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.
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) [
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 [
The backtracking search optimization algorithm (BSA) [
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 [
The remainder of this paper is organized as follows. Section
The backtracking search optimization algorithm is a new stochastic search technique developed recently [
BSA employs a random mutation strategy that used only one direction individual for each target individual, formulated as follows:
BSA also uses a nonuniform and more complex crossover strategy. There are two steps in the crossover process. Firstly, a binary integer-values matrix (
Initiate Generate If Generate a vector containing a random permutation of the integers Generate Else Generate a random integer End If
For If Else End If End For
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
Initiate the population While (Stop Condition doesn’t meet) Perform the first type selection: range between 0 and 1. Permute arbitrary changes in position of Generate the Generate the population Perform the second type selection: select the population with better fitness from Update the best solution. End While Output the best solution.
DE is a powerful evolutionary algorithm for global optimization over continuous space. When being used to solve optimization problems, it evolves a population of
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 “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
The crossover operator is performed to generate a trial vector
Finally, DE uses a greedy mechanism to select the better vector from each pair of
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.
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., (
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, (
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:
Note that the probability equation is similar to the selection probability in DE with ranking-based mutation operators [
It is worth pointing out that our previous work [
According to the above descriptions, the pseudocode of HBD is described in Algorithm
Initiate the population While (Stop Condition doesn’t meet) Perform the first type selection: range between 0 and 1. Permute arbitrary changes in position of Generate the Generate the population Perform the second type selection: select the population with better fitness from Update the best solution. //Invoke DE with exploitive strategy Select One Individual according to its probability: Optimize If (fitness( End If Update the best solution. End While Output the best solution.
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 [
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
To evaluate the performance of algorithms, we use
To show the effect of the proposed algorithm, Table
Error values obtained by BSA and HBD for 30-dimensional CEC-2013 benchmark functions.
BSA | HBD | |||
---|---|---|---|---|
AVGEr ± STDEr | AVGEr ± STDEr |
|
||
|
|
1.36 |
= | 0.359375 |
|
1.22 |
|
+ | 0.000014 |
|
7.52 |
|
+ | 0.045010 |
|
1.25 |
|
+ | 0.000016 |
|
0.00 |
0.00 |
= | 1.000000 |
|
3.04 |
|
+ | 0.001721 |
|
7.39 |
|
+ | 0.000101 |
|
|
2.10 |
= | 0.142532 |
|
2.70 |
|
+ | 0.000081 |
|
1.78 |
|
+ | 0.009417 |
|
|
3.58 |
= | 0.062500 |
|
8.41 |
|
= | 0.396679 |
|
1.44 |
|
= | 0.061480 |
|
3.70 |
|
+ | 0.034670 |
|
3.73 |
|
= | 0.097970 |
|
1.31 |
1.31 |
= | 0.903627 |
|
3.09 |
3.09 |
= | 0.840072 |
|
1.20 |
|
+ | 0.000980 |
|
|
1.23 |
= | 0.312970 |
|
1.15 |
|
= | 0.057836 |
|
|
2.95 |
= | 0.431762 |
|
|
4.48 |
= | 0.443172 |
|
4.46 |
|
= | 0.103553 |
|
2.32 |
|
= | 0.157770 |
|
2.87 |
|
= | 0.087527 |
|
2.00 |
2.00 |
+ | 0.000020 |
|
8.80 |
|
+ | 0.003822 |
|
3.00 |
3.00 |
+ | 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
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,
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.
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-
Error values obtained by BSA and HBD for 10- and 50-dimensional CEC-2013 benchmark functions.
|
BSA | HBD |
|
BSA | HBD | |||||
---|---|---|---|---|---|---|---|---|---|---|
AVGEr ± STDEr | AVGEr ± STDEr |
|
AVGEr ± STDEr | AVGEr ± STDEr |
|
|||||
|
0.00 |
0.00 |
= | 1.000000 | 1.69 |
|
= | 0.582031 | ||
|
8.59 |
|
+ | 0.000065 | 2.77 |
|
+ | 0.000014 | ||
|
5.46 |
|
+ | 0.000012 | 4.95 |
|
+ | 0.000140 | ||
|
5.04 |
|
+ | 0.000014 | 3.22 |
|
+ | 0.000023 | ||
|
0.00 |
0.00 |
= | 1.000000 | 3.06 |
|
= | 0.250000 | ||
|
|
2.75 |
= | 0.220852 | 5.42 |
|
+ | 0.000046 | ||
|
6.96 |
|
+ | 0.045010 | 8.48 |
|
= | 0.756995 | ||
|
2.04 |
2.04 |
= | 0.989266 | 2.11 |
2.11 |
= | 0.562928 | ||
|
3.54 |
|
+ | 0.006313 |
|
5.50 |
= | 0.657069 | ||
|
8.49 |
|
= | 0.989266 | 3.99 |
|
+ | 0.000018 | ||
|
0.00 |
0.00 |
= | 1.000000 | 7.96 |
|
= | 0.062500 | ||
|
1.10 |
|
= | 0.599802 | 1.92 |
|
+ | 0.007423 | ||
|
1.68 |
|
= | 0.736617 | 3.25 |
|
+ | 0.001569 | ||
|
1.31 |
|
+ | 0.000296 | 2.17 |
|
= | 0.287862 | ||
|
6.10 |
|
+ | 0.022988 | 8.25 |
|
= | 0.142532 | ||
|
6.54 |
|
= | 0.989266 | 1.88 |
|
= | 0.427339 | ||
|
|
8.37 |
= | 0.427339 |
|
5.45 |
= | 0.924971 | ||
|
2.67 |
|
+ | 0.000665 | 2.65 |
|
+ | 0.039554 | ||
|
2.45 |
|
= | 0.657069 |
|
2.80 |
= | 0.312970 | ||
|
2.92 |
|
+ | 0.019941 | 2.09 |
|
= | 0.819095 | ||
|
3.16 |
|
= | 0.444824 |
|
8.72 |
= | 0.675764 | ||
|
1.49 |
|
= | 0.544910 |
|
6.10 |
= | 0.544910 | ||
|
8.60 |
|
= | 0.818641 | 9.53 |
|
= | 0.599802 | ||
|
|
1.56 |
= | 0.716423 | 2.70 |
|
+ | 0.001569 | ||
|
1.93 |
|
= | 0.241820 | 3.79 |
|
+ | 0.013817 | ||
|
1.13 |
|
= | 0.109386 | 2.00 |
2.00 |
+ | 0.000012 | ||
|
|
3.32 |
= | 0.676637 | 1.53 |
|
+ | 0.028314 | ||
|
2.20 |
|
= | 0.802856 | 4.00 |
4.00 |
+ | 0.000001 | ||
|
9/19/0 | 13/15/0 |
In the case of
When
In summary, it suggests that the advantage of HBD over BSA is stable when the dimensionality of problems increases.
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
Error values obtained by cHBD, HBD, and bHBD for CEC-2013 benchmark functions at
cHBD | HBD | bHBD | |||||
---|---|---|---|---|---|---|---|
AVGEr ± STDEr |
|
AVGEr ± STDEr |
|
AVGEr ± STDEr | |||
|
|
= | 0.125000 | 1.36 |
= | 0.328125 |
|
|
|
= | 0.676637 | 3.15 |
+ | 0.000240 | 7.07 |
|
|
= | 0.618641 | 4.38 |
= | 0.637733 |
|
|
|
= | 0.736617 | 5.05 |
+ | 0.006848 | 6.55 |
|
0.00 |
= | 1.000000 | 0.00 |
= | 1.000000 | 0.00 |
|
1.12 |
= | 0.326049 | 1.12 |
= | 0.562928 |
|
|
5.69 |
= | 0.142532 | 5.03 |
+ | 0.003822 | 6.58 |
|
|
= | 0.264150 | 2.10 |
= | 0.287862 |
|
|
2.61 |
+ | 0.000126 | 2.12 |
+ | 0.000446 | 2.65 |
|
|
= | 0.509755 | 9.15 |
= | 0.300241 |
|
|
|
− | 0.015625 | 3.58 |
− | 0.015625 |
|
|
8.19 |
= | 0.427339 | 8.09 |
= | 0.065311 | 9.01 |
|
|
= | 0.798248 | 1.29 |
= | 0.174210 | 1.39 |
|
4.01 |
+ | 0.032428 | 2.83 |
= | 0.082653 | 3.88 |
|
3.80 |
+ | 0.017253 | 3.50 |
+ | 0.014889 | 3.81 |
|
|
= | 0.967806 | 1.31 |
= | 0.924971 |
|
|
3.10 |
= | 0.165837 | 3.09 |
= | 0.989266 | 3.09 |
|
1.17 |
+ | 0.000665 | 9.46 |
+ | 0.021418 | 1.10 |
|
1.27 |
= | 0.427339 | 1.23 |
= | 0.924971 |
|
|
1.12 |
= | 0.443172 | 1.11 |
= | 0.121828 | 1.14 |
|
3.18 |
= | 0.300009 | 2.95 |
+ | 0.040267 | 3.49 |
|
5.28 |
= | 0.367385 | 4.48 |
= | 0.696425 | 4.58 |
|
4.37 |
= | 0.121828 | 4.16 |
+ | 0.039554 | 4.48 |
|
2.28 |
= | 0.861162 | 2.28 |
= | 0.946369 | 2.28 |
|
2.85 |
= | 0.054374 | 2.80 |
= | 0.275832 | 2.85 |
|
2.00 |
= | 0.637733 | 2.00 |
+ | 0.022988 | 2.00 |
|
7.68 |
= | 0.736617 | 7.52 |
= | 0.092631 | 8.36 |
|
|
= | 0.161513 | 3.00 |
= | 0.808365 | 3.00 |
|
|
|
From Table
Additionally, Table
Results of the multiple-problem Wilcoxon test for HBD, cHBD, and bHBD for
Algorithm |
|
|
|
|
|
---|---|---|---|---|---|
HBD versus cHBD | 235 | 143 | 0.269095 | = | = |
HBD versus bHBD | 276 | 75 | 0.010695 | + | + |
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
Table
Error values obtained by aHBD, HBD, and paHBD for CEC-2013 benchmark functions at
aHBD | paHBD | BSADE | HBD | |||||||
---|---|---|---|---|---|---|---|---|---|---|
AVGEr ± STDEr |
|
AVGEr ± STDEr |
|
AVGEr ± STDEr |
|
AVGEr ± STDEr | ||||
|
1.29 |
+ | 0.000292 | 5.53 |
+ | 0.030946 |
|
= | 0.666016 | 1.36 |
|
|
− | 0.000012 |
|
− | 0.000012 |
|
− | 0.000012 | 3.15 |
|
1.13 |
= | 0.924971 |
|
= | 0.201222 | 5.21 |
= | 0.696425 | 4.38 |
|
|
− | 0.000012 |
|
− | 0.000012 |
|
− | 0.000012 | 5.05 |
|
4.26 |
+ | 0.007813 |
|
= | 0.062500 | 0.00 |
= | 1.000000 | 0.00 |
|
|
− | 0.017253 |
|
− | 0.003822 | 1.18 |
= | 0.396679 | 1.12 |
|
|
= | 0.097970 |
|
= | 0.492633 | 5.87 |
= | 0.097970 | 5.03 |
|
|
= | 0.300241 |
|
= | 0.509755 |
|
= | 0.210872 | 2.10 |
|
2.18 |
= | 0.861162 | 2.18 |
= | 0.637733 | 2.60 |
+ | 0.000296 | 2.12 |
|
|
− | 0.014889 |
|
− | 0.022988 | 2.04 |
+ | 0.000126 | 9.15 |
|
3.03 |
+ | 0.000012 | 3.06 |
+ | 0.000083 |
|
= | 0.062500 | 3.58 |
|
|
= | 0.264150 | 8.12 |
= | 0.967806 | 8.64 |
= | 0.241820 | 8.09 |
|
1.38 |
= | 0.509755 | 1.35 |
= | 0.367385 | 1.43 |
= | 0.054374 | 1.29 |
|
5.85 |
+ | 0.000012 | 7.23 |
+ | 0.000266 | 7.33 |
+ | 0.000808 | 2.83 |
|
|
= | 0.339479 |
|
= | 0.861162 |
|
= | 0.381860 | 3.50 |
|
1.49 |
+ | 0.008041 |
|
= | 0.676637 |
|
= | 0.882352 | 1.31 |
|
3.92 |
+ | 0.000012 | 3.17 |
+ | 0.000023 | 3.17 |
+ | 0.000025 | 3.09 |
|
1.05 |
= | 0.121828 |
|
= | 0.300241 |
|
= | 0.411840 | 9.46 |
|
2.10 |
+ | 0.000036 | 1.44 |
+ | 0.011000 | 1.47 |
+ | 0.000602 | 1.23 |
|
|
− | 0.004530 | 1.13 |
= | 0.562928 | 1.13 |
= | 0.427339 | 1.11 |
|
3.26 |
= | 0.121477 | 3.41 |
= | 0.061407 |
|
= | 0.894867 | 2.95 |
|
2.47 |
+ | 0.000014 | 5.78 |
= | 0.492633 | 6.63 |
+ | 0.002947 | 4.48 |
|
|
= | 0.252813 | 4.21 |
= | 0.339479 | 4.19 |
= | 0.819095 | 4.16 |
|
2.36 |
+ | 0.008705 |
|
= | 0.562928 | 2.35 |
+ | 0.009417 | 2.28 |
|
|
= | 0.103553 | 2.80 |
= | 0.756995 | 2.86 |
= | 0.069337 | 2.80 |
|
2.11 |
+ | 0.002470 | 2.00 |
− | 0.000016 | 2.00 |
− | 0.000012 | 2.00 |
|
|
= | 0.882352 | 7.58 |
= | 0.989266 | 9.19 |
+ | 0.000891 | 7.52 |
|
3.00 |
+ | 0.016377 | 3.00 |
+ | 0.038947 | 3.00 |
= | 1.000000 | 3.00 |
|
|
|
|
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
Results of the multiple-problem Wilcoxon test for HBD, aHBD, paHBD, and BSADE for
Algorithm |
|
|
|
|
|
---|---|---|---|---|---|
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
In HBD, a linear model seen (
Error values obtained by qHBD, HBD, and sHBD for CEC-2013 benchmark functions at
qHBD | HBD | sHBD | |||||
---|---|---|---|---|---|---|---|
AVGEr ± STDEr |
|
AVGEr ± STDEr |
|
AVGEr ± STDEr | |||
|
|
= | 0.078125 | 1.36 |
= | 0.429688 |
|
|
3.56 |
= | 0.798248 | 3.15 |
= | 0.736617 |
|
|
4.94 |
= | 0.618641 | 4.38 |
= | 0.756995 | 3.57 |
|
|
= | 0.736617 | 5.05 |
= | 0.903627 | 5.20 |
|
2.84 |
= | 1.000000 | 0.00 |
= | 1.000000 | 0.00 |
|
|
= | 0.381860 | 1.12 |
= | 0.861162 |
|
|
5.68 |
= | 0.191898 | 5.03 |
= | 0.756995 | 5.19 |
|
|
= | 0.509755 | 2.10 |
= | 0.165837 |
|
|
2.38 |
+ | 0.039554 | 2.12 |
= | 0.231167 | 2.24 |
|
9.15 |
= | 0.264150 | 9.15 |
= | 0.527183 | 9.80 |
|
4.38 |
= | 0.986328 | 3.58 |
= | 0.366699 |
|
|
|
= | 0.073565 | 8.09 |
= | 0.374558 |
|
|
|
= | 0.989266 | 1.29 |
= | 0.411840 |
|
|
3.14 |
= | 0.618641 | 2.83 |
= | 0.051087 | 3.66 |
|
3.60 |
= | 0.903627 | 3.50 |
= | 0.676637 |
|
|
|
= | 0.967806 | 1.31 |
= | 0.696425 |
|
|
3.10 |
= | 0.492633 | 3.09 |
= | 0.946369 | 3.09 |
|
9.93 |
= | 0.396679 | 9.46 |
= | 0.840072 | 9.52 |
|
|
= | 0.736617 | 1.23 |
= | 0.210872 | 1.26 |
|
|
= | 0.165837 | 1.11 |
= | 0.191898 | 1.14 |
|
3.10 |
= | 0.480701 | 2.95 |
= | 0.165492 | 3.28 |
|
|
= | 0.443172 | 4.48 |
= | 0.287862 |
|
|
4.16 |
= | 0.924971 | 4.16 |
= | 0.840072 |
|
|
2.31 |
= | 0.637733 | 2.28 |
= | 0.777543 | 2.28 |
|
|
− | 0.042207 | 2.80 |
= | 0.157770 |
|
|
2.00 |
= | 0.777543 | 2.00 |
= | 0.840072 | 2.00 |
|
7.82 |
= | 0.287862 | 7.52 |
= | 0.459336 | 7.90 |
|
3.00 |
= | 1.000000 | 3.00 |
− | 0.025347 | 3.00 |
|
|
|
Results of the multiple-problem Wilcoxon test for HBD, qHBD, and sHBD for
Algorithm |
|
|
|
|
|
---|---|---|---|---|---|
HBD versus qHBD | 238 | 140 | 0.239106 | = | = |
HBD versus sHBD | 143 | 208 | 0.409125 | = | = |
From Table
Firstly, HBD is compared with 6 non-BSA approaches in [
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 | −7.159 |
915.4958100611630000 | 1133.7536009808600000 | 1075.5292326436900000 | 1094.6823697304900000 | 1033.4073460688400000 |
Std. | 121.3539576317800000 | 4.387 |
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 |
Results of the multiple-problem Wilcoxon test for seven algorithms for CEC2005 functions at
Algorithm |
|
|
|
|
|
---|---|---|---|---|---|
HBD versus SPSO2011 |
|
63.05 | 0.007453 | = | + |
HBD versus CMAES |
|
49.10 | 0.002279 | + | + |
HBD versus ABC |
|
145.07 | 0.639106 | = | = |
HBD versus JDE |
|
49.00 | 0.002259 | + | + |
HBD versus CLPSO |
|
34.00 | 0.000545 | + | + |
HBD versus SADE |
|
90.50 | 0.052709 | = | + |
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 |
|
From Table
Secondly, to appreciate the actual performance of the proposed algorithm, HBD is in comparison with the other five algorithms identified as NBIPOP-aCMA [
Table
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 | |
|
|
|
|
|
|
1.36 |
|
|
1.59 |
3.38 |
8.78 |
2.12 |
3.15 |
|
|
2.40 |
2.88 |
5.16 |
1.65 |
4.38 |
|
|
4.78 |
3.86 |
6.02 |
1.70 |
5.05 |
|
|
|
5.42 |
|
1.40 |
|
|
|
2.99 |
3.79 |
1.09 |
8.29 |
1.12 |
|
2.31 |
6.39 |
8.79 |
5.12 |
|
5.03 |
|
|
|
|
|
|
2.10 |
|
|
1.85 |
2.88 |
2.95 |
6.30 |
2.12 |
|
|
2.29 |
3.40 |
1.32 |
2.16 |
9.15 |
|
3.04 |
2.36 |
1.05 |
|
5.84 |
3.58 |
|
|
5.64 |
1.04 |
6.44 |
1.15 |
8.09 |
|
|
1.23 |
1.94 |
1.15 |
1.31 |
1.29 |
|
8.10 |
7.04 |
3.99 |
1.55 |
3.20 |
|
|
|
3.42 |
3.81 |
3.55 |
5.16 |
3.50 |
|
|
8.48 |
1.31 |
1.03 |
2.39 |
1.31 |
|
3.44 |
5.26 |
1.16 |
|
1.02 |
|
|
|
6.81 |
1.21 |
9.01 |
1.82 |
9.46 |
|
2.23 |
3.12 |
9.51 |
1.71 |
5.40 |
|
|
1.29 |
1.20 |
1.35 |
|
1.13 |
|
|
|
3.11 |
3.09 |
3.18 |
3.19 |
2.95 |
|
8.38 |
8.59 |
4.30 |
8.41 |
2.50 |
|
|
|
3.57 |
4.83 |
4.18 |
5.81 |
4.16 |
|
|
2.48 |
2.67 |
2.51 |
2.02 |
2.28 |
|
|
2.49 |
2.99 |
2.75 |
2.30 |
2.80 |
|
|
2.95 |
2.86 |
2.60 |
2.18 |
2.00 |
|
4.69 |
7.76 |
1.00 |
9.10 |
|
7.52 |
|
|
4.01 |
4.01 |
3.33 |
3.00 |
3.00 |
Average ranking of six algorithms by the Friedman test for CEC2013 functions at
Algorithm | NBIPOP-aCMA | fk-PSO | SPSO2011 | SPSOABC | PVADE | HBD |
---|---|---|---|---|---|---|
Ranking |
|
3.61 | 5.29 | 3.34 | 3.95 |
|
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.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.