=Paper=
{{Paper
|id=Vol-3304/paper28
|storemode=property
|title=An Adaptive Differential Evolution Based on Multi-subpopulation for Global Optimization
|pdfUrl=https://ceur-ws.org/Vol-3304/paper28.pdf
|volume=Vol-3304
|authors=Zhoucheng Bao,Kaige Chen,Tingting Pang,Zuling Wang
}}
==An Adaptive Differential Evolution Based on Multi-subpopulation for Global Optimization==
https://ceur-ws.org/Vol-3304/paper28.pdf
An Adaptive Differential Evolution Based on Multi-
subpopulation for Global Optimization
Zhoucheng Bao1, Kaige Chen1, Tingting Pang1, Zuling Wang2
1
HangZhou Normal University, Hangzhou, China
2
Chongqing Vocational Institute of Safety and Technology, Chongqing, China
Abstract
This paper proposed a multiple-subpopulation based on differential evolution with an adaptive
parameter control and population reduction, called EMPADE, for global optimization. In
EMPADE, an adaptive parameter control method based on population diversity and fitness
value is proposed to set the parameters F. Further, a mutation strategy named DE/current-
pbad/1 with archives is designed. In addition, a population size reduction strategy is devised to
adaptively eliminate poor performance Individuals. The proposed algorithm has been tested by
solving CEC2014 benchmark problems. The experimental results show that the EMPADE
algorithm strategy is feasible and effective, and has greater competitiveness compared with
other related algorithms.
Keywords 1
Adaptive, Differential evolution, Population size control, Multiple-population
1. Introduction
Differential evolution (DE), initially introduced by Storn and Price [1], has become a powerful tool
to solve global optimization problems [2]. Applications of DE for real problems includes pattern
recognition [3], power systems [4] and vehicle routing [5].
The selection of mutation operator is concerned by many researchers. For example, in JADE [6], a
parameter adaptation strategy along with DE/current-to-pbest mutation strategy is employed to generate
offspring. CoDE CoDE adopts multiple compound mutation operators (DE/rand/1, DE/rand/2 and
DE/current-to-rand/1) and DE variants of parameter control mode [8]. In the above methods, mutation
strategies are performed on a single population, rather than multiple populations.
Recently, multi-population based DE algorithms have also been proposed. These algorithms generally
divide initial population into multiple sub-populations and share information among different sub-
populations through individual migration. For example, in MPEDE [9] and EDEV[10], the authors tried
to divide the population into a large reward sub-population and three small indicator sub-populations. In
these methods, three different mutation strategies are employed in an ensemble manner. In SAMDE[11],
the entire population is divided into three subpopulations of the same size, each employing a different
mutation strategy. After each generation, the population is randomly reorganized. However, random
recombination of the population could lead to lead to unreasonable allocation of computing resources.
Further, parameter control in these methods does not consider the diversity and performance of sub-
populations, restricting their performance [6].
In this paper, we propose multiple-population based differential evolution with adaptive parameter
control and population reduction strategies, called EMPADE, for global optimization. In each generation,
three subpopulations with the same size will be divided according to the Euclidean distance, and the
ICBASE2022@3rd International Conference on Big Data & Artificial Intelligence & Software Engineering, October 21-
23, 2022, Guangzhou, China
EMAIL: baozc1031@163.com(Zhoucheng Bao); 2020112011013@stu.hznu.edu.cn(Kaige Chen);
2020112011025@stu.hznu.edu.cn(Tingting Pang); zulingw@163.com(Zuling Wang)
ORCID: 0000-0001-5461-800 (Zhoucheng Bao); 0000-0002-0432-4454(Kaige Chen); 0000-0002-0923-8803(Tingting Pang); 0000-0003-
4850-9976 (Zuling Wang)
Β© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
225
three subpopulations will use different and fixed mutation operators, that is, similar individuals will form
a new population in each generation, so that the mutation operator can act on individuals. A mutation
strategy named DE/current-pbad/1 has also been devised and performed on the top-ranked
subpopulation. At the same time, a parameter adaptive method based on the diversity and fitness value
of subpopulation is proposed, which can better address different problems. Additionally, a population
reduction strategy is proposed to adaptively remove unpromising individuals. Experiments show that the
performance of EMPADE is better or competitive than related method.
The rest of this paper is arranged as follows. The details of EMPADE are described in Section 2.
Section 3 gives the related experiments and results of EMAPDE algorithm. Finally, we conclude this
paper in Section 4.
2. Proposed Algorithm
In this section, we introduce the detailed process and strategies of our algorithm. First, initializing a
population randomly, calculating the Euclidean distance of each individual in the population to the best
individual. Then, dividing the population into three sub-populations of the same size according to the
sort, in which the top-ranked subpopulation has a low diversity, and the bottom-ranked subpopulation
has high diversity. The first third uses a devised mutation strategy called DE/current-to-pbad/1, which
can improve the subpopulation diversity. The middle part uses mutation strategy DE/current-to-rand/1,
which is able to converge fast while maintaining diversity. The last third uses mutation strategy
DE/current-to-pbest/1, which can accelerate convergence. The program runs repeatedly until the
termination conditions are met: three populations are run in parallel to search the space. At the end of
each generation, update the control parameters F and CR and employing the proposed adaptive
elimination operation to update the population size. This is followed by re-sorting and redividing the
population into three sub-populations according to Euclidean distance. The overview of the proposed
algorithm is shown in Algorithm I.
In the following subsections, we will describe the details of a cooperative mechanism strategy based
on multi group mutation in Section 2.1, a parameter adaptive strategy for multi population in Section 2.2,
and an adaptive elimination operation in Section 2.3.
Algorithm I EMPADE algorithm.
1) Set the initial population size ππ , minimum population ππ and probability of excellent
individual(pi).
2) Calculate the Euclidean distance of each individual to the best individual and sort the
individuals.
3) Set the initial parameters of each subpopulation mF, mCR, ππ = ππ /3, p = 1, 2,3.
4) Assign different mutation operators to three subpopulations.
5) Run the following processes circularly until the termination conditions are met.
a) Information sharing: the individuals with the best fitness values of three sub populations
are replaced by the best ones in the whole population
b) The three sub populations Pi evolve in parallel with different mutation mi operators and
return toπ , πππ‘πππ π , ππ
c) Calculate the proportional parameters of control mF and mCr of each sub population (ππΉπ
and ππΆπ ).
d) Adjust population size.
e) Sort the individuals according to Euclidean distance, and redivide three subpopulations of
the same size.
6) Output results after calculating resource consumption.
226
2.1. Cooperative mechanism of adaptive multi group
When the entire population is divided into three equally sized sub-populations according to their
Euclidean distances to the best individual, the highest ranked population will have the least diversity. If
traditional mutation operators such as DE/rand/1 and DE/rand/2 are used, it cannot help to improve its
diversity, since the Euclidean distance among individuals in the subpopulation is small.
In view of the fact that and cannot maintain diversity in the subpopulation, a new mutation operator,
called DE/current-to-pbad with archive, is proposed. Compared with the current subpopulation, the
recently explored inferior solutions can improve the diversity of subpopulation. π , β π , is good
for individuals to evolve toward diversity, and random π , β π , makes the individual change value
larger, which is good for jumping out of the local optimum. In DE/current-to-pbad/1, a mutation vector
is generated in the following manner.
π, = π, +πΉ β π , β π, + ππππ π , β π , , i β 1,2, β¦ , ππ (1)
where π , represents an individual randomly selected from the last ππ individuals of the current
sub population, ππ β (0,1).
In summary, we calculate and sort the Euclidean distance for each generation, and divide three
subpopulations. And P1 (βDE/current-pbad/1β for the top ranking), P2(βDE/current-to-rand/1β for the
general ranking) and P3(βDE/current-to-pbest/1β for the worst ranking).
2.2. Parameter adaptive strategy for multi-subpopulation
The parameter control of each subpopulation is very meaningful for improving the performance of
the algorithm. In EMPADE, a adaptive parameter control mechanism based on multi subpopulation is
designed that control parameters of each subpopulation can be adaptively changed according to the
population diversity and the best fitness value. The adaptive scheme can coordinate the exploration and
exploitation capabilities of each subpopulation to generate better offspring in the process of evolution.
The updates of ππΉ and ππΆπ are shown as follows, respectively. (And in the paper, p represents the
p subpopulation, and p is 1,2,3)
π
ππΉ = β ππππ ππΉ (2)
2
β
π
ππΉ = β ππππ ππΉ (3)
2
where π·π β (0,1)is the proportion of each sub population to the diversity of all sub-populations,
and ππΉ β (0,1) is the ratio of the best fitness value of the current subpopulation to the sum of the best
fitness values of the three sub-populations. When the diversity of population π·π is small, (F)
increased and ππΉ β ππΉ (CR) decreased, then mF will increase and mCr will decrease to increase the
diversity of subpopulation.
2.3. Adaptive elimination operation
The current multi population algorithms often migrate individuals or copy individuals in different
subpopulations, and the poorly performing individuals in the entire population are not eliminated.
Although the diversity of population can be maintained, the overall performance of the algorithm will be
affected. The linear population size reduction scheme proposed in LSHADE was proven to be excellent
scheme for population size adaptation[12][13]. However, the rapid decline of the population at the
beginning may lead to the loss of diversity of the population. Here, we designed a parabolic population
227
elimination operation to eliminate under-performing individuals at each generation while maintaining
the diversity of population at the early stage. The adaptive elimination operation is calculated as:
ππ β ππ
ππ = πππ’ππ β ππππ + ππ (4)
max _ππππ
where ππ represents the minimum population, ππ represents the original population. The
poor individuals in the whole population will be randomly selected and removed.
3. Experiments &Results
All the algorithms in the experiment are run on the same computer, and are aimed at the cec2014
dataset. We will run each function 51 times, record its average value and standard deviation, and mark
the best one as bold. The rank sum test of the algorithm takes 0.05 as the confidence interval. The sign
β+β means better than the correlation algorithm, β-β means worse than the correlation algorithm, and
β=βmeans similar to the correlation algorithm. The error of the experimental results is f-f *, where f *
represents the best fitness value of the function.
3.1. Parameter Settings
Table I lists the parameter settings in our algorithm, and the parameters in the comparison algorithm
are all the parameters that get the best results according to the relevant papers.
Table I
Experimental parameter setting of EMPADE
Parameters ππ ππ ππΉ ππΆπ pi
Value 9 *Dim 15 0.5 0.5 0.11
3.2. The proposed strategies
The strategy mechanisms are been explored by comparing EMPADE with EMPADE 1, in which a
cooperative mechanism strategy of adaptive multi group mutation operator is employed. EMPADE 2
uses an adaptive elimination operation based on EMPADE 1 and EMPADE uses a parameter adaptive
strategy for multi population based on EMPADE 2. The results on 30D are shown in Table II. Compared
with other algorithms, 25 of the 30 benchmark functions have the best results. The EMPADE could be
significantly better than three algorithms to be compared, especially on unimodal and hybrid functions.
The results therefore indicated the strategies proposed are effective.
3.3. Comparisons with related algorithms
EMPADE is compared to four DE variants: JADE[6], EPSDE[7], MPEDE[9], EDEV[10]. And Table
II show the comparison results of the four algorithms. Looking at the experimental results on 30D,
β
EMPADE achieves the best mean values on 22 functions β
out of 30 functions. For unimodal functions
F1βF3, EMPADE shows the best performance, the best mean fitness value can be found on all functions.
EPSDE, MPEDE and EDEV can find the best mean fitness value on F2 and F3. And JADE can also find
the best mean fitness value on F2. For the multi-modal benchmark functions F4-F16, compared with
other algorithms, EMPADE still has the best performance and can be more competitive on eleven
functions (F4, F5, F6, F7, F8, F9, F11, F12, F14, F15, F16), and finds the best fitness values on F4, F7
and F8. JADE also has a good performance and can achieve good results on ten functions (F4, F5, F7,
F8, F10, F11, F12, F13, F14, F16), and finds the best fitness values on F4 and F8. EPSDE, MPEDE and
EDEV have competitive performance on one function (F8), three functions (F6, F7, F8) and three
functions (F7, F8, F13), respectively, and they can all find the best mean fitness value on F8. For the
hybrid functions F17-F22, EMPADE is the most competitive, better than other related algorithms in all
228
functions. For the complex composition functions F23-F30, EDEV achieves the best results, with
excellent performance on 7 functions (F23, F25, F26, F27, F28, F29 and F30). JADE, EPSDE, MPEDE
and EMPADE perform well on one function (F27), three functions (F23, F25, F29), one function (F26)
and two functions (F24, F26), respectively.
Based on the above experiments and results, it can be concluded that EMPADE is better or
competitive than related algorithms. especially in low dimensional unimodal functions, multimodal
functions and hybrid functions.
Table II
Comparison results within 30D
EMPADE 1 EMPADE 2 EMPADE
Functions Mean (Std) Mean (Std) Mean (Std)
F1 2.28E-05(1.34E-04) 6.67E-07(2.26E-06) 0.00E+00(0.00E+00)
F2 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00)
F3 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00)
F4 1.26E+00(8.88E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00)
F5 2.04E+01(5.41E-02) 2.02E+01(6.66E-0) 2.01E+01(1.23E-01)
F6 3.04E+00(2.05E+00) 2.47E+00(1.33E+00) 1.17E+00(1.32E+00)
F7 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00)
F8 3.22E-05(5.83E-05) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00)
F9 2.22E+01(6.12E+00) 2.19E+01(5.23E+00) 2.34E+01(6.24E+00)
F10 1.45E+01(4.09E+00) 2.45E-01(5.85E-02) 8.04E-02(4.02E-02)
F11 2.25E+03(3.85E+02) 1.70E+03(3.57E+02) 1.72E+03(3.86E+02)
F12 4.65E-01(6.78E-02) 1.69E-01(5.88E-02) 1.61E-01(8.10E-02)
F13 2.38E-01(3.30E-02) 2.06E-01(3.95E-02) 2.18E-01(5.09E-02)
F14 2.22E-01(2.52E-02) 2.14E-01(2.44E-02) 2.17E-01(3.54E-02)
F15 3.75E+00(5.78E-01) 2.54E+00(5.97E-01) 2.58E+00(6.51E-01)
F16 9.86E+00(4.12E-01) 9.30E+00(5.48E-01) 9.21E+00(6.16E-01)
F17 2.64E+02(1.55E+02) 2.64E+02(1.46E+02) 1.81E+02(1.53E+02)
F18 1.08E+01(3.90E+00) 1.09E+01(3.88E+00) 9.73E+00(5.20E+00)
F19 4.03E+00(4.68E-01) 3.25E+00(4.67E-01) 2.95E+00(4.19E-01)
F20 1.15E+01(4.02E+00) 9.57E+00(3.46E+00) 5.81E+00(2.03E+00)
F21 1.19E+02(8.70E+01) 1.04E+02(8.73E+01) 8.55E+01(7.92E+01)
F22 1.02E+02(5.88E+01) 7.08E+01(6.28E+01) 6.36E+01(6.24E+01)
F23 3.15E+02(2.76E-12) 3.15E+02(2.76E-12) 3.15E+02(2.76E-12)
F24 2.24E+02(7.36E-01) 2.24E+02(7.15E-01) 2.24E+02(2.01E+00)
F25 2.03E+02(1.53E-01) 2.03E+02(1.41E-01) 2.03E+02(1.14E-01)
F26 1.00E+02(3.24E-02) 1.00E+02(4.12E-02) 1.00E+02(5.22E-02)
F27 3.59E+02(3.57E+01) 3.59E+02(3.57E+01) 3.81E+02(3.07E+01)
F28 7.61E+02(4.09E+01) 7.57E+02(4.32E+01) 8.03E+02(4.98E+01)
F29 6.87E+02(1.20E+02) 6.97E+02(9.17E+01) 7.15E+02(7.85E-01)
F30 5.72E+02(1.78E+02) 5.63E+02(1.36E+02) 7.41E+02(3.75E+02)
20/7/3 10/17/3 +/-/=
4. Conclusions
This paper proposes an adaptive DE based on multi-subpopulation. In the proposed method, we
design an adaptive parameter control scheme, which can comprehensively consider the optimal solution
and diversity of each sub-population. This strategy can improve the robustness of multiple population
based on DE. Further, a mechanism of eliminating poor individuals is proposed to preserve the
population diversity at the early stage as well as improve the convergence speed of the population. In
addition, we propose a mutation strategy to solve the problem of insufficient diversity of the top-ranked
sub-populations when obtaining sub-populations based on Euclidean distance. The experimental results
229
clearly show the significance of the proposed strategy, and the obtained algorithm can achieve better or
comparable performance than related method.
Table III
Comparison results with related methods in 30D
JADE EPSDE MPEDE EDEV EMPADE
Functions Mean (Std) Mean (Std) Mean (Std) Mean (Std) Mean (Std)
F1 2.36E+03(2.89E+03) 8.12E+04(4.66E+05) 6.19E-06(4.42E-05) 6.33E+03(1.71E+04) 0.00E+00(0.00E+00)
F2 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)
F3 2.16E-11(8.08E-11) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00)
F4 0.00E+00(0.00E+00) 3.03E+00(2.21E+00) 5.67E-09(3.40E-08) 2.23E-01(8.07E-01) 0.00E+00(0.00E+00)
F5 2.03E+01(3.44E-02) 2.04E+01(3.42E-02) 2.04E+01(5.41E-02) 2.04E+01(5.06E-02) 2.01E+01(1.23E-01)
F6 9.15E+00(2.59E+00) 1.89E+01(1.67E+00) 7.53E-01(1.02E+00) 6.01E+00(4.15E+00) 1.17E+00(1.32E+00)
F7 3.98E-12(2.80E-11) 1.35E-03(3.95E-03) 1.45E-04(1.04E-03) 0.00E+00(0.00E+00 0.00E+00(0.00E+00)
F8 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)
F9 2.61E+01(4.09E+00) 4.29E+01(6.61E+00) 2.69E+01(6.51E+00) 3.44E+01(4.62E+00) 2.34E+01(6.24E+00)
F10 7.35E-03(1.43E-02) 2.27E-01(1.83E-01) 1.32E+00(7.00E-01) 1.61E+00(1.14E+01) 8.04E-02(4.02E-02)
F11 1.73E+03(2.11E+02) 3.49E+03(4.45E+02) 2.43E+03(3.37E+02) 2.95E+03(7.29E+02) 1.72E+03(3.86E+02)
F12 2.71E-01(2.87E-02) 5.12E-01(5.41E-02) 4.74E-01(8.78E-02) 6.71E-01(1.71E-01) 1.61E-01(8.10E-02)
F13 2.03E-01(3.62E-02) 2.58E-01(4.38E-02) 2.11E-01(3.85E-02) 2.06E-01(3.19E-02) 2.18E-01(5.09E-02)
F14 2.28E-01(3.38E-02) 3.00E-01(8.39E-02) 2.37E-01(2.77E-02) 2.31E-01(3.34E-02) 2.17E-01(3.54E-02)
F15 3.16E+00(3.27E-01) 5.37E+00(7.27E-01) 4.04E+00(5.31E-01) 4.13E+00(4.98E-01) 2.58E+00(6.51E-01)
F16 9.40E+00(3.83E-01) 1.12E+01(4.09E-01) 1.00E+01(5.37E-01) 1.01E+01(3.76E-01) 9.21E+00(6.16E-01)
F17 9.97E+02(3.81E+02) 5.00E+04(5.06E+04) 2.13E+02(1.47E+02) 9.09E+02(6.23E+02) 1.81E+02(1.53E+02)
F18 6.03E+01(2.99E+01) 2.02E+02(3.62E+02) 1.54E+01(7.11E+00) 2.55E+01(1.74E+01) 9.73E+00(5.20E+00)
F19 4.58E+00(7.63E-01) 1.31E+01(1.24E+00) 3.78E+00(6.85E-01) 4.70E+00(1.56E+00) 2.95E+00(4.19E-01)
F20 2.20E+03(2.98E+03) 9.07E+01(1.55E+02) 9.06E+00(2.96E+00) 1.50E+01(3.61E+00) 5.81E+00(2.03E+00)
F21 2.79E+02(1.53E+02) 1.53E+04(3.15E+04) 1.04E+02(9.60E+01) 3.55E+02(1.79E+02) 8.55E+01(7.92E+01)
F22 1.33E+02(7.57E+01) 2.06E+02(8.50E+01) 1.03E+02(7.17E+01) 7.48E+01(5.44E+01) 6.36E+01(6.24E+01)
F23 3.15E+02(3.21E-12) 3.14E+02(1.38E-12) 3.15E+02(3.21E-12)) 3.14E+02(0.00E+00) 3.15E+02(2.76E-12)
F24 2.25E+02(3.05E+00) 2.29E+02(5.66E+00) 2.25E+02(2.54E+00) 2.25E+02(2.61E+00) 2.24E+02(2.01E+00)
F25 2.05E+02(1.70E+00) 2.00E+02(4.42E-01) 2.03E+02(3.01E-01) 2.00E+02(1.38E+00) 2.03E+02(1.14E-01)
F26 1.02E+02(1.40E+01) 1.00E+02(4.82E-02) 1.00E+02(2.84E-02) 1.00E+02(3.77E-02) 1.00E+02(5.22E-02)
F27 3.42E+02(4.92E+01) 8.24E+02(1.05E+02) 3.73E+02(4.50E+01)) 3.50E+02(4.47E+01) 3.81E+02(3.07E+01)
F28 7.89E+02(4.09E+01) 3.93E+02(1.19E+01) 8.34E+02(3.28E+01) 3.80E+02(6.16E+00) 8.03E+02(4.98E+01)
F29 7.77E+02(1.97E+02) 2.14E+02(1.26E+00) 6.82E+02(1.33E+02) 2.14E+02(1.11E+00) 7.15E+02(7.85E-01)
F30 1.36E+03(4.76E+02) 5.91E+02(1.38E+02) 6.08E+02(1.85E+02) 4.23E+02(1.41E+02) 7.41E+02(3.75E+02)
19/8/3 22/4/4 19/11/0 17/6/7 +/-/=
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