. The comparison can be found in Section 4.4.

According to the above descriptions, the pseudocode of HBD is described in Algorithm 3.

Initiate the population and the historical population randomly sampled from search space.

While (Stop Condition doesn’t meet)

    Perform the first type selection: in the case of , where   and are drawn from uniformly distribution with the

    range between 0 and 1.

    Permute arbitrary changes in position of .

    Generate the mutant according to (1).

    Generate the population based on Algorithm 1.

    Perform the second type selection: select the population with better fitness from and .

    Update the best solution.

    //Invoke DE with exploitive strategy

    Select One Individual according to its probability: .

    Optimize with the help of DE, and get

    If (fitness( <= fitness())

    End If

    Update the best solution.

End While

Output the best solution.

Algorithm 3: HBD.

4. Experimental Verifications

In this section, to verify the performance of HBD, we carry out comprehensive experimental tests on a suit of 28 benchmark functions proposed in the CEC-2013 competition [31]. These 28 benchmark functions include 5 unimodal functions , 15 basic multimodal functions , and 8 composition functions . More details about 28 functions can be found in [31].

To make a fair comparison, we use the same parameters for BSA and HBD, unless a change is mentioned. Each algorithm is performed 25 times for each function with the dimensions , 30, and 50, respectively. The population size of each algorithm is when and , while it is 30 in the case of . The maximum function evaluations are . The mutation factor and the crossover factor are 0.8 and 0.9 for HBD, respectively. In addition, we use the boundary handling method given in [17].

To evaluate the performance of algorithms, we use Error as an evaluation indicator first. Error, which is the function error value for the solution obtained by the algorithms, is defined as , where is the global optimum of function. In addition, the average and standard deviation of the best error values, presented as “,” are used in the different tables. Second, the convergence graphs are employed to show the mean error values of the best solutions at iteration process over the total run. Third, a Wilcoxon signed-rank test at the 5% significance level () is used to show the significant differences between two algorithms. The “+” symbol shows that the null hypothesis is rejected at the 5% significant level and HBD outperforms BSA, the “−” symbol says that the null hypothesis is rejected at the 5% significant level and BSA exceeds HBD, and the “” symbol reveals that the null hypothesis is accepted at the 5% significant level and HBD ties BSA. Additionally, we also give the total number of statistical significant cases at the bottom of each table.

4.1. The Effect of HBD

To show the effect of the proposed algorithm, Table 1 lists the average error values obtained by BSA and HBD for 30-dimentional benchmark functions. For unimodal functions , HBD overall obtains better average error values than BSA does. For instance, HBD gains the global optimum on and brings solutions with high quality to in terms of average error values. HBD exhibits a little inferiority to BSA for , but these two approaches are not significant. For 15 basic multimodal functions , with the help of average error values, HBD brings superior solutions to 10 out of 15 functions, equal ones to 2 out of 15 functions, and inferior ones to 3 out of 15 functions. However, according to the results of Wilcoxon test, they are not significant for HBD and BSA for 3 functions in which HBD gains lower solution quality. For composition functions , HBD and BSA draw a tie on and by the aid of average error values; however, HBD significantly outperforms BSA according to the results of Wilcoxon test. Moreover, according to average error values, HBD performs better than BSA in , , , and but worse than BSA in and . Nevertheless, two algorithms almost are not significant for these 8 composition functions in terms of the results of Wilcoxon test. Summarily, according to “+//−,” HBD wins and ties BSA on 12 and 16 out of 28 benchmark functions, respectively.

Table 1: Error values obtained  by  BSA and HBD for 30-dimensional CEC-2013 benchmark functions.

BSA
HBD

AVGEr  ± STDEr
AVGEr  ± STDEr

value

4.17E − 30 ± 1.40E − 29
1.36E − 29 ± 4.17E − 29
=
0.359375

1.22E + 06 ± 5.54E + 05
3.15E + 05 ± 1.52E + 05
+
0.000014

7.52E + 06 ± 8.54E + 06
4.38E + 06 ± 8.11E + 06
+
0.045010

1.25E + 04 ± 3.47E + 03
5.05E + 03 ± 2.23E + 03
+
0.000016

0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
=
1.000000

3.04E + 01 ± 2.54E + 01
1.12E + 01 ± 1.47E + 01
+
0.001721

7.39E + 01 ± 1.03E + 01
5.03E + 01 ± 1.61E + 01
+
0.000101

2.09E + 01 ± 5.95E − 02
2.10E + 01 ± 3.17E − 02
=
0.142532

2.70E + 01 ± 2.29E + 00
2.12E + 01 ± 4.26E + 00
+
0.000081

1.78E − 01 ± 1.34E − 01
9.15E − 02 ± 5.25E − 02
+
0.009417

7.96E − 02 ± 2.75E − 01
3.58E − 01 ± 6.34E − 01
=
0.062500

8.41E + 01 ± 1.51E + 01
8.09E + 01 ± 1.53E + 01
=
0.396679

1.44E + 02 ± 2.37E + 01
1.29E + 02 ± 2.90E + 01
=
0.061480

3.70E + 00 ± 1.68E + 00
2.83E + 00 ± 1.82E + 00
+
0.034670

3.73E + 03 ± 4.27E + 02
3.50E + 03 ± 4.82E + 02
=
0.097970

1.31E + 00 ± 2.14E − 01
1.31E + 00 ± 2.38E − 01
=
0.903627

3.09E + 01 ± 1.97E − 01
3.09E + 01 ± 1.79E − 01
=
0.840072

1.20E + 02 ± 1.80E + 01
9.46E + 01 ± 2.11E + 01
+
0.000980

1.16E + 00 ± 1.79E − 01
1.23E + 00 ± 2.28E − 01
=
0.312970

1.15E + 01 ± 4.55E − 01
1.11E + 01 ± 5.89E − 01
=
0.057836

2.78E + 02 ± 6.63E + 01
2.95E + 02 ± 7.95E + 01
=
0.431762

4.45E + 01 ± 1.91E + 01
4.48E + 01 ± 1.35E + 01
=
0.443172

4.46E + 03 ± 5.40E + 02
4.16E + 03 ± 5.07E + 02
=
0.103553

2.32E + 02 ± 1.17E + 01
2.28E + 02 ± 8.92E + 00
=
0.157770

2.87E + 02 ± 1.41E + 01
2.80E + 02 ± 8.80E + 00
=
0.087527

2.00E + 02 ± 2.17E − 02
2.00E + 02 ± 6.81E − 03
+
0.000020

8.80E + 02 ± 1.49E + 02
7.52E + 02 ± 1.33E + 02
+
0.003822

3.00E + 02 ± 1.65E − 13
3.00E + 02 ± 1.32E − 13
+
0.016377

12/16/0

In order to further show the convergence speed of HBD, the convergence curves of two algorithms for six selected benchmark functions are given in Figure 1.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: The convergence curves of BSA and HBD for selected benchmark functions.

It is observed that the selected functions can be divided into four groups, and overall the convergence performance of HBD is better than BSA. For example, for the first group of functions, for example, and in which HBD has significantly better average error values than BSA, HBD converges faster than BSA in terms of the convergence curves seen in Figures 1(c) and 1(f). For and belong to the second group where HBD cannot bring the solutions with higher quality significantly, HBD still converges faster than BSA does. Third, for in which both of the two algorithms reach the global optimum, convergence performance of HBD is better compared to BSA. Additionally, HBD outperforms BSA according to the convergence curves seen in Figure 1(a), although the average error values optimized by HBD are inferior but not significant to BSA.

All in all, HBD overall outperforms BSA in terms of solution quality and convergence speed. This is because DE with exploitive mutation strategy enhances the exploitation capability of HBD, and it does not expend too much function evaluations.

4.2. Scalability of HBD

In this section, to analyze the performance of HBD affected by the problem dimensionality, a scalability study is investigated, respectively, on the 28 functions with 10- and 50- due to their definition up to 50- [31]. The results are tabulated in Table 2.

Table 2: Error values obtained by BSA and HBD for 10- and 50-dimensional CEC-2013 benchmark functions.

BSA
HBD

BSA
HBD

AVGEr  ± STDEr
AVGEr  ± STDEr

value
AVGEr  ± STDEr
AVGEr  ± STDEr

value

0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
=
1.000000

1.69E − 29 ± 3.04E − 29
1.48E − 29 ± 4.26E − 29
=
0.582031

8.59E + 04 ± 2.30E + 05
4.75E + 03 ± 5.65E + 03
+
0.000065

2.77E + 06 ± 8.09E + 05
1.11E + 06 ± 4.38E + 05
+
0.000014

5.46E + 04 ± 1.36E + 05
1.44E + 02 ± 6.95E + 02
+
0.000012

4.95E + 07 ± 4.00E + 07
1.25E + 07 ± 1.20E + 07
+
0.000140

5.04E + 02 ± 3.66E + 02
5.67E + 01 ± 1.10E + 02
+
0.000014

3.22E + 04 ± 5.16E + 03
2.04E + 04 ± 4.88E + 03
+
0.000023

0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
=
1.000000

3.06E − 38 ± 1.53E − 37
0.00E + 00 ± 0.00E + 00
=
0.250000

1.60E + 00 ± 3.25E + 00
2.75E + 00 ± 4.50E + 00
=
0.220852

5.42E + 01 ± 1.87E + 01
4.42E + 01 ± 7.78E − 01
+
0.000046

6.96E + 00 ± 6.85E + 00
1.73E + 00 ± 2.49E + 00
+
0.045010

8.48E + 01 ± 1.02E + 01
8.38E + 01 ± 1.06E + 01
=
0.756995

2.04E + 01 ± 7.79E − 02
2.04E + 01 ± 5.55E − 02
=
0.989266

2.11E + 01 ± 3.33E − 02
2.11E + 01 ± 2.32E − 02
=
0.562928

3.54E + 00 ± 8.43E − 01
2.47E + 00 ± 1.29E + 00
+
0.006313

5.44E + 01 ± 3.73E + 00
5.50E + 01 ± 3.85E + 00
=
0.657069

8.49E − 02 ± 3.51E − 02
7.75E − 02 ± 5.50E − 02
=
0.989266

3.99E − 01 ± 2.11E − 01
1.14E − 01 ± 6.51E − 02
+
0.000018

0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
=
1.000000

7.96E − 02 ± 2.75E − 01
0.00E + 00 ± 0.00E + 00
=
0.062500

1.10E + 01 ± 3.31E + 00
1.09E + 01 ± 3.48E + 00
=
0.599802

1.92E + 02 ± 2.87E + 01
1.68E + 02 ± 2.65E + 01
+
0.007423

1.68E + 01 ± 7.18E + 00
1.60E + 01 ± 7.27E + 00
=
0.736617

3.25E + 02 ± 3.81E + 01
2.88E + 02 ± 4.17E + 01
+
0.001569

1.31E − 01 ± 5.44E − 02
1.28E − 01 ± 8.94E − 02
+
0.000296

2.17E + 01 ± 5.08E + 00
2.12E + 01 ± 4.56E + 00
=
0.287862

6.10E + 02 ± 1.78E + 02
4.67E + 02 ± 1.74E + 02
+
0.022988

8.25E + 03 ± 4.91E + 02
8.13E + 03 ± 4.40E + 02
=
0.142532

6.54E − 01 ± 1.92E − 01
6.23E − 01 ± 2.50E − 01
=
0.989266

1.88E + 00 ± 2.67E − 01
1.82E + 00 ± 2.86E − 01
=
0.427339

7.77E + 00 ± 2.88E + 00
8.37E + 00 ± 2.87E + 00
=
0.427339

5.44E + 01 ± 6.11E − 01
5.45E + 01 ± 6.85E − 01
=
0.924971

2.67E + 01 ± 4.85E + 00
2.05E + 01 ± 4.06E + 00
+
0.000665

2.65E + 02 ± 2.44E + 01
2.45E + 02 ± 3.23E + 01
+
0.039554

2.45E − 01 ± 9.51E − 02
2.15E − 01 ± 1.19E − 01
=
0.657069

2.67E + 00 ± 3.30E − 01
2.80E + 00 ± 2.59E − 01
=
0.312970

2.92E + 00 ± 3.60E − 01
2.57E + 00 ± 4.67E − 01
+
0.019941

2.09E + 01 ± 7.34E − 01
2.08E + 01 ± 4.44E − 01
=
0.819095

3.16E + 02 ± 1.32E + 02
3.04E + 02 ± 1.21E + 02
=
0.444824

6.94E + 02 ± 4.53E + 02
8.72E + 02 ± 3.27E + 02
=
0.675764

1.49E + 01 ± 4.29E + 00
1.42E + 01 ± 4.11E + 00
=
0.544910

5.96E + 01 ± 1.39E + 01
6.10E + 01 ± 9.87E + 00
=
0.544910

8.60E + 02 ± 1.53E + 02
8.06E + 02 ± 2.04E + 02
=
0.818641

9.53E + 03 ± 6.62E + 02
9.41E + 03 ± 7.87E + 02
=
0.599802

1.50E + 02 ± 3.69E + 01
1.56E + 02 ± 4.44E + 01
=
0.716423

2.70E + 02 ± 1.04E + 01
2.58E + 02 ± 1.09E + 01
+
0.001569

1.93E + 02 ± 2.44E + 01
1.84E + 02 ± 3.39E + 01
=
0.241820

3.79E + 02 ± 1.40E + 01
3.65E + 02 ± 2.10E + 01
+
0.013817

1.13E + 02 ± 5.23E + 00
1.11E + 02 ± 1.87E + 01
=
0.109386

2.00E + 02 ± 7.33E − 02
2.00E + 02 ± 4.44E − 02
+
0.000012

3.13E + 02 ± 2.39E + 01
3.32E + 02 ± 4.73E + 01
=
0.676637

1.53E + 03 ± 1.89E + 02
1.39E + 03 ± 2.01E + 02
+
0.028314

2.20E + 02 ± 1.00E + 02
2.04E + 02 ± 1.02E + 02
=
0.802856

4.00E + 02 ± 1.33E − 13
4.00E + 02 ± 1.43E − 13
+
0.000001

9/19/0

13/15/0

In the case of , according to average error values shown in Table 2, HBD exhibits superiority in the majority of functions while inferiority in a handful of ones. Additionally, in terms of the total of “+//−,” HBD wins and ties BSA in 9 and 19 out of 28 functions, respectively.

When , HBD still can bring solutions with higher quality than BSA does in most of benchmark functions. Moreover, HBD outperforms and ties BSA in 13 and 15 out of 28 functions, respectively.

In summary, it suggests that the advantage of HBD over BSA is stable when the dimensionality of problems increases.

4.3. The Effect of Mutation Strategy

In HBD, the “DE/best/1” mutation strategy is used to enhance the exploitation capability of HBD in default. To show the performance of HBD influenced by other exploitive mutation strategies, the experiments are carried on benchmark functions and the results are listed in Table 3 where cHBD and bHBD mean that HBD uses “DE/current-to-best/1” and “DE/best/2,” respectively. The results obtained by cHBD and bHBD, which are highly accurate compared to those obtained by HBD, are marked in bold.

Table 3: Error values obtained by cHBD, HBD, and bHBD for CEC-2013 benchmark functions at .

cHBD
HBD
bHBD

AVGEr  ± STDEr

value
AVGEr  ± STDEr

value
AVGEr  ± STDEr

1.01E − 30 ± 3.49E − 30
=
0.125000
1.36E − 29 ± 4.17E − 29
=
0.328125
6.31E − 30 ± 1.67E − 29

2.97E + 05 ± 1.51E + 05
=
0.676637
3.15E + 05 ± 1.52E + 05
+
0.000240
7.07E + 05 ± 3.84E + 05

3.77E + 06 ± 7.51E + 06
=
0.618641
4.38E + 06 ± 8.11E + 06
=
0.637733
4.25E + 06 ± 7.27E + 06

4.72E + 03 ± 1.90E + 03
=
0.736617
5.05E + 03 ± 2.23E + 03
+
0.006848
6.55E + 03 ± 2.23E + 03

0.00E + 00 ± 0.00E + 00
=
1.000000
0.00E + 00 ± 0.00E + 00
=
1.000000
0.00E + 00 ± 0.00E + 00

1.12E + 01 ± 1.34E + 01
=
0.326049
1.12E + 01 ± 1.47E + 01
=
0.562928
8.98E + 00 ± 5.93E + 00

5.69E + 01 ± 1.50E + 01
=
0.142532
5.03E + 01 ± 1.61E + 01
+
0.003822
6.58E + 01 ± 1.69E + 01

2.09E + 01 ± 4.71E − 02
=
0.264150
2.10E + 01 ± 3.17E − 02
=
0.287862
2.09E + 01 ± 4.67E − 02

2.61E + 01 ± 3.12E + 00
+
0.000126
2.12E + 01 ± 4.26E + 00
+
0.000446
2.65E + 01 ± 3.29E + 00

8.18E − 02 ± 5.50E − 02
=
0.509755
9.15E − 02 ± 5.25E − 02
=
0.300241
8.06E − 02 ± 4.77E − 02

0.00E + 00 ± 0.00E + 00
−
0.015625
3.58E − 01 ± 6.34E − 01
−
0.015625
0.00E + 00 ± 0.00E + 00

8.19E + 01 ± 1.61E + 01
=
0.427339
8.09E + 01 ± 1.53E + 01
=
0.065311
9.01E + 01 ± 1.78E + 01

1.26E + 02 ± 2.58E + 01
=
0.798248
1.29E + 02 ± 2.90E + 01
=
0.174210
1.39E + 02 ± 2.67E + 01

4.01E + 00 ± 1.94E + 00
+
0.032428
2.83E + 00 ± 1.82E + 00
=
0.082653
3.88E + 00 ± 2.08E + 00

3.80E + 03 ± 4.02E + 02
+
0.017253
3.50E + 03 ± 4.82E + 02
+
0.014889
3.81E + 03 ± 2.84E + 02

1.29E + 00 ± 2.45E − 01
=
0.967806
1.31E + 00 ± 2.38E − 01
=
0.924971
1.30E + 00 ± 2.09E − 01

3.10E + 01 ± 1.75E − 01
=
0.165837
3.09E + 01 ± 1.79E − 01
=
0.989266
3.09E + 01 ± 1.55E − 01

1.17E + 02 ± 1.58E + 01
+
0.000665
9.46E + 01 ± 2.11E + 01
+
0.021418
1.10E + 02 ± 1.99E + 01

1.27E + 00 ± 2.22E − 01
=
0.427339
1.23E + 00 ± 2.28E − 01
=
0.924971
1.22E + 00 ± 2.55E − 01

1.12E + 01 ± 6.80E − 01
=
0.443172
1.11E + 01 ± 5.89E − 01
=
0.121828
1.14E + 01 ± 4.65E − 01

3.18E + 02 ± 8.04E + 01
=
0.300009
2.95E + 02 ± 7.95E + 01
+
0.040267
3.49E + 02 ± 8.29E + 01

5.28E + 01 ± 2.67E + 01
=
0.367385
4.48E + 01 ± 1.35E + 01
=
0.696425
4.58E + 01 ± 2.05E + 01

4.37E + 03 ± 4.03E + 02
=
0.121828
4.16E + 03 ± 5.07E + 02
+
0.039554
4.48E + 03 ± 4.85E + 02

2.28E + 02 ± 7.11E + 00
=
0.861162
2.28E + 02 ± 8.92E + 00
=
0.946369
2.28E + 02 ± 8.83E + 00

2.85E + 02 ± 2.02E + 01
=
0.054374
2.80E + 02 ± 8.80E + 00
=
0.275832
2.85E + 02 ± 1.03E + 01

2.00E + 02 ± 8.66E − 03
=
0.637733
2.00E + 02 ± 6.81E − 03
+
0.022988
2.00E + 02 ± 9.50E − 03

7.68E + 02 ± 1.45E + 02
=
0.736617
7.52E + 02 ± 1.33E + 02
=
0.092631
8.36E + 02 ± 1.67E + 02

2.94E + 02 ± 2.96E + 01
=
0.161513
3.00E + 02 ± 1.32E − 13
=
0.808365
3.00E + 02 ± 1.36E − 13

4/23/1

9/18/1

From Table 3, in terms of the average error values, bHBD shows the higher accuracy compared to HBD for a few functions since “DE/best/2” usually exhibits better exploration than “DE/best/1” because of one more difference of randomly selected individuals in the former [23]. cHBD also gains higher accuracy of solutions than HBD does for a handful of functions because “DE/current-best/1,” a modification combining “DE/best/1” and “DE/rand/1” [28], shows better exploration than “DE/best/1.” In other words, for a few functions, “DE/best/2” and “DE/current-best/1” can balance the exploration and exploitation capabilities of HBD better. For example, bHBD and cHBD bring the solutions with higher quality to , , , , , and ; in particular, they reach the global optimum. However, for most of the functions, HBD with “DE/best/1” performs better than cHBD and bHBD.

Additionally, Table 4 reports the results of the multiple-problem Wilcoxon test which was done similarly in [29, 32] between HBD and its variants for all functions. We can see from Table 4 that HBD is significantly better than bHBD and HBD gets higher value than value although two values are not significant. Therefore, HBD uses “DE/best/1” in the tradeoff.

Table 4: Results of the multiple-problem Wilcoxon test for HBD, cHBD, and bHBD for at .

Algorithm

-value

HBD versus cHBD
235
143
0.269095
=
=

HBD versus bHBD
276
75
0.010695
+
+

4.4. The Effect of Hybrid Schema

In this section, we analyze the performance of HBD affected by the hybrid schema. Firstly, to show the effect of more than one individual optimized by DE, the algorithm, called aHBD which uses DE to optimize the whole population, is used to compare with HBD. Secondly, we add a probability on aHBD to control the use of DE and propose paHBD. In paHBD, if the random number drawn from uniform distribution between 0 and 1 is less than , then DE is invoked. The is defined as follows:

where is the number of function evaluations which had been spent and is the maximum number of function evaluations. Additionally, BSADE is compared with HBD to show their differences.

Table 5 lists the error values obtained by aHBD, paHBD, BSADE, and HBD for 28 functions at . It can be observed that HBD wins, ties, and loses aHBD in 10, 12, and 6 out of 28 functions in terms of “+//−,” respectively. It says that optimizing more individuals using DE costs more function evaluations when DE is embedded behind BSA directly, resulting in reducing the iteration process cycles and then getting poor performance for most functions. Regarding BSADE, since BSA and DE are invoked in different stages where they cannot exchange the population information, it is clear that this schema cannot balance the exploitation and exploration well. Thus, compared with BSADE, HBD brings solutions with higher accuracy for most functions. Moreover, HBD wins, ties, and loses BSADE in 8, 17, and 3 out of 28 functions with the help of “+//−,” respectively. However, paHBD uses the probability to control the use of DE. In this case, it can decrease the cost of function evaluation at early evolution stage. Thus, paHBD is almost similar to HBD according to “+//−.”

Table 5: Error values obtained by aHBD, HBD, and paHBD for CEC-2013 benchmark functions at .

aHBD
paHBD
BSADE
HBD

AVGEr  ± STDEr

value
AVGEr  ± STDEr

value
AVGEr  ± STDEr

value
AVGEr  ± STDEr

1.29E − 28 ± 1.18E − 28
+
0.000292
5.53E − 29 ± 8.39E − 29
+
0.030946
1.11E − 29 ± 4.11E − 29
=
0.666016
1.36E − 29 ± 4.17E − 29

5.01E + 04 ± 2.88E + 04
−
0.000012
5.64E + 04 ± 3.39E + 04
−
0.000012
6.80E + 04 ± 3.33E + 04
−
0.000012
3.15E + 05 ± 1.52E + 05

1.13E + 07 ± 3.20E + 07
=
0.924971
1.74E + 06 ± 2.61E + 06
=
0.201222
5.21E + 06 ± 1.24E + 07
=
0.696425
4.38E + 06 ± 8.11E + 06

5.61E + 01 ± 5.41E + 01
−
0.000012
2.40E + 02 ± 2.17E + 02
−
0.000012
4.02E + 02 ± 2.24E + 02
−
0.000012
5.05E + 03 ± 2.23E + 03

4.26E − 16 ± 1.48E − 15
+
0.007813
2.84E − 16 ± 1.42E − 15
=
0.062500
0.00E + 00 ± 0.00E + 00
=
1.000000
0.00E + 00 ± 0.00E + 00

5.01E + 00 ± 5.53E + 00
−
0.017253
4.67E + 00 ± 3.66E + 00
−
0.003822
1.18E + 01 ± 1.25E + 01
=
0.396679
1.12E + 01 ± 1.47E + 01

4.31E + 01 ± 1.62E + 01
=
0.097970
4.74E + 01 ± 1.50E + 01
=
0.492633
5.87E + 01 ± 1.63E + 01
=
0.097970
5.03E + 01 ± 1.61E + 01

2.09E + 01 ± 4.16E − 02
=
0.300241
2.09E + 01 ± 6.05E − 02
=
0.509755
2.09E + 01 ± 4.74E − 02
=
0.210872
2.10E + 01 ± 3.17E − 02

2.18E + 01 ± 3.56E + 00
=
0.861162
2.18E + 01 ± 3.42E + 00
=
0.637733
2.60E + 01 ± 2.82E + 00
+
0.000296
2.12E + 01 ± 4.26E + 00

6.25E − 02 ± 3.76E − 02
−
0.014889
6.11E − 02 ± 3.92E − 02
−
0.022988
2.04E − 01 ± 1.19E − 01
+
0.000126
9.15E − 02 ± 5.25E − 02

3.03E + 01 ± 1.53E + 01
+
0.000012
3.06E + 00 ± 2.05E + 00
+
0.000083
7.96E − 02 ± 2.75E − 01
=
0.062500
3.58E − 01 ± 6.34E − 01

7.43E + 01 ± 1.98E + 01
=
0.264150
8.12E + 01 ± 1.39E + 01
=
0.967806
8.64E + 01 ± 2.23E + 01
=
0.241820
8.09E + 01 ± 1.53E + 01

1.38E + 02 ± 2.88E + 01
=
0.509755
1.35E + 02 ± 2.66E + 01
=
0.367385
1.43E + 02 ± 1.76E + 01
=
0.054374
1.29E + 02 ± 2.90E + 01

5.85E + 01 ± 5.92E + 01
+
0.000012
7.23E + 00 ± 4.53E + 00
+
0.000266
7.33E + 00 ± 4.72E + 00
+
0.000808
2.83E + 00 ± 1.82E + 00

3.45E + 03 ± 4.72E + 02
=
0.339479
3.49E + 03 ± 5.12E + 02
=
0.861162
3.39E + 03 ± 3.92E + 02
=
0.381860
3.50E + 03 ± 4.82E + 02

1.49E + 00 ± 3.48E − 01
+
0.008041
1.14E + 00 ± 5.92E − 01
=
0.676637
1.23E + 00 ± 5.19E − 01
=
0.882352
1.31E + 00 ± 2.38E − 01

3.92E + 01 ± 9.55E + 00
+
0.000012
3.17E + 01 ± 7.18E − 01
+
0.000023
3.17E + 01 ± 5.55E − 01
+
0.000025
3.09E + 01 ± 1.79E − 01

1.05E + 02 ± 2.28E + 01
=
0.121828
8.78E + 01 ± 2.07E + 01
=
0.300241
8.84E + 01 ± 1.83E + 01
=
0.411840
9.46E + 01 ± 2.11E + 01

2.10E + 00 ± 1.09E + 00
+
0.000036
1.44E + 00 ± 2.94E − 01
+
0.011000
1.47E + 00 ± 2.13E − 01
+
0.000602
1.23E + 00 ± 2.28E − 01

1.07E + 01 ± 5.66E − 01
−
0.004530
1.13E + 01 ± 4.56E − 01
=
0.562928
1.13E + 01 ± 6.33E − 01
=
0.427339
1.11E + 01 ± 5.89E − 01

3.26E + 02 ± 9.08E + 01
=
0.121477
3.41E + 02 ± 9.20E + 01
=
0.061407
2.81E + 02 ± 6.67E + 01
=
0.894867
2.95E + 02 ± 7.95E + 01

2.47E + 02 ± 2.57E + 02
+
0.000014
5.78E + 01 ± 3.61E + 01
=
0.492633
6.63E + 01 ± 2.99E + 01
+
0.002947
4.48E + 01 ± 1.35E + 01

3.86E + 03 ± 5.21E + 02
=
0.252813
4.21E + 03 ± 5.58E + 02
=
0.339479
4.19E + 03 ± 4.20E + 02
=
0.819095
4.16E + 03 ± 5.07E + 02

2.36E + 02 ± 8.94E + 00
+
0.008705
2.27E + 02 ± 9.76E + 00
=
0.562928
2.35E + 02 ± 9.21E + 00
+
0.009417
2.28E + 02 ± 8.92E + 00

2.75E + 02 ± 1.32E + 01
=
0.103553
2.80E + 02 ± 1.02E + 01
=
0.756995
2.86E + 02 ± 1.34E + 01
=
0.069337
2.80E + 02 ± 8.80E + 00

2.11E + 02 ± 3.75E + 01
+
0.002470
2.00E + 02 ± 3.02E − 03
−
0.000016
2.00E + 02 ± 1.85E − 03
−
0.000012
2.00E + 02 ± 6.81E − 03

7.42E + 02 ± 1.07E + 02
=
0.882352
7.58E + 02 ± 1.52E + 02
=
0.989266
9.19E + 02 ± 1.49E + 02
+
0.000891
7.52E + 02 ± 1.33E + 02

3.00E + 02 ± 1.90E − 13
+
0.016377
3.00E + 02 ± 1.95E − 13
+
0.038947
3.00E + 02 ± 1.65E − 13
=
1.000000
3.00E + 02 ± 1.32E − 13

10/12/6

6/17/5
8/17/3

In addition, we also perform the multiple-problem Wilcoxon test for HBD, aHBD, paHBD, and BSADE for 28 functions and list the results in Table 6.

Table 6: Results of the multiple-problem Wilcoxon test for HBD, aHBD, paHBD, and BSADE for at .

Algorithm

value

HBD versus aHBD
226
180
0.600457
=
=

HBD versus paHBD
197
209
0.891321
=
=

HBD versus BSADE
262.5
143.5
0.175450
=
=

It can be found from Table 6 that HBD is not significant to aHBD, paHBD, and BSADE. But HBD gets higher values than values, compared with aHBD and BSADE, respectively. But HBD obtains slightly lower value than value in comparison with paHBD. This is because HBD brings weakly lower accurate solutions on , , , and , resulting in higher ranking. Nevertheless, it indicates that the hybrid schema used in HBD is a reasonable choice.

4.5. The Effect of Probability Model

In HBD, a linear model seen (10) is used to select one individual to optimize. It is worth pointing out that other models, for example, nonlinear, can also be adopted in our algorithm. In this section, we do not seek the optimal probability model but only analyze the performance influenced by different models. Thus, two models, as similarly used in [29, 33], are employed to study the performance affected by other models. They are the quadratic model and the sinusoidal model, formulated as seen in (13) and (14), respectively. The average error values and the results of the multiple-problem Wilcoxon test are reported in Tables 7 and 8, respectively, where qHBD is HBD with the quadratic model and sHBD means HBD with the sinusoidal one. Consider

Table 7: Error values obtained by qHBD, HBD, and sHBD for CEC-2013 benchmark functions at .

qHBD
HBD
sHBD

AVGEr  ± STDEr

value
AVGEr  ± STDEr

value
AVGEr  ± STDEr

5.05E − 31 ± 2.52E − 30
=
0.078125
1.36E − 29 ± 4.17E − 29
=
0.429688
6.18E − 30 ± 2.22E − 29

3.56E + 05 ± 2.89E + 05
=
0.798248
3.15E + 05 ± 1.52E + 05
=
0.736617
2.87E + 05 ± 1.36E + 05

4.94E + 06 ± 7.32E + 06
=
0.618641
4.38E + 06 ± 8.11E + 06
=
0.756995
3.57E + 06 ± 6.05E + 06

4.69E + 03 ± 1.64E + 03
=
0.736617
5.05E + 03 ± 2.23E + 03
=
0.903627
5.20E + 03 ± 2.96E + 03

2.84E − 16 ± 1.42E − 15
=
1.000000
0.00E + 00 ± 0.00E + 00
=
1.000000
0.00E + 00 ± 0.00E + 00

1.05E + 01 ± 1.31E + 01
=
0.381860
1.12E + 01 ± 1.47E + 01
=
0.861162
9.06E + 00 ± 6.44E + 00

5.68E + 01 ± 1.66E + 01
=
0.191898
5.03E + 01 ± 1.61E + 01
=
0.756995
5.19E + 01 ± 1.93E + 01

2.09E + 01 ± 6.58E − 02
=
0.509755
2.10E + 01 ± 3.17E − 02
=
0.165837
2.09E + 01 ± 5.52E − 02

2.38E + 01 ± 3.78E + 00
+
0.039554
2.12E + 01 ± 4.26E + 00
=
0.231167
2.24E + 01 ± 3.57E + 00

9.15E − 02 ± 6.35E − 02
=
0.264150
9.15E − 02 ± 5.25E − 02
=
0.527183
9.80E − 02 ± 6.25E − 02

4.38E − 01 ± 9.56E − 01
=
0.986328
3.58E − 01 ± 6.34E − 01
=
0.366699
1.99E − 01 ± 4.06E − 01

7.44E + 01 ± 1.79E + 01
=
0.073565
8.09E + 01 ± 1.53E + 01
=
0.374558
7.83E + 01 ± 1.32E + 01

1.26E + 02 ± 2.53E + 01
=
0.989266
1.29E + 02 ± 2.90E + 01
=
0.411840
1.26E + 02 ± 2.94E + 01

3.14E + 00 ± 1.74E + 00
=
0.618641
2.83E + 00 ± 1.82E + 00
=
0.051087
3.66E + 00 ± 1.60E + 00

3.60E + 03 ± 4.07E + 02
=
0.903627
3.50E + 03 ± 4.82E + 02
=
0.676637
3.49E + 03 ± 3.98E + 02

1.29E + 00 ± 2.55E − 01
=
0.967806
1.31E + 00 ± 2.38E − 01
=
0.696425
1.28E + 00 ± 2.15E − 01

3.10E + 01 ± 1.86E − 01
=
0.492633
3.09E + 01 ± 1.79E − 01
=
0.946369
3.09E + 01 ± 2.11E − 01

9.93E + 01 ± 1.75E + 01
=
0.396679
9.46E + 01 ± 2.11E + 01
=
0.840072
9.52E + 01 ± 1.75E + 01

1.20E + 00 ± 2.23E − 01
=
0.736617
1.23E + 00 ± 2.28E − 01
=
0.210872
1.26E + 00 ± 2.04E − 01

1.09E + 01 ± 7.81E − 01
=
0.165837
1.11E + 01 ± 5.89E − 01
=
0.191898
1.14E + 01 ± 5.24E − 01

3.10E + 02 ± 8.68E + 01
=
0.480701
2.95E + 02 ± 7.95E + 01
=
0.165492
3.28E + 02 ± 8.02E + 01

4.19E + 01 ± 1.90E + 01
=
0.443172
4.48E + 01 ± 1.35E + 01
=
0.287862
4.24E + 01 ± 1.86E + 01

4.16E + 03 ± 4.19E + 02
=
0.924971
4.16E + 03 ± 5.07E + 02
=
0.840072
4.12E + 03 ± 3.91E + 02

2.31E + 02 ± 1.02E + 01
=
0.637733
2.28E + 02 ± 8.92E + 00
=
0.777543
2.28E + 02 ± 9.52E + 00

2.76E + 02 ± 9.42E + 00
−
0.042207
2.80E + 02 ± 8.80E + 00
=
0.157770
2.75E + 02 ± 1.43E + 01

2.00E + 02 ± 6.05E − 03
=
0.777543
2.00E + 02 ± 6.81E − 03
=
0.840072
2.00E + 02 ± 6.31E − 03

7.82E + 02 ± 1.48E + 02
=
0.287862
7.52E + 02 ± 1.33E + 02
=
0.459336
7.90E + 02 ± 1.21E + 02

3.00E + 02 ± 1.14E − 13
=
1.000000
3.00E + 02 ± 1.32E − 13
−
0.025347
3.00E + 02 ± 8.76E − 14

1/26/1

0/27/1

Table 8: Results of the multiple-problem Wilcoxon test for HBD, qHBD, and sHBD for at .

Algorithm

value

HBD versus qHBD
238
140
0.239106
=
=

HBD versus sHBD
143
208
0.409125
=
=

From Table 7, we can find that qHBD can bring higher solutions to 11 out of 28 functions compared with HBD, although the results they obtain are not significant in terms of “+//−.” In addition, qHBD gets lower values than values HBD gained, though they are not significant at the 5% and 10% significance level. It says that the linear model is a reasonable choice compared with the quadratic model. However, it is not the optimal one compared with the sinusoidal model. For instance, sHBD wins, ties, and loses HBD in 1, 27, and 0 out of 28 functions according to “+//−.” Moreover, sHBD has higher values than values HBD does though they are not significant at the 5% and 10% significance level.

4.6. Compared with Other Algorithms

Firstly, HBD is compared with 6 non-BSA approaches in [17], namely, PSO2011 [34], CMAES [35, 36], ABC [7], JDE [37], CLPSO [38], and SADE [39]. Moreover, to compare fair and conveniently, we use the 25 functions and the parameters which are employed and suggested in [17]. More details about these 25 functions can be found in CEC-2005 competition [40]. Table 9 lists the minimal fitness and average fitness of 7 approaches, where the results of 6 non-BSA algorithms are adopted from [17] directly. In addition, the results of multiple-problem Wilcoxon test and Friedman test similarly done in [29] for the seven algorithms are listed in Tables 10 and 11, respectively.

Table 9: Fitness obtained by HBD and 6 non-BSAs for CEC-2005 functions at .

PSO2011
CMAES
ABC
JDE
CLPSO
SADE
HBD

Mean
−450.0000000000000000
−450.0000000000000000
−450.0000000000000000
−450.0000000000000000
−450.0000000000000000
−450.0000000000000000
−450.0000000000000000

Std.
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000

Mean
−450.0000000000000000
−450.0000000000000000
−449.9999999999220000
−450.0000000000000000
−418.8551838547760000
−450.0000000000000000
−450.0000000000000000

Std.
0.0000000000000350
0.0000000000000000
0.0000000002052730
0.0000000000000615
51.0880511039985000
0.0000000000000000
0.0000000000000000

Mean
−44.5873911956554000
−450.0000000000000000
387131.2441213970000000
−197.9999999999850000
62142.0000000000000000
245.0483283713550000
−449.9999999999980000

Std.
458.5794120016290000
0.0000000000000000
166951.7336592640000000
391.5169437474990000
34796.1785167236000000
790.6056596723160000
0.0000000000012208

Mean
−450.0000000000000000
77982.4567046980000000
140.4509447125110000
−414.0000000000000000
−178.8320689185280000
−450.0000000000000000
−450.0000000000000000

Std.
0.0000000000000460
131376.7365456010000000
217.2646715063190000
55.9309919639279000
394.8667499339530000
0.0000000000000000
0.0000000000000000

Mean
−310.0000000000000000
−310.0000000000000000
−291.5327549384120000
−271.0000000000000000
333.4108259915760000
−309.9999999999960000
−309.9999999999990000

Std.
0.0000000000000000
0.0000000000000000
17.6942171217937000
60.5919079609218000
512.6920837704510000
0.0000000000133965
0.0000000000017057

Mean
393.4959999056240000
390.5315438816460000
391.2531452421960000
231.3986579112350000
405.5233436479650000
390.2657719408230000
390.2657719408230000

Std.
16.0224965900462000
1.3783433976373800
3.7254660805238600
247.2968415284400000
10.7480096852869000
1.0114275384776600
1.0114275384776500

Mean
1091.0644335162500000
1087.2645466786700000
1087.0459486286000000
1141.0459486286000000
1087.0459486286000000
1087.0459486286000000
−179.9480783793820000

Std.
3.4976948942723200
0.5365230018001780
0.0000000000005585
83.8964879458918000
0.0000000000004264
0.0000000000004814
0.0353036104861447

Mean
−119.8190232990920000
−119.9261073509850000
−119.7446063439080000
−119.4459380180300000
−119.9300269839980000
−119.7727713703720000
−119.8110293720060000

Std.
0.0720107560874199
0.1554021446157740
0.0623866434489108
0.0927418223065644
0.0417913553101429
0.1248514853682450
0.0915374510176448

Mean
−324.6046006320200000
−306.5782069681560000
−330.0000000000000000
−329.8673387923880000
−329.4361898676470000
−329.9668346980970000
−329.9668346980960000

Std.
2.5082306041521000
21.9475396048756000
0.0000000000000000
0.3440030182812760
0.6229063711904190
0.1816538397880230
0.1816538397880230

Mean
−324.3311322538170000
−314.7871102989330000
−306.7949047862760000
−319.6763749798700000
−321.7278926895280000
−322.9689591871600000
−320.3489045380250000

Std.
3.0072222933667300
8.3115989308305500
5.1787864195870400
4.9173541245304800
1.8971778613701300
2.8254645254663600
4.1899978130687500

Mean
92.5640111212146000
90.7642785704506000
94.8428485804138000
93.2972315784963000
94.6109567642977000
91.6859083842723000
92.0330962077418000

Std.
1.5827416781636900
26.4613831425879000
0.6869412813090850
1.8766951726453600
0.6689129174038950
0.9033073777915270
1.4570152623440700

Mean
18611.3142254809000000
−70.0486708747625000
−337.3273080760500000
400.3240208136310000
−447.8870804905020000
−394.5206365378250000
−453.7580906206240000

Std.
12508.7866126316000000
637.4585182420270000
56.5730759032367000
688.3344299264300000
11.8934815947019000
128.6353424718180000
7.9399805117226300

Mean
−129.2373581503910000
−128.7850616923410000
−129.8343428775830000
−129.6294851450880000
−129.8382867796110000
−129.7129164862680000
−129.7795909188150000

Std.
0.5986210944493790
0.6157633658946230
0.0408016481905455
0.1054759371085400
0.0372256921835666
0.0875456568200232
0.0974559125367515

Mean
−298.2835926212850000
−295.1290938304830000
−296.9323391084610000
−296.8839733969750000
−297.5119726691150000
−297.8403738182600000
−297.1054029448800000

Std.
0.5587676271753680
0.1634039984609270
0.2251930667702880
0.4330673614598290
0.3440115280624180
0.4536801689800720
0.3062178045203870

Mean
417.4613663019860000
492.5045364088000000
120.0000000000000000
326.6601114362900000
131.3550392249760000
234.2689845349590000
134.6767426915020000

Std.
153.9215808771580000
181.5709657779580000
0.0000000000000188
174.6877238188330000
26.1407360548431000
150.7595974059750000
23.3648038768225000

Mean
221.4232628350220000
455.4454684594550000
258.8582688922670000
231.1806131539990000
231.5547154800990000
222.0256674919140000
231.8426524963750000

Std.
12.2450207482898000
254.3583511786970000
11.8823213189685000
13.5473380962764000
11.5441451076421000
6.1841489800660300
10.3007095087283000

Mean
217.3338617866620000
681.0349114021570000
265.0370119084380000
228.7309024901770000
240.3635189964930000
221.1801916743850000
230.6398805937190000

Std.
20.6685850658838000
488.0618274343640000
12.4033917090208000
12.3682716268631000
14.8435137485293000
5.7037006844690500
10.7176191104135000

Mean
668.9850326105730000
926.9488078829420000
513.8925774904480000
743.9859973770210000
892.4391527217660000
845.4504613493740000
626.6666666666660000

Std.
275.8071370273340000
174.1027182659660000
31.0124861524005000
175.6497294240330000
79.1422224454971000
120.8505129523180000
245.0662589267800000

Mean
708.2979222913040000
831.2324139697050000
500.5478931040730000
776.5150806087790000
863.8929608090610000
809.7183195902260000
673.6943739862640000

Std.
256.2419561521300000
289.7296413284470000
31.2240894705539000
160.7307526692470000
96.5618989087194000
147.3158109824600000
240.7952379798540000

Mean
711.2970397614200000
876.9306161887680000
483.2984167460740000
761.2954767038960000
844.6391674419360000
810.5227124472170000
568.3524718683710000

Std.
258.9317052508320000
289.7296413284470000
99.3976740616107000
163.4084080635650000
113.6848457105400000
104.7139423525340000
264.1320553106420000

Mean
1117.8857079625100000
1258.1065536572400000
659.5351969346130000
959.3735119754180000
911.4640642691360000
990.8546718748010000
887.0165496528500000

Std.
311.0011859260640000
359.7382897536570000
98.5410511961986000
240.5568407069990000
238.3180009803040000
235.1014092849970000
130.7951954796500000

Mean
1094.8305116977000000
−7.159E + 49
915.4958100611630000
1133.7536009808600000
1075.5292326436900000
1094.6823697304900000
1033.4073460688400000

Std.
121.3539576317800000
4.387E + 50
242.1993331983530000
42.1171260000361000
166.9355145236330000
87.9884000140656000
170.5780088721150000

Mean
1304.3661550124000000
1159.9280867973000000
830.2290165794410000
1167.9040488743800000
1070.4327462836400000
1105.2511774948600000
985.5892101103390000

Std.
262.1065863453340000
742.1215416320490000
60.2286903507069000
236.7325108248320000
203.0676627074300000
190.6172874229610000
140.3217672954560000

Mean
500.0000000000000000
653.3355378428050000
460.0000000000000000
510.0000000000000000
493.3333333333400000
490.0000000000000000
460.0000000000000000

Std.
103.7237710925280000
302.5312999719650000
0.0000000000016493
113.7147065368360000
137.2973951415090000
91.5385729888094000
0.0000000000000000

Mean
1107.9038127876700000
1401.6553278264300000
930.4565414149210000
1072.9924659809200000
1258.5157766524700000
1074.3695435628600000
632.0799172389040000

Std.
127.9566489362040000
253.2428066220210000
87.9959072391079000
2.2606058314671500
241.4024507676890000
2.8314182838917800
3.5935258481745500

Table 10: Results of the multiple-problem Wilcoxon test for seven algorithms for CEC2005 functions at .

Algorithm

value

HBD versus SPSO2011
261.95
63.05
0.007453
=
+

HBD versus CMAES
275.90
49.10
0.002279
+
+

HBD versus ABC
179.93
145.07
0.639106
=
=

HBD versus JDE
276.00
49.00
0.002259
+
+

HBD versus CLPSO
291.00
34.00
0.000545
+
+

HBD versus SADE
234.50
90.50
0.052709
=
+

Table 11: Average ranking of seven algorithms by the Friedman test for CEC2005 functions at .

Algorithm
SPSO2011
CMAES
ABC
JDE
CLPSO
SADE
HBD

Ranking
4.06
5.24
3.58
4.44
4.60
3.42
2.66

From Table 9, we find that each algorithm does well in some functions according to its average error value. For instance, PSO2011, CMAES, ABC, JDE, CLPSO, SADE, and HBD perform better in 8, 5, 9, 3, 3, 3, and 7 out of 25 functions, respectively. However, Table 10 shows that HBD gets higher values than values in all cases. This suggests that HBD is better than the other 6 algorithms. Moreover, for Wilcoxon test at and in three cases, there are significant differences for CEC2005 functions. Furthermore, with respect to the average rankings of different algorithms by the Friedman test, it can be seen clearly from Table 11 that HBD offers the best overall performance, while SADE is the second best, followed by ABC, PSO2011, CLPSO, JDE, and CMAES.

Secondly, to appreciate the actual performance of the proposed algorithm, HBD is in comparison with the other five algorithms identified as NBIPOP-aCMA [41], fk-PSO [42], SPSO2011 [43], SPSOABC [44], and PVADE [45], which were presented during the CEC-2013 Special Session & Competition on Real-Parameter Single Objective Optimization.

Table 12 lists the average error values which are dealt with from [46], and the average rankings of the six algorithms by the Friedman test for CEC-2013 functions at are given in Table 13. Since NBIPOP-aCMA is one of top three performing algorithms for CEC-2013 functions [47], seen from Table 12, it shows the promising performance in almost all of functions. Other algorithms bring solutions with higher accuracy in a handful of functions. For example, fk-PSO, SPSO2011, SPSOABC, PVADE, and HBD yield the better performance on 3, 2, 6, 4, and 5 out of 28 functions in terms of the average error values. However, according to the average rankings of different algorithms by the Friedman test in Table 13, we can find that NBIPOP-aCMA is the best, and HBD offers the second best overall performance, followed by SPSOABC, fk-PSO, PVADE, and PSO2011.

Table 12: Error values obtained by HBD and 5 compared algorithms for CEC-2013 benchmark functions at .

NBIPOP-aCMA
fk-PSO
SPSO2011
SPSOABC
PVADE
HBD

AVGEr  ± STDEr
AVGEr  ± STDEr
AVGEr  ± STDEr
AVGEr  ± STDEr
AVGEr  ± STDEr
AVGEr  ± STDEr

0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
1.36E − 29 ± 4.17E − 29

0.00E + 00 ± 0.00E + 00
1.59E + 06 ± 8.03E + 05
3.38E + 05 ± 1.67E + 05
8.78E + 05 ± 1.69E + 06
2.12E + 06 ± 1.56E + 06
3.15E + 05 ± 1.52E + 05

0.00E + 00 ± 0.00E + 00
2.40E + 08 ± 3.71E + 08
2.88E + 08 ± 5.24E + 08
5.16E + 07 ± 8.00E + 07
1.65E + 03 ± 2.83E + 03
4.38E + 06 ± 8.11E + 06

0.00E + 00 ± 0.00E + 00
4.78E + 02 ± 1.96E + 02
3.86E + 04 ± 6.70E + 03
6.02E + 03 ± 2.30E + 03
1.70E + 04 ± 2.85E + 03
5.05E + 03 ± 2.23E + 03

0.00E + 00 ± 0.00E + 00
0.00E + 00 ± 0.00E + 00
5.42E − 04 ± 4.91E − 05
0.00E + 00 ± 0.00E + 00
1.40E − 07 ± 1.86E − 07
0.00E + 00 ± 0.00E + 00

0.00E + 00 ± 0.00E + 00
2.99E + 01 ± 1.76E + 01
3.79E + 01 ± 2.83E + 01
1.09E + 01 ± 1.09E + 01
8.29E + 00 ± 5.82E + 00
1.12E + 01 ± 1.47E + 01

2.31E + 00 ± 6.05E + 00
6.39E + 01 ± 3.09E + 01
8.79E + 01 ± 2.11E + 01
5.12E + 01 ± 2.04E + 01
1.29E + 00 ± 1.22E + 00
5.03E + 01 ± 1.61E + 01

2.09E + 01 ± 4.80E − 02
2.09E + 01 ± 6.28E − 02
2.09E + 01 ± 5.89E − 02
2.09E + 01 ± 4.92E − 02
2.09E + 01 ± 4.82E − 02
2.10E + 01 ± 3.17E − 02

3.30E + 00 ± 1.38E + 00
1.85E + 01 ± 2.69E + 00
2.88E + 01 ± 4.43E + 00
2.95E + 01 ± 2.62E + 00
6.30E + 00 ± 3.27E + 00
2.12E + 01 ± 4.26E + 00

0.00E + 00 ± 0.00E + 00
2.29E − 01 ± 1.32E − 01
3.40E − 01 ± 1.48E − 01
1.32E − 01 ± 6.23E − 02
2.16E − 02 ± 1.36E − 02
9.15E − 02 ± 5.25E − 02

3.04E + 00 ± 1.41E + 00
2.36E + 01 ± 8.76E + 00
1.05E + 02 ± 2.74E + 01
0.00E + 00 ± 0.00E + 00
5.84E + 01 ± 1.11E + 01
3.58E − 01 ± 6.34E − 01

2.91E + 00 ± 1.38E + 00
5.64E + 01 ± 1.51E + 01
1.04E + 02 ± 3.54E + 01
6.44E + 01 ± 1.48E + 01
1.15E + 02 ± 1.14E + 01
8.09E + 01 ± 1.53E + 01

2.78E + 00 ± 1.45E + 00
1.23E + 02 ± 2.19E + 01
1.94E + 02 ± 3.86E + 01
1.15E + 02 ± 2.24E + 01
1.31E + 02 ± 1.24E + 01
1.29E + 02 ± 2.90E + 01

8.10E + 02 ± 3.60E + 02
7.04E + 02 ± 2.38E + 02
3.99E + 03 ± 6.19E + 02
1.55E + 01 ± 6.13E + 00
3.20E + 03 ± 4.38E + 02
2.83E + 00 ± 1.82E + 00

7.65E + 02 ± 2.95E + 02
3.42E + 03 ± 5.16E + 02
3.81E + 03 ± 6.94E + 02
3.55E + 03 ± 3.04E + 02
5.16E + 03 ± 3.19E + 02
3.50E + 03 ± 4.82E + 02

4.40E − 01 ± 9.26E − 01
8.48E − 01 ± 2.20E − 01
1.31E + 00 ± 3.59E − 01
1.03E + 00 ± 2.01E − 01
2.39E + 00 ± 2.66E − 01
1.31E + 00 ± 2.38E − 01

3.44E + 01 ± 1.87E + 00
5.26E + 01 ± 7.11E + 00
1.16E + 02 ± 2.02E + 01
3.09E + 01 ± 1.23E − 01
1.02E + 02 ± 1.17E + 01
3.09E + 01 ± 1.79E − 01

6.23E + 01 ± 4.56E + 01
6.81E + 01 ± 9.68E + 00
1.21E + 02 ± 2.46E + 01
9.01E + 01 ± 8.95E + 00
1.82E + 02 ± 1.20E + 01
9.46E + 01 ± 2.11E + 01

2.23E + 00 ± 3.41E − 01
3.12E + 00 ± 9.83E − 01
9.51E + 00 ± 4.42E + 00
1.71E + 00 ± 4.68E − 01
5.40E + 00 ± 8.10E − 01
1.23E + 00 ± 2.28E − 01

1.29E + 01 ± 5.98E − 01
1.20E + 01 ± 9.26E − 01
1.35E + 01 ± 1.11E + 00
1.11E + 01 ± 7.60E − 01
1.13E + 01 ± 3.28E − 01
1.11E + 01 ± 5.89E − 01

1.92E + 02 ± 2.72E + 01
3.11E + 02 ± 7.92E + 01
3.09E + 02 ± 6.80E + 01
3.18E + 02 ± 7.53E + 01
3.19E + 02 ± 6.26E + 01
2.95E + 02 ± 7.95E + 01

8.38E + 02 ± 4.60E + 02
8.59E + 02 ± 3.10E + 02
4.30E + 03 ± 7.67E + 02
8.41E + 01 ± 3.90E + 01
2.50E + 03 ± 3.86E + 02
4.48E + 01 ± 1.35E + 01

6.67E + 02 ± 2.90E + 02
3.57E + 03 ± 5.90E + 02
4.83E + 03 ± 8.23E + 02
4.18E + 03 ± 5.62E + 02
5.81E + 03 ± 5.04E + 02
4.16E + 03 ± 5.07E + 02

1.62E + 02 ± 3.00E + 01
2.48E + 02 ± 8.11E + 00
2.67E + 02 ± 1.25E + 01
2.51E + 02 ± 1.43E + 01
2.02E + 02 ± 1.40E + 00
2.28E + 02 ± 8.92E + 00

2.20E + 02 ± 1.11E + 01
2.49E + 02 ± 7.82E + 00
2.99E + 02 ± 1.05E + 01
2.75E + 02 ± 9.76E + 00
2.30E + 02 ± 2.08E + 01
2.80E + 02 ± 8.80E + 00

1.58E + 02 ± 3.00E + 01
2.95E + 02 ± 7.06E + 01
2.86E + 02 ± 8.24E + 01
2.60E + 02 ± 7.62E + 01
2.18E + 02 ± 4.01E + 01
2.00E + 02 ± 6.81E − 03

4.69E + 02 ± 7.38E + 01
7.76E + 02 ± 7.11E + 01
1.00E + 03 ± 1.12E + 02
9.10E + 02 ± 1.62E + 02
3.26E + 02 ± 1.14E + 01
7.52E + 02 ± 1.33E + 02

2.69E + 02 ± 7.35E + 01
4.01E + 02 ± 3.48E + 02
4.01E + 02 ± 4.76E + 02
3.33E + 02 ± 2.32E + 02
3.00E + 02 ± 2.24E − 02
3.00E + 02 ± 1.32E − 13

Table 13: Average ranking of six algorithms by the Friedman test for CEC2013 functions at .

Algorithm
NBIPOP-aCMA
fk-PSO
SPSO2011
SPSOABC
PVADE
HBD

Ranking
1.80
3.61
5.29
3.34
3.95
3.02

5. Conclusion

In this paper, we presented a hybrid BSA, called HBD, which combined BSA and DE with exploitive mutation strategy. At each iteration process, DE was embedded behind the BSA algorithm to optimize one individual which was selected according to its probability in order to enhance the convergence of BSA and to bring solutions with higher quality.

Comprehensive experiments have been carried out in 28 benchmark functions proposed in CEC-2013 competition. The experimental results reveal that the hybridization of BSA and DE provides the high effectiveness and efficiency in most of functions, contributing to solutions with higher accuracy, faster convergence speed, and more stable scalability. HBD was also compared with other evolutionary algorithms and has shown its promising performance.

There are several interesting directions for future work. Experimentally, the linear probability model used to select one individual to optimize is a reasonable but not optimal one; thus, firstly, the comprehensive tests will be performed on various probability models in HBD. Secondly, although experimental results have shown that HBD owns the stable scalability, we plan to investigate HBD for large-scale optimization problems. Last but not least, we plan to apply HBD to some real-world optimization problems for further examinations.

Conflict of Interests

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

Acknowledgments

The authors are very grateful to the editor and the anonymous reviewers for their constructive comments and suggestions to this paper. This work was supported by the NSFC Joint Fund with Guangdong of China under Key Project U1201258, the Shandong Natural Science Funds for Distinguished Young Scholar under Grant no. JQ201316, and the Natural Science Foundation of Fujian Province of China under Grant no. 2013J01216.

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