Differential evolution with Gaussian mutation and dynamic parameter adjustment

Abstract

Differential evolution (DE) is a remarkable evolutionary algorithm for global optimization over continuous search space, whose performance is significantly influenced by its mutation operator and control parameters (scaling factor and crossover rate). In order to enhance the performance of DE, we adopt a novel Gaussian mutation operator and a modified common mutation operator to collaboratively produce new mutant vectors, and employ a periodic function and a Gaussian function to generate the required values of scaling factor and crossover rate, respectively. In the proposed variant of DE (denoted by GPDE), the two adopted mutation operators are adaptively applied to generate the corresponding mutant vector of each individual based on their own cumulative scores, the periodic scaling factor can provide a better balance between exploration ability and exploitation ability, and the Gaussian function-based crossover rate will possess fluctuant value, which possibly enhance the population diversity. To verify the performance of proposed GPDE, a suite of thirty benchmark functions and four real-world problems are applied to conduct the simulation experiment. The simulation results demonstrate that the proposed GPDE performs significantly better than five state-of-the-art DE variants and other two meta-heuristics algorithms.

Full text

Soft Comput
DOI 10.1007/s00500-017-2885-z
METHODOLOGIES AND APPLICATION
Differential evolution with Gaussian mutation and dynamic
parameter adjustment
Gaoji Sun1·Yanfei Lan2·Ruiqing Zhao2
© Springer-Verlag GmbH Germany 2017
Abstract Differential evolution (DE) is a remarkable evo-
lutionary algorithm for global optimization over continuous
search space, whose performance is significantly influenced
by its mutation operator and control parameters (scaling fac-
tor and crossover rate). In order to enhance the performance
of DE, we adopt a novel Gaussian mutation operator and a
modified common mutation operator to collaboratively pro-
duce new mutant vectors, and employ a periodic function and
a Gaussian function to generate the required values of scaling
factor and crossover rate, respectively. In the proposed variant
of DE (denoted by GPDE), the two adopted mutation oper-
ators are adaptively applied to generate the corresponding
mutant vector of each individual based on their own cumu-
lative scores, the periodic scaling factor can provide a better
balance between exploration ability and exploitation ability,
and the Gaussian function-based crossover rate will possess
fluctuant value, which possibly enhance the population diver-
sity. To verify the performance of proposed GPDE, a suite
of thirty benchmark functions and four real-world problems
are applied to conduct the simulation experiment. The simu-
lation results demonstrate that the proposed GPDE performs
significantly better than five state-of-the-art DE variants and
other two meta-heuristics algorithms.
Communicated by V. Loia.
BGaoji Sun
gsun@zjnu.edu.cn
BYanfei Lan
lanyf@tju.edu.cn
1College of Economic and Management, Zhejiang Normal
University, Jinhua 321004, China
2Institute of Systems Engineering, Tianjin University, Tianjin
300072, China
Keywords Differential evolution ·Gaussian mutation ·
Dynamic parameter adjustment ·Evolutionary computation ·
Global optimization
1 Introduction
Differential evolution (DE), proposed by Storn and Price
(Storn and Price 1997), is a simple yet efficient evolution-
ary algorithm (EA) for the global numerical optimization.
Due to its simple structure and ease of use, DE has been
successfully applied to solve many real-world problems,
including decision-making (Zhang et al. 2010), dynamic
scheduling (Tang et al. 2014), parameter optimization (Gong
and Cai 2014), spam detection (Idris et al. 2014), system fault
diagnosis (Zhao et al. 2014), motion estimation (Cuevas et al.
2013) and so on. More details on the recent research about
DE can be found in the literature reviews (Neri and Tirronen
2010;Das and Suganthan 2011a) and the references therein.
Three evolutionary operators (mutation, crossover and
selection) and three control parameters (population size,
scaling factor and crossover rate) are included in the orig-
inal DE algorithm, which have significant influence on its
performance. Thus, many researchers have been engaged
in improving DE by designing new evolutionary operators,
combining multiple operators and adopting adaptive or self-
adaptive strategies for those control parameters. Although
various variants of DE have been proposed, there still exists
a big room for improvement, owing to the thorny work of bal-
ancing the global exploration ability and local exploitation
ability (Lin and Gen 2009;ˇ
Crepinšek et al. 2013).
Using Gaussian function can randomly produce new solu-
tions around a given position, which may provide an excellent
exploitation ability. Meanwhile, periodic or fluctuant param-
eter adjustment strategy possibly can achieve a good balance
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G. Sun et al.
between the exploitation operation around the already-found
good solutions and the exploration operation for seeking
out non-visited regions in the search space. Inspired by
the above observations, we design a novel Gaussian muta-
tion operator (which, respectively, takes the position of the
best individual among three randomly selected individuals
and the distance between the other two as the mean and
standard deviation of Gaussian distribution) and a modified
common mutation operator (denoted by DE/rand-worst/1) to
collaboratively produce the new potential position for every
individual, and the collaborative rule between them relies on
their own cumulative scores during the evolutionary process.
In addition, the scaling factor adopts a cosine function to real-
ize the objective of adjusting its value periodically, and the
crossover rate employs a Gaussian function to dynamically
adjust the population diversity during the evolutionary pro-
cess. At last, a novel DE variant is proposed via combining
the above-mentioned Gaussian mutation operator, DE/rand-
worst/1 and parameter adjustment strategies, which is called
GPDE for short. A suite of 30 benchmark functions with
different dimensions and four real-world optimization prob-
lems are applied to evaluate the performance of GPDE, and
its performance is compared to five excellent DE variants
and two up-to-date meta-heuristics algorithms. The com-
parative results show that GPDE obviously outperforms the
seven compared algorithms. Moreover, the parameter anal-
ysis expresses that the adopted control parameters within
GPDE are robust.
The remainder of this paper is organized as follows. Sec-
tion 2briefly introduces the basic operators of original DE
algorithm. Section 3reviews some currently related works
on DE. Section 4provides a detailed description of the pro-
posed GPDE algorithm and its overall procedure. Section 5
presents the comparison between GPDE and seven compared
algorithms. Section 6draws the conclusions.
2 Differential evolution
DE is a population-based stochastic search algorithm which
simulates the natural evolutionary process via mutation,
crossover and selection to move its population toward the
global optimum. The DE algorithm mainly contains the fol-
lowing four operations.
2.1 Initialization operation
Similar to other EAs, DE searches for a global optimum in
the D-dimensional real parameter space with a population of
vectors xi=[xi,1,xi,2,...,xi,D],i=1,2,...,NP, where
NP is the population size. An initial population should cover
the entire search space by uniformly randomizing individuals
between the prescribed lower bounds L=L1,L2,...,LD
and upper bounds U=U1,U2,...,UD.The jth compo-
nent of the ith individual can be initialized as follows,
xi,j=Lj+rand[0,1]·Uj−Lj,(1)
where rand[0,1]represents a uniformly distributed random
number within the interval [0,1]and is used throughout the
paper.
2.2 Mutation operation
After the initialization operation, DE employs a mutation
operation to produce a mutant vector vi=vi,1,v
i,2,...,v
i,D
for each target vector xi. The followings are five most fre-
quently used mutation operators implemented in various DE
algorithms.
(1) DE/rand/1
vi=xr1+F·(xr2−xr3). (2)
(2) DE/best/1
vi=xbest +F·(xr1−xr2). (3)
(3) DE/current-to-best/1
vi=xi+F·(xbest −xr1)+F·(xr2−xr3). (4)
(4) DE/best/2
vi=xbest +F·(xr1−xr2)+F·(xr3−xr4). (5)
(5) DE/rand/2
vi=xr1+F·(xr2−xr3)+F·(xr4−xr5). (6)
The indices r1,r2,r3,r4and r5in the above equations
are mutually exclusive integers randomly generated from
set {1,2,...,NP}and are also different from the index
i. The parameter Fis called the scaling factor, which is
a positive real number for scaling the difference vectors.
The vector xbest =(xbest,1,xbest,2,...,xbest,D)is the
best individual in the current population.
2.3 Crossover operation
After the mutation operation, DE performs a binomial
crossover operator on target vector xiand its correspond-
ing mutant vector vito produce a trial vector ui=
ui,1,ui,2,...,ui,D. This process can be expressed as
ui,j=vi,j,if rand[0,1]≤CR or j=jrand 
xi,j,otherwise.(7)
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Differential evolution with Gaussian mutation and dynamic parameter adjustment
The crossover rate CR is a user-specified constant within the
interval (0,1)in original DE, which controls the fraction
of trial vector components inherited from the mutant vec-
tor. The index jrand is an integer randomly chosen from set
{1,2,...,D}, which is used to ensure that the trial vector
has at least one component different from the target vector.
2.4 Selection operation
After the crossover operation, a selection operation is exe-
cuted between the trial vector and the target vector according
to their fitness values f(·), and the better one will survive
to the next generation. Without loss of generality, we only
consider minimization problems. Specifically, the selection
operator can be expressed as follows:
xi=ui,if f(ui)≤f(xi)
xi,otherwise.(8)
From the expression of selection operator (8), it is easy to
see that the population of DE either gets better or remains
the same in fitness status, but never deteriorates.
3 Related work
In the past decades, many meta-heuristic algorithms have
been proposed, such genetic algorithm (Goldberg 1989), dif-
ferential evolution (Storn and Price 1997), particle swarm
optimization (Kennedy et al. 2001), ant colony optimiza-
tion (Dorigo and Blum 2005), joint operations algorithm (Sun
et al. 2016) and so on. These meta-heuristic algorithms have
been successfully applied in various fields, such as produc-
tion planning (Lan et al. 2012), procurement planning (Sun
et al. 2010), location problems (Wang and Watada 2012),
workforce planning (Yang et al. 2017). Among these meta-
heuristic algorithms, DE has shown outstanding performance
in solving many test functions and real-world problems, but
its performance highly depends on the selected evolutionary
operators and the values of control parameters. To over-
come these drawbacks, many variants have been proposed
to improve the performance of DE. In this section, we only
provide a brief overview of the enhanced approaches which
is related to our work.
There are lots of researchers who tried to enhance DE
via designing new mutation operators or combining mul-
tiple operators. Qin et al. (2009) proposed a self-adaptive
DE (SADE) which focuses on the mutation operator selec-
tion and crossover rate of DE. Zhang and Sanderson (2009)
presented a self-adaptive DE with optional external archive
(JADE) which employs a new mutation operator called
“DE/current-to-pbest.” Han et al. (2013) introduced a group-
based DE variant (GDE) which divides the population into
two groups and each group employs a different mutation
operator. Wang et al. (2013) proposed a modified Gaus-
sian bare-bones DE variant (MGBDE) which combines two
mutation operators, and one of the mutation operators is
designed based on Gaussian distribution. Das et al. (2009)
presented two kinds of topological neighborhood models
and embedded them into the mutation operators of DE.
Gong et al. (2011a) introduced a simple strategy adapta-
tion mechanism (SaM) which can be used for coordinating
different mutation operators. Many other DE variants also
adopted new designed mutation operator or multi-mutation
operator strategies with different searching features, such as
NDi-DE (Cai and Wang 2013), MS-DE (Wang et al. 2014),
CoDE (Wang et al. 2011), HLXDE (Cai and Wang 2015),
MDE_pBX (Islam et al. 2012), AdapSS-JADE (Gong et al.
2011b), IDDE (Sun et al. 2017).
Some other researchers applied parameter adjustment to
improve the performance of DE. For instance, Draa et al.
(2015) introduced sinusoidal differential evolution (SinDE),
which adopts two sinusoidal formulas to adjust the values of
scaling factor and crossover rate. Brest et al. (2006) proposed
a self-adaptive scheme for the DE’s control parameters. Liu
and Lampinen (2005) applied fuzzy logic controllers to adapt
the value of crossover rate. Zhu et al. (2013) adopted an adap-
tive population tuning scheme to enhance DE. Ghosh et al.
(2011) introduced a control parameter adaptation strategy,
which is based on the fitness values of individuals in DE
population. Yu et al. (2014) proposed a two-level adaptive
parameter control strategy, which is based on the optimiza-
tion states and the fitness values of individuals. Sarker et al.
(2014) introduced a new mechanism to dynamically select
the best performing combinations of control parameters,
which is based on the success rate of each parameter com-
bination. Karafotias et al. (2015) provided a comprehensive
overview about the parameter control in evolutionary algo-
rithms.
Actually, many aforementioned references simultane-
ously utilize new evolutionary operators and adaptive control
parameters to enhance the performance of DE, including
SADE (Qin et al. 2009), JADE (Zhang and Sanderson 2009),
MGBDE (Wang et al. 2013) and CoDE (Wang et al. 2011).
In addition, Mallipeddi et al. (2011) employed a pool of
distinct mutation operators along with a pool of values for
each control parameter which coexists and competes to pro-
duce offsprings during the evolutionary process. Yang et al.
(2015) proposed a mechanism named auto-enhanced popu-
lation diversity to automatically enhance the performance of
DE, which is based on the population diversity at the dimen-
sional level. Biswas et al. (2015) presented an improved
information-sharing mechanism among the individuals to
enhance the niche behavior of DE. Tang et al. (2015)intro-
duced a novel variant of DE with an individual-dependent
mechanism which includes an individual-dependent param-
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G. Sun et al.
eter setting and mutation operator. However, these DE
variants still cannot resolve the problems of premature con-
vergence or stagnation when handling complex optimization
problems.
4 Description of GPDE
In this section, we firstly provide a detailed description of
the new Gaussian mutation operator, the modified common
mutation operator and the cooperative rule between them,
and then summarize the overall procedure of GPDE.
4.1 Gaussian mutation operator
Gaussian distribution is very important and often used in
statistics and natural sciences to represent real-valued ran-
dom variable, which can be denoted by N(μ, σ 2), where
μand σare its mean and standard deviation, respectively.
It is well known that there is 3-σrule which exists in Gaus-
sian distribution. Specifically, about 68% of the values drawn
from the Gaussian distribution N(μ, σ 2)are within the inter-
val [μ−σ, μ +σ]; about 95% of the values lie within the
interval [μ−2σ, μ +2σ]; and about 99.7% are within the
interval [μ−3σ, μ +3σ].
The 3-σrule of Gaussian distribution provides a wonder-
ful chance to control the hunting zone which depends on the
requirement of considered problem. Actually, Gaussian dis-
tribution has been widely used to adjust the values of control
parameters, such as SADE (Qin et al. 2009), MGBDE (Wang
et al. 2013), DEGL (Das et al. 2009) and MDE_pBE (Islam
et al. 2012), but rarely applied to generate new mutation
operator. In order to take full advantage of Gaussian dis-
tribution, we propose the following novel mutation operator
which combines crossover operator to directly produce the
new trial vector (denoted by ug
i=(ug
i,1,ug
i,2,...,ug
i,D))for
the ith individual, i=1,2,...,NP,
ug
i,j=⎧
⎪
⎨
⎪
⎩
Nxr1,j,xr2,j−xr3,j2,if j=jrand
or rand[0,1]≤CRi
t,
xi,j,otherwise,
(9)
where the indices r1,r2,r3are mutually exclusive inte-
gers randomly generated from the set {1,2,...,NP}and
are also different from the base index i. Note that the
r1th individual is the best one among the three randomly
selected individuals, and the novel Gaussian mutation opera-
tor Nxr1,j,xr2,j−xr3,j2in formula (9), respectively,
takes the position xr1,jof the best one and the distance
|xr2,j−xr3,j|between the other two as the mean and standard
deviation; meanwhile, it will only be executed when meeting
the triggering condition j=jrand or rand[0,1]≤CRi
t,
which means that the proposed Gaussian mutation opera-
tor only transfers a certain proportional dimensions of each
corresponding individual to new positions around the best
selected one. Furthermore, if the new position of one dimen-
sion is within one standard deviation |xr2,j−xr3,j|away
from the mean xr1,j, we can call that it executes the exploita-
tion operation in this dimension; otherwise, it carries out the
exploration operation. Therefore, according to the 3–σrule,
the designed Gaussian mutation operator can simultaneously
conduct exploitation and exploration works, especially the
former one. In addition, dynamic parameter CRi
texpresses
the crossover rate of the ith individual in the tth generation,
which can be computed by
CRi
t=N(0.5,V), i=1,2,...,NP,t=1,2,...,T,
(10)
where Vindicates the variance of Gaussian distribution
N(0.5,V), and it is applied to control the fluctuation of
crossover rate, and Tis the maximum allowable generation.
It should be pointed out that Vis a user-specified constant
and it has to simultaneously ensure that the value of dynamic
crossover rate CRi
thas a certain extent of fluctuation and
almost falls into the range [0,1], and thus its reasonable
interval is [0.01,0.1]. Actually, the individuals employing
different crossover rates in the same generation can poten-
tially enhance the population diversity.
4.2 DE/rand-worst/1
In mutation operator, the base individual can be taken as the
center point of the searching area, the difference vector is
applied to set the searching direction, and the scale factor is
employed to control the step size. In a general way, a bet-
ter base individual has a higher probability to produce better
offsprings, a more proper direction induces a more efficient
searching behavior of the population, and a periodic scaling
factor has potential advantage in balancing exploration abil-
ity and exploitation ability. Based on the aforementioned con-
siderations, we incorporate the fitness information of selected
individuals and a periodic scaling factor to modify the
most popular mutation operator (DE/rand/1). The obtained
modified mutation operator DE/rand/1 (denoted by DE/rand-
worst/1) which combines crossover operator is applied to
directly produce the new trial vector (denoted by um
i=
(um
i,1,um
i,2,...,um
i,D), i=1,2,...,NP) of each individual,
the corresponding formula can be described as follows,
um
i,j=⎧
⎨
⎩
xr
1,j+Ft·(xr
2,j−xr
3,j), if j=jrand or
rand[0,1]≤CRi
t,
xi,j,otherwise,
(11)
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Differential evolution with Gaussian mutation and dynamic parameter adjustment
where the indices r
1,r
2and r
3are mutually exclusive inte-
gers randomly chosen from the set {1,2,...,NP}, which are
also different from the index i. Moreover, the r
3th individual
is the worst one among the three randomly selected indi-
viduals, which can ensure that the base individual is not the
worst one and the searching direction is relatively better. In
addition, note that the mutation operator DE/rand-worst/1 in
formula (11) has the same triggering condition with Gaussian
mutation operator in formula (9). About the periodic scal-
ing factor, we apply a cosine function to realize the periodic
adjustment strategy, which can be expressed via the follow-
ing formula,
Ft=cos(t·FR ·π),(12)
where Ftis the value of scaling factor in the tth generation,
and FR represents the frequency of cosine function, which is a
user-specified constant and applied to adjust the turnover rate
between the exploration and exploitation operations. Usually,
a smaller frequency FR corresponds to a smaller turnover
rate.
4.3 Cooperative rule
Up to now, two mutation operators (Gaussian and DE/rand-
worst/1) have been introduced, which combine a same
crossover operator to produce the new trial vector uifor
each individual, and then the cooperative rule between them
becomes a burning problem. A natural and reasonable rule is
that adaptively executing one of the two mutation operators in
terms of their own performance. To evaluate the performance
of adopted mutation operators during the evolutionary pro-
cess, we introduce a new concept called “cumulative score”
into the mutation operation. For the two adopted mutation
operators, their cumulative scores during the evolutionary
process can be obtained via the following three steps. Firstly,
the values of their initial cumulative scores are set (denoted
by CSg
0and CSm
0) to 0.5. Secondly, suppose that the values
of their historical cumulative scores are CSg
t−1and CSm
t−1,
and then their single-period scores in the current generation
(denoted by Sg
tand Sm
t) can be obtained via the following
two formulas, respectively.
Sg
t=⎧
⎨
⎩
Cg
t
Ng
t
,if Ng
t>0,
CSg
t−1
t,otherwise,
(13)
Sm
t=Cm
t
Nm
t,if Nm
t>0,
CSm
t−1
t,otherwise,
(14)
where the indices Ng
tand Nm
trepresent the numbers of new
trial vectors produced by Gaussian mutation operator and
DE/rand-worst/1 in the tth generation, respectively. Actually,
the value of Ng
talways is equals to NP −Nm
t, owing to the
fact that the population only executes NP times of mutation
operators in one generation. And the indices Cg
tand Cm
t,
respectively, represent the success times of the two opera-
tors’ execution, where the concept of success means that the
new produced trial vector is better than the original target
vector. Note that formulas (13) and (14) express that the cur-
rent single-period score of each adopted mutation operator
is equal to its current success rate when executing at least
once in the current generation, otherwise takes its average
value of historical cumulative score. Thirdly, after obtaining
the current single-period scores of the two adopted mutation
operators, their current cumulative scores can be updated by
the following two formulas,
CSg
t=CSg
t−1+Sg
t,(15)
CSm
t=CSm
t−1+Sm
t.(16)
Now, the value of a parameter involved in cooperative
rule can be derived in terms of the two mutation operators’
cumulative scores, which can be calculated by,
CSt=CSg
t
CSg
t+CSm
t
,(17)
where parameter CStis applied to control the selection prob-
ability of Gaussian mutation operator in the next generation.
Furthermore, the detailed cooperative rule can be described
as follows,
ui=ug
i,if rand[0,1]<CSt,
um
i,otherwise.(18)
The cooperative rule (18) shows that the chance of execut-
ing the adopted two mutation operators relies on their own
cumulative scores, and the one with higher cumulative score
has more chance to produce the trial vectors. After the new
trial vector produced, GPDE will compare the fitness values
of each individual xiand its new trial vector uiand then
produce the offspring via the selection operator (8).
4.4 The overall procedure of GPDE
We have provided a detailed description of Gaussian muta-
tion operator, DE/rand-worst/1, and the cooperative rule
between them. Now, we summarize the overall procedure
of GPDE into Algorithm 1.
5 Comparison and result analysis
In this section, we firstly provide the test functions, real-world
problems and compared DE algorithms, secondly present the
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G. Sun et al.
Tabl e 1 Summary of the IEEE
CEC 2014 benchmark functions Type No. Functions f∗
Unimodal f1Rotated high conditioned elliptic function 100
f2Rotated bent cigar function 200
f3Rotated discus function 300
Multimodal f4Shifted and rotated Rosenbrock function 400
f5Shifted and rotated Ackley’s function 500
f6Shifted and rotated Weierstrass function 600
f7Shifted and rotated Griewank’s function 700
f8Shifted Rastrigin’s function 800
f9Shifted and rotated Rastrigin’s function 900
f10 Shifted Schwefel’s function 1000
f11 Shifted and rotated Schwefel’s function 1100
f12 Shifted and rotated Katsuura function 1200
f13 Shifted and rotated HappyCat function 1300
f14 Shifted and rotated HGBat function 1400
f15 Shifted and rotated Expanded Griewank’s
plus Rosenbrock’s function
1500
f16 Shifted and rotated Expanded Scaffer’s F6
function
1600
Hybrid f17 Hybrid function 1 1700
f18 Hybrid function 2 1800
f19 Hybrid function 3 1900
f20 Hybrid function 4 2000
f21 Hybrid function 5 2100
f22 Hybrid function 6 2200
Composition f23 Composition function 1 2300
f24 Composition function 2 2400
f25 Composition function 3 2500
f26 Composition function 4 2600
f27 Composition function 5 2700
f28 Composition function 6 2800
f29 Composition function 7 2900
f30 Composition function 8 3000
Search space: [−100,100]D
comparative results between GPDE and the other seven algo-
rithms, and analyze the effects of control parameters on the
performance of GPDE at last.
5.1 Test functions and real-world problems
In order to evaluate the performance of GPDE, we apply a set
of 30 well-known test functions from IEEE CEC 2014 (Liang
et al. 2013) and four real-world problems to conduct the com-
parative experiment. Specifically, based on the characteristics
of the 30 test functions, they can be divided into four classes,
which are summarized in Table 1. Moreover, the adopted
test functions are carried out in the comparative experiment
when their dimensions are equal to 30, 50 and 100, respec-
tively. In addition, the four real-world problems (denoted by
rf1,rf2,rf3and rf4, respectively) are widely used to evaluate
the performance of various algorithms, which are applica-
tions to parameter estimation for frequency-modulated sound
waves (Das and Suganthan 2011b), spread spectrum radar
poly-phase code design (Das and Suganthan 2011b), sys-
tems of linear equations (García-Martínez et al. 2008), and
parameter optimization for polynomial fitting problem (Her-
rera and Lozano 2000), respectively.
5.2 Compared algorithms and parameter configurations
In our comparative experiment, GPDE is compared with five
excellent DE variants, including SADE (Qin et al. 2009),
JADE (Zhang and Sanderson 2009), GDE (Han et al. 2013),
MGBDE (Wang et al. 2013) and SinDE (Draa et al. 2015).
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Differential evolution with Gaussian mutation and dynamic parameter adjustment
Tabl e 2 Comparative results on functions f1−f15 with D=30
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
f1Mean 3.22e+04 3.76e+04 6.85e+03 5.57e+03 1.92e+06 8.63e+04 9.12e+05 5.21e+04
Std. 2.17e+04 3.66e+04 6.41e+03 3.45e+03 1.11e+06 6.15e+04 6.40e+05 3.51e+04
+/=/−− − − − + + + −
f2Mean 4.84e−20 3.08e−20 3.06e−22 3.92e−15 2.90e−24 7.49e−01 8.87e+03 1.35e−23
Std. 2.39e−19 9.38e−20 3.80e−22 1.95e−14 1.01e−23 1.12e+00 9.26e+03 2.17e−22
+/=/−+ + + + = + + −
f3Mean 9.82e−10 1.34e−01 3.56e−03 2.08e−18 2.18e−12 4.32e+00 1.95e+04 5.42e−25
Std. 4.91e−09 6.05e−01 9.98e−03 9.54e−18 9.18e−12 8.66e+00 1.76e+04 1.83e−24
+/=/−+ + + + + + + −
f4Mean 1.39e+01 1.94e+01 5.48e+00 9.95e−05 1.43e+01 1.15e+01 4.09e+01 2.99e+00
Std. 2.72e+01 3.09e+01 1.89e+01 4.48e−04 2.28e+01 2.54e+01 5.16e+01 1.49e+01
+/=/−+ + + − + + + −
f5Mean 2.04e+01 2.00e+01 2.09e+01 2.02e+01 2.06e+01 2.09e+01 2.00e+01 2.00e+01
Std. 4.74e−02 3.99e−03 1.48e−01 3.92e−02 4.77e−02 1.21e−01 9.03e−03 6.53e−06
+/=/−+ + + + + + + −
f6Mean 8.93e+00 1.14e+01 8.25e+00 2.22e+01 1.93e−02 4.03e+00 1.33e+01 1.33e+00
Std. 2.03e+00 1.75e+00 2.85e+00 3.64e+00 9.27e−02 2.02e+00 3.13e+00 1.16e+00
+/=/−+ + + + − + + −
f7Mean 1.89e−02 2.69e−02 1.30e−02 1.12e−02 0.00e+00 2.55e−02 2.32e−02 2.17e−03
Std. 2.10e−02 2.17e−02 1.75e−02 1.44e−02 0.00e+00 4.00e−02 2.63e−02 4.17e−03
+/=/−+ + + + = + + −
f8Mean 4.78e+00 3.98e−02 6.38e+01 1.29e+02 4.70e−01 2.60e+01 0.00e+00 9.79e+00
Std. 2.54e+00 1.99e−01 1.58e+01 3.17e+01 5.69e−01 7.64e+00 0.00e+00 3.72e+00
+/=/−− − + + − + − −
f9Mean 4.59e+01 4.97e+01 7.46e+01 1.56e+02 3.58e+01 3.13e+01 7.34e+01 3.46e+01
Std. 1.09e+01 8.71e+00 2.74e+01 2.95e+01 7.51e+00 1.02e+01 1.87e+01 9.26e+00
+/=/−+ + + + = = + −
f10 Mean 3.40e+00 6.98e+00 1.80e+03 1.25e+03 9.31e+00 2.42e+03 4.49e−03 1.25e+02
Std. 2.09e+00 2.40e+01 6.43e+02 7.96e+02 4.77e+00 1.18e+03 8.05e−03 9.65e+01
+/=/−− − + + − + − −
f11 Mean 2.42e+03 2.04e+03 5.06e+03 2.85e+03 2.35e+03 6.58e+03 2.07e+03 1.97e+03
Std. 5.45e+02 2.43e+02 1.60e+03 6.29e+02 4.08e+02 4.87e+02 3.70e+02 4.71e+02
+/=/−+ = + + + + = −
f12 Mean 5.91e−01 1.28e−01 1.77e+00 3.36e−01 8.04e−01 1.90e+00 8.49e−02 1.49e−01
Std. 8.56e−02 2.61e−02 8.70e−01 3.67e−02 1.24e−01 3.80e−01 1.85e−02 7.84e−02
+/=/−+ = + + + + − −
f13 Mean 2.80e−01 3.09e−01 3.64e−01 4.40e−01 2.06e−01 3.10e−01 3.27e−01 2.40e−01
Std. 5.43e−02 5.90e−02 7.23e−02 7.72e−02 5.27e−02 6.09e−02 9.95e−02 6.84e−02
+/=/−+ + + + = + + −
f14 Mean 2.41e−01 2.50e−01 2.95e−01 2.58e−01 2.42e−01 2.82e−01 2.06e−01 2.22e−01
Std. 4.47e−02 1.01e−01 9.12e−02 5.78e−02 2.60e−02 4.64e−02 1.32e−01 3.40e−02
+/=/−= = + + + + − −
f15 Mean 4.57e+00 1.23e+01 8.12e+00 1.33e+01 4.81e+00 6.57e+00 7.15e+00 3.75e+00
Std. 1.31e+00 6.69e+00 2.95e+00 2.37e+00 9.80e−01 5.03e+00 3.14e+00 9.44e−01
+/=/−+ + + + + + + −
123

G. Sun et al.
Tabl e 3 Comparative results on functions f16 −f30 with D=30
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
f16 Mean 1.03e+01 1.02e+01 1.10e+01 1.07e+01 1.00e+01 1.24e+01 9.76e+00 9.64e+00
Std. 4.16e−01 3.56e−01 1.12e+00 6.49e−01 5.22e−01 2.49e−01 6.60e−01 7.85e−01
+/=/−++++++=−
f17 Mean 5.95e+03 6.29e+04 1.91e+04 1.71e+03 1.25e+05 1.37e+04 7.32e+05 3.84e+03
Std. 4.17e+03 6.21e+04 3.03e+04 7.57e+02 1.20e+05 9.70e+03 4.57e+05 3.53e+03
+/=/−+++−+++−
f18 Mean 8.34e+02 7.06e+02 7.43e+01 1.03e+02 5.15e+02 2.70e+03 5.53e+03 2.16e+01
Std. 1.22e+03 9.62e+02 1.90e+02 3.60e+01 6.94e+02 3.24e+03 5.52e+03 9.20e+00
+/=/−+++++++−
f19 Mean 4.23e+00 1.11e+01 4.73e+00 2.33e+01 3.71e+00 6.54e+00 1.14e+01 3.45e+00
Std. 1.23e+00 1.66e+01 1.20e+00 2.55e+01 7.18e−01 8.46e+00 1.84e+01 1.18e+00
+/=/−+++++++−
f20 Mean 1.08e+02 1.37e+03 2.88e+01 7.80e+01 2.57e+01 3.32e+02 3.34e+04 1.71e+01
Std. 1.42e+02 2.06e+03 2.19e+01 4.38e+01 2.85e+01 2.95e+02 1.77e+04 1.12e+01
+/=/−+++++++−
f21 Mean 4.84e+03 5.74e+03 3.29e+03 8.50e+02 9.23e+03 9.11e+03 4.50e+05 3.63e+03
Std. 4.42e+03 7.17e+03 5.36e+03 3.90e+02 7.52e+03 7.06e+03 3.82e+05 4.61e+03
+/=/−++=−+++−
f22 Mean 1.55e+02 2.10e+02 4.82e+02 7.17e+02 7.25e+01 1.52e+02 6.73e+02 2.90e+02
Std. 8.61e+01 7.94e+01 2.09e+02 2.74e+02 6.28e+01 1.30e+02 1.78e+02 1.41e+02
+/=/−−=++−−+−
f23 Mean 3.15e+02 3.15e+02 3.15e+02 3.15e+02 3.15e+02 3.15e+02 3.15e+02 3.15e+02
Std. 1.38e−13 3.48e−13 6.13e−13 1.21e−12 1.24e−13 1.78e−13 3.27e−05 1.04e−13
+/=/−=======−
f24 Mean 2.28e+02 2.30e+02 2.35e+02 2.42e+02 2.23e+02 2.36e+02 2.27e+02 2.27e+02
Std. 5.41e+00 3.87e+00 7.27e+00 1.18e+01 8.57e−01 7.48e+00 3.12e+00 4.51e+00
+/=/−=+++−+=−
f25 Mean 2.10e+02 2.12e+02 2.04e+02 2.20e+02 2.04e+02 2.04e+02 2.09e+02 2.04e+02
Std. 2.07e+00 1.47e+00 1.19e+00 6.22e+00 6.64e−01 8.41e−01 4.60e+00 7.80e−01
+/=/−++=+==+−
f26 Mean 1.12e+02 1.56e+02 1.00e+02 1.56e+02 1.00e+02 1.00e+02 1.42e+02 1.08e+02
Std. 3.31e+01 5.05e+01 9.32e−02 5.04e+01 3.99e−02 7.14e−02 6.27e+01 2.76e+01
+/=/−=+=+−=+−
f27 Mean 4.36e+02 4.29e+02 4.28e+02 8.61e+02 3.02e+02 4.14e+02 6.08e+02 3.34e+02
Std. 5.74e+01 6.28e+01 5.93e+01 3.11e+02 7.08e+00 4.31e+01 1.42e+02 3.49e+01
+/=/−++++−++−
f28 Mean 9.12e+02 9.19e+02 9.56e+02 2.60e+03 7.94e+02 8.65e+02 1.22e+03 7.93e+02
Std. 4.37e+01 6.51e+01 6.23e+01 7.61e+02 3.23e+01 6.06e+01 4.64e+02 2.60e+01
+/=/−++++=++−
f29 Mean 6.83e+02 7.84e+02 4.80e+02 7.18e+02 1.32e+03 3.48e+05 1.02e+06 6.32e+02
Std. 2.64e+02 2.69e+02 2.73e+02 1.26e+02 2.39e+02 1.73e+06 2.82e+06 1.96e+02
+/=/−=+−=+++−
f30 Mean 1.96e+03 2.05e+03 1.11e+03 2.14e+03 8.10e+02 1.26e+03 3.17e+03 1.62e+03
Std. 6.22e+02 5.50e+02 3.00e+02 7.18e+02 1.69e+02 4.43e+02 8.42e+02 7.04e+02
+/=/−++−+−=+−
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
Tabl e 4 Comparative results on functions f1−f15 with D=50
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
f1Mean 1.81e+05 7.59e+04 4.26e+05 6.96e+04 2.96e+06 5.85e+05 2.52e+06 9.52e+05
Std. 7.78e+04 3.67e+04 1.89e+05 3.61e+04 1.02e+06 2.12e+05 1.00e+06 3.05e+05
+/=/−−−−−+−+−
f2Mean 4.72e+03 4.03e+03 2.27e+01 1.37e−10 4.15e+03 3.27e+03 9.70e+03 1.75e+00
Std. 4.07e+03 5.19e+03 3.58e+01 4.09e−10 2.76e+03 3.92e+03 9.99e+03 3.35e+00
+/=/−+++−+++−
f3Mean 1.91e+01 3.88e−02 3.09e+01 2.70e−05 5.81e+02 5.67e+03 3.90e+04 1.45e+00
Std. 2.32e+01 1.04e−01 3.14e+01 4.57e−05 4.24e+02 2.11e+03 1.59e+04 4.09e+00
+/=/−+−+−+++−
f4Mean 6.24e+01 6.26e+01 2.41e+01 1.50e+01 9.28e+01 7.40e+01 7.88e+01 4.37e+01
Std. 3.62e+01 2.72e+01 3.31e+01 3.83e+01 3.51e+00 3.22e+01 3.09e+01 3.58e+01
+/=/−++−−+++−
f5Mean 2.07e+01 2.01e+01 2.11e+01 2.04e+01 2.08e+01 2.11e+01 2.00e+01 2.00e+01
Std. 3.48e−02 8.33e−03 5.23e−02 2.87e−02 3.49e−02 4.32e−02 1.06e−03 4.73e−06
+/=/−+++++++−
f6Mean 1.95e+01 2.54e+01 1.53e+01 4.19e+01 4.07e−02 4.77e+00 2.39e+01 4.97e+00
Std. 3.73e+00 2.37e+00 2.82e+00 4.31e+00 1.34e−01 2.50e+00 6.37e+00 2.99e+00
+/=/−++++−=+−
f7Mean 8.70e−03 2.16e−02 9.03e−03 6.24e−03 1.18e−16 4.60e−03 1.72e−02 6.57e−04
Std. 9.17e−03 4.29e−02 9.96e−03 6.16e−03 1.29e−16 5.88e−03 2.44e−02 2.55e−03
+/=/−++++=++−
f8Mean 9.42e+00 6.63e−02 1.11e+02 2.58e+02 1.10e+01 4.24e+01 0.00e+00 1.57e+01
Std. 3.62e+00 2.57e−01 2.01e+01 4.42e+01 4.33e+00 1.24e+01 0.00e+00 3.93e+00
+/=/−−−++−+−−
f9Mean 9.15e+01 1.04e+02 1.09e+02 2.95e+02 7.04e+01 7.81e+01 1.49e+02 6.76e+01
Std. 1.15e+01 1.46e+01 2.81e+01 3.68e+01 1.92e+01 7.51e+01 3.34e+01 9.61e+00
+/=/−++++==+−
f10 Mean 2.50e+00 3.05e+00 3.46e+03 3.37e+03 1.57e+02 9.45e+03 2.56e−03 2.03e+02
Std. 1.07e+00 9.81e−01 1.13e+03 1.93e+03 8.15e+01 3.01e+03 6.98e−03 1.46e+02
+/=/−−−++=+−−
f11 Mean 7.04e+03 4.07e+03 1.26e+04 5.65e+03 4.95e+03 1.33e+04 4.23e+03 4.52e+03
Std. 4.82e+02 4.54e+02 2.13e+03 6.00e+02 7.01e+02 2.71e+02 5.64e+02 6.60e+02
+/=/−+=++=+=−
f12 Mean 7.82e−01 1.57e−01 3.15e+00 4.10e−01 1.34e+00 3.12e+00 8.73e−02 1.54e−01
Std. 9.98e−02 1.51e−02 4.66e−01 3.78e−02 1.25e−01 4.71e−01 2.73e−02 6.97e−02
+/=/−+=++++−−
f13 Mean 4.17e−01 4.56e−01 5.06e−01 5.46e−01 3.46e−01 4.01e−01 4.28e−01 3.44e−01
Std. 5.86e−02 6.73e−02 9.63e−02 1.12e−01 5.18e−02 5.08e−02 6.72e−02 6.10e−02
+/=/−++++=++−
f14 Mean 3.14e−01 2.85e−01 3.50e−01 3.60e−01 2.48e−01 3.23e−01 3.48e−01 2.46e−01
Std. 3.21e−02 1.76e−02 1.57e−01 1.30e−01 3.08e−02 2.99e−02 2.44e−01 2.42e−02
+/=/−++++=+=−
f15 Mean 1.44e+01 3.03e+01 1.85e+01 2.77e+01 8.38e+00 2.32e+01 1.28e+01 6.86e+00
Std. 3.35e+00 7.10e+00 1.20e+01 4.11e+00 1.56e+00 1.20e+01 5.75e+00 1.81e+00
+/=/−+++++++−
123

G. Sun et al.
Tabl e 5 Comparative results on functions f16 −f30 with D=50
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
f16 Mean 1.98e+01 1.86e+01 2.22e+01 1.92e+01 2.00e+01 2.23e+01 1.74e+01 1.86e+01
Std. 2.50e−01 4.04e−01 3.04e−01 4.33e−01 6.39e−01 1.34e−01 8.29e−01 9.38e−01
+/=/−+=++++−−
f17 Mean 2.23e+04 9.29e+04 5.48e+04 1.06e+04 3.94e+05 5.19e+04 6.64e+05 5.41e+04
Std. 1.41e+04 7.40e+04 3.40e+04 4.95e+03 2.26e+05 2.49e+04 4.44e+05 2.85e+04
+/=/−−+=−+=+−
f18 Mean 4.02e+02 8.88e+02 1.50e+02 6.03e+02 3.53e+02 5.89e+02 2.01e+03 4.28e+01
Std. 3.23e+02 7.82e+02 1.56e+02 1.20e+03 3.13e+02 7.44e+02 1.28e+03 3.36e+01
+/=/−+++++++−
f19 Mean 1.36e+01 3.40e+01 9.41e+00 1.92e+01 9.58e+00 2.38e+01 1.48e+01 6.82e+00
Std. 5.79e+00 2.14e+01 4.50e+00 2.18e+00 7.10e−01 2.03e+01 2.35e+00 1.34e+00
+/=/−+++++++−
f20 Mean 2.39e+02 6.87e+02 4.80e+02 1.81e+02 2.16e+02 1.12e+03 6.45e+04 1.17e+02
Std. 6.34e+01 1.37e+03 5.13e+02 4.31e+01 1.53e+02 2.91e+02 2.38e+04 1.45e+02
+/=/−+++++++−
f21 Mean 2.65e+04 4.22e+04 2.93e+04 3.02e+03 2.62e+05 4.97e+04 8.82e+05 3.13e+04
Std. 1.94e+04 4.93e+04 2.48e+04 1.81e+03 1.49e+05 3.74e+04 8.11e+05 3.80e+04
+/=/−=+=−+++−
f22 Mean 4.22e+02 5.80e+02 1.36e+03 1.23e+03 2.57e+02 5.87e+02 1.26e+03 2.14e+02
Std. 1.04e+02 1.07e+02 4.14e+02 3.62e+02 1.49e+02 3.61e+02 2.80e+02 1.64e+02
+/=/−++++=++−
f23 Mean 3.44e+02 3.44e+02 3.44e+02 3.44e+02 3.44e+02 3.44e+02 3.44e+02 3.44e+02
Std. 1.55e−13 2.36e−13 1.87e−13 5.02e−13 1.15e−13 1.04e−13 3.46e−12 8.73e−14
+/=/−=======−
f24 Mean 2.72e+02 2.78e+02 2.81e+02 3.02e+02 2.64e+02 2.76e+02 2.58e+02 2.63e+02
Std. 6.57e+00 3.95e+00 3.36e+00 1.13e+01 3.28e+00 4.35e+00 3.10e+00 6.01e+00
+/=/−++++=+=−
f25 Mean 2.10e+02 2.28e+02 2.08e+02 2.38e+02 2.10e+02 2.08e+02 2.15e+02 2.08e+02
Std. 1.01e+01 2.16e+00 2.06e+00 4.72e+00 1.29e+00 1.83e+00 3.63e+00 2.23e+00
+/=/−=+=+==+−
f26 Mean 1.54e+02 1.20e+02 1.00e+02 1.07e+02 1.00e+02 1.07e+02 1.75e+02 1.09e+02
Std. 5.15e+01 4.12e+01 7.72e−02 2.57e+01 2.36e−02 2.57e+01 9.01e+01 3.51e+01
+/=/−++++==+−
f27 Mean 7.70e+02 7.75e+02 7.05e+02 1.58e+03 3.22e+02 4.57e+02 1.05e+03 4.32e+02
Std. 8.80e+01 1.53e+02 1.05e+02 1.48e+02 2.13e+01 6.65e+01 1.75e+02 3.60e+01
+/=/−++++−=+−
f28 Mean 1.42e+03 1.61e+03 1.45e+03 5.36e+03 1.09e+03 1.15e+03 2.12e+03 1.19e+03
Std. 1.14e+02 2.46e+02 1.10e+02 1.03e+03 3.62e+01 6.78e+01 6.65e+02 5.11e+01
+/=/−++++−=+−
f29 Mean 1.05e+03 9.94e+02 8.36e+02 8.96e+02 1.89e+03 1.36e+03 2.48e+03 1.27e+03
Std. 1.93e+02 1.17e+02 2.51e+02 2.04e+02 3.13e+02 3.54e+02 6.52e+02 2.25e+02
+/=/−−−−−+=+−
f30 Mean 1.06e+04 1.14e+04 9.89e+03 1.18e+04 8.99e+03 9.12e+03 1.24e+04 9.07e+03
Std. 1.68e+03 1.71e+03 5.53e+02 9.07e+02 2.86e+02 4.41e+02 1.48+03e 4.86e+02
+/=/−++++==+−
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
Tabl e 6 Comparative results on functions f1−f15 with D=100
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
f1Mean 8.96e+05 4.70e+05 8.29e+06 4.38e+05 2.19e+07 4.73e+06 1.92e+07 9.59e+06
Std. 1.45e+05 1.93e+05 3.20e+06 1.51e+05 5.50e+06 9.62e+05 6.89e+06 2.74e+06
+/=/−− − = − + − + −
f2Mean 1.31e+04 6.38e+03 1.74e+04 2.80e−10 1.06e+04 1.27e+04 3.67e+04 1.23e+01
Std. 8.76e+03 1.08e+04 2.13e+04 4.04e−10 6.32e+03 1.02e+04 3.80e+04 1.50e+01
+/=/−+ + + − + + + −
f3Mean 7.66e+01 8.65e+00 5.22e+03 2.51e−02 3.22e+03 5.46e+04 5.01e+04 3.45e+02
Std. 6.52e+01 2.83e+00 4.07e+03 6.79e−02 1.34e+03 1.10e+04 2.24e+04 2.62e+02
+/=/−− − + − + + + −
f4Mean 1.71e+02 1.62e+02 1.89e+02 1.35e+02 1.59e+02 1.69e+02 2.09e+02 1.60e+02
Std. 4.05e+01 4.80e+01 3.38e+01 5.92e+01 2.57e+01 3.63e+01 3.14e+01 2.77e+01
+/=/−= = + − = = + −
f5Mean 2.10e+01 2.03e+01 2.13e+01 2.07e+01 2.12e+01 2.13e+01 2.00e+01 2.00e+01
Std. 2.30e−02 1.75e−02 2.31e−02 2.06e−02 2.46e−02 3.04e−02 1.18e−03 9.75e−07
+/=/−+ + + + + + + −
f6Mean 6.44e+01 7.23e+01 4.56e+01 1.03e+02 4.79e+00 8.30e+00 6.35e+01 6.36e+00
Std. 5.35e+00 3.68e+00 7.45e+00 5.42e+00 2.59e+00 2.67e+00 1.07e+01 3.85e+00
+/=/−+ + + + = + + −
f7Mean 2.63e−03 6.39e−03 6.07e−03 3.61e−03 7.52e−11 2.00e−16 3.78e−03 3.77e−16
Std. 6.20e−03 1.06e−02 9.15e−03 6.02e−03 6.49e−11 1.47e−16 5.88e−03 2.44e−16
+/=/−+ + + + + = + −
f8Mean 1.92e+01 4.17e+00 2.32e+02 5.84e+02 6.22e+01 2.42e+02 5.09e−09 3.92e+01
Std. 5.89e+00 9.84e−01 4.44e+01 4.27e+01 8.16e+00 1.76e+02 3.75e−09 6.47e+00
+/=/−− − + + + + − −
f9Mean 2.77e+02 2.72e+02 3.30e+02 6.57e+02 1.36e+02 8.17e+02 4.21e+02 1.45e+02
Std. 2.84e+01 2.04e+01 1.65e+02 7.80e+01 2.61e+01 7.54e+01 1.23e+02 2.72e+01
+/=/−+ + + + = + + −
f10 Mean 1.26e+02 1.49e+01 6.69e+03 1.01e+04 5.53e+03 2.69e+04 1.40e−05 5.73e+02
Std. 1.97e+01 2.49e+00 1.25e+03 3.99e+03 7.98e+02 5.42e+02 3.37e−05 2.32e+02
+/=/−− − + + + + − −
f11 Mean 2.05e+04 1.06e+04 3.03e+04 1.41e+04 1.53e+04 3.03e+04 1.12e+04 1.22e+04
Std. 5.94e+02 6.56e+02 5.24e+02 1.21e+03 2.10e+03 4.83e+02 1.04e+03 2.12e+03
+/=/−+ = + + + + = −
f12 Mean 1.62e+00 2.91e−01 4.01e+00 6.80e−01 2.49e+00 4.05e+00 1.19e−01 3.18e−01
Std. 1.51e−01 3.69e−02 1.74e−01 5.26e−02 1.30e−01 1.68e−01 3.19e−02 1.22e−01
+/=/−+ = + + + + − −
f13 Mean 4.67e−01 4.85e−01 6.53e−01 5.90e−01 5.33e−01 5.40e−01 5.30e−01 4.71e−01
Std. 3.88e−02 5.58e−02 8.10e−02 8.60e−02 4.26e−02 4.92e−02 6.22e−02 6.06e−02
+/=/−= = + + + + + −
f14 Mean 3.21e−01 2.88e−01 3.47e−01 3.67e−01 2.83e−01 3.34e−01 3.33e−01 2.80e−01
Std. 1.97e−02 2.10e−02 3.77e−02 1.07e−01 2.40e−02 2.64e−02 1.97e−01 1.74e−02
+/=/−+ = + + = + + −
f15 Mean 4.09e+01 6.61e+01 8.70e+01 6.95e+01 2.26e+01 7.43e+01 4.36e+01 1.59e+01
Std. 1.12e+01 1.82e+01 1.80e+01 7.08e+00 5.45e+00 1.79e+00 2.14e+01 2.52e+00
+/=/−+ + + + + + + −
123

G. Sun et al.
Tabl e 7 Comparative results on functions f16 −f30 with D=100
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
f16 Mean 4.38e+01 4.14e+01 4.66e+01 4.21e+01 4.48e+01 4.65e+01 3.93e+01 4.13e+01
Std. 3.77e−01 3.46e−01 2.57e−01 4.73e−01 3.32e−01 2.69e−01 1.07e+00 8.40e−01
+/=/−+=++++−−
f17 Mean 1.50e+05 3.27e+05 4.82e+05 8.97e+04 2.49e+06 1.92e+06 3.98e+06 1.01e+06
Std. 4.37e+04 1.63e+05 2.07e+05 4.39e+04 1.09e+06 5.58e+05 1.99e+06 5.12e+05
+/=/−−−−−+++−
f18 Mean 8.02e+02 6.84e+02 7.02e+02 1.50e+03 2.61e+02 5.46e+02 4.71e+03 5.74e+02
Std. 8.63e+02 4.40e+02 7.69e+02 1.64e+03 3.12e+02 5.69e+02 4.68e+03 1.08e+03
+/=/−++++−=+−
f19 Mean 8.39e+01 1.01e+02 9.83e+01 8.95e+01 9.02e+01 8.51e+01 1.02e+02 8.91e+01
Std. 3.13e+01 4.27e+01 1.86e+01 3.56e+01 8.56e−01 1.99e+01 1.95e+01 9.74e−01
+/=/−=++=+=+−
f20 Mean 7.14e+02 1.09e+03 3.21e+03 4.54e+02 6.32e+03 1.83e+04 1.08e+05 5.71e+02
Std. 2.09e+02 1.71e+03 1.17e+03 1.29e+02 1.63e+03 5.75e+03 4.45e+04 8.60e+01
+/=/−+++−+++−
f21 Mean 7.28e+04 1.34e+05 1.87e+05 2.96e+04 1.98e+06 1.29e+06 3.02e+06 3.85e+05
Std. 5.04e+04 8.18e+04 7.22e+04 1.54e+04 6.00e+05 5.08e+05 1.16e+06 1.64e+05
+/=/−−−−−+++−
f22 Mean 1.38e+03 1.41e+03 3.65e+03 2.69e+03 1.35e+03 4.03e+03 2.62e+03 1.16e+03
Std. 2.94e+02 2.58e+02 9.48e+02 4.76e+02 3.96e+02 2.20e+02 5.07e+02 2.59e+02
+/=/−+++++++−
f23 Mean 3.48e+02 3.48e+02 3.48e+02 3.48e+02 3.48e+02 3.48e+02 3.49e+02 3.48e+02
Std. 4.07e−13 6.44e−12 5.33e−05 3.19e−12 1.11e−04 3.73e−12 7.64e−01 5.08e−13
+/=/−======+−
f24 Mean 3.78e+02 3.99e+02 4.06e+02 4.52e+02 3.62e+02 3.78e+02 3.31e+02 3.64e+02
Std. 6.30e+00 7.55e+00 8.35e+00 2.28e+01 1.69e+00 4.21e+00 1.05e+01 2.65e+00
+/=/−++++−+−−
f25 Mean 2.00e+02 2.57e+02 2.45e+02 2.88e+02 2.49e+02 2.45e+02 2.54e+02 2.39e+02
Std. 6.26e−14 8.37e+00 9.57e+00 1.66e+01 4.19e+00 1.19e+01 1.17e+01 4.76e+00
+/=/−−+=++=+−
f26 Mean 2.00e+02 2.00e+02 2.01e+02 2.00e+02 2.01e+02 2.00e+02 2.02e+02 2.01e+02
Std. 2.49e−02 7.47e−03 1.10e−01 1.24e−02 2.09e−01 6.94e−02 4.20e−01 1.77e−01
+/=/−−−=−=−+−
f27 Mean 1.43e+03 1.37e+03 1.40e+03 3.12e+03 3.06e+02 4.43e+02 2.09e+03 3.74e+02
Std. 1.01e+02 2.15e+02 1.63e+02 2.39e+02 1.24e+01 3.76e+01 3.55e+02 5.91e+01
+/=/−++++−++−
f28 Mean 3.20e+03 4.72e+03 2.91e+03 1.39e+04 2.20e+03 1.95e+03 3.87e+03 2.08e+03
Std. 3.34e+02 6.34e+02 3.15e+02 1.92e+03 4.63e+01 3.32e+02 8.04e+02 1.78e+02
+/=/−+++++=+−
f29 Mean 1.37e+03 1.45e+03 1.98e+03 1.23e+03 2.81e+03 5.68e+03 4.37e+03 1.91e+03
Std. 2.51e+02 1.85e+02 4.05e+02 2.18e+02 4.80e+02 2.50e+03 8.66e+02 1.41e+02
+/=/−−−=−+++−
f30 Mean 8.17e+03 8.59e+03 6.01e+03 7.45e+03 9.53e+03 6.64e+03 1.91e+04 8.90e+03
Std. 9.43e+02 1.58e+03 9.47e+02 2.02e+03 1.25e+03 1.44e+03 6.97e+03 8.01e+02
+/=/−−=−−=−+−
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
Algorithm 1 The overall procedure of GPDE
1: Set the values of parameters NP,FR,Vand T;
2: Initialize NP individuals with random positions via the formula (1);
3: for (t=1;t<=T;t++),do
4: Compute the value of scaling factor Ftin the tth generation via
the formula (12);
5: Compute the value of parameter CStin the tth generation via the
formula (17);
6: for (i=1;i<=NP;i++),do
7: Compute the crossover rate CRi
tof the ith individual in the tth
generation via the formula (10);
8: if rand[0,1]<CStthen
9: Generate the new trial vector via the formula (9) included
Gaussian mutation operator;
10: else
11: Generate the new trial vector via the formula (11) containing
mutation operator DE/rand-worst/1;
12: end if;
13: Update the ith individual via the selection operator (8);
14: Replace the best individual xbest by the new individual xiif xi
is better than xbest ;
15: end for
16: end for
17: Output the position of the best individual as the global optimal
solution.
To be specific, SADE and JADE are two state-of-the-art DE
variants, and GDE and MGBDE are recently proposed vari-
ants both of which adopt two different mutation operators;
in particular, MGBDE employs a similar Gaussian muta-
tion operator with GPDE, and SinDE is an up-to-date DE
variant, which applies two sinusoidal functions to adjust
the values of mutation scaling factor and crossover rate.
These selected DE variants not only have outstanding perfor-
mance, but also have some similar aspects to our proposed
GPDE, that is why we take them as the comparison object. In
addition, two good performing state-of-the-art meta-heuristic
algorithms, i.e., cooperative coevolving particle swarm opti-
mization with random grouping [denoted by CCPSO2 (Li
and Yao 2012] and collective resource-based artificial bee
colony with decentralized tasking (denoted by C-ABC (Bose
et al. 2014) for short), are used to enrich the comparative
experiment.
For all the aforementioned compared algorithms, except
the population sizes NP, which, respectively, are equal to
Dand 5Dfor test functions and real-world problems, the
other involved control parameters keep the same with their
corresponding literature. In GPDE, there are only three user-
specified control parameters, including population size NP,
periodic adjustment parameter FR and the variance Vof
crossover rate. Note that the values of FR and Vare, respec-
tively, set to 0.05 and 0.1, and the value of NP always takes
the same as its competitors, and all these three values will
keep no change for all the adopted test functions and real-
world problems. In addition, all the compared algorithms are
Tabl e 8 Comparative results on real-world problems rf1−rf4
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
rf1Mean 1.05e+00 9.95e+00 6.35e+00 1.26e+01 4.68e−01 2.23e+00 1.24e+01 1.34e+00
Std. 3.66e+00 4.74e+00 6.87e+00 4.41e+00 2.34e+00 4.62e+00 7.67e+00 3.72e+00
+/=/−=+++==+−
rf2Mean 1.93e+00 1.79e+00 2.38e+00 1.95e+00 1.78e+00 2.39e+00 1.80e+00 1.63e+00
Std. 8.64e−02 9.98e−02 9.90e−02 1.56e−01 1.69e−01 9.67e−02 1.96e−01 2.19e−01
+/=/−+++++++−
rf3Mean 1.76e−04 4.38e−09 4.63e−14 1.09e+01 7.07e+00 7.98e+00 2.66e+02 2.82e+00
Std. 4.55e−04 1.49e−08 7.08e−14 6.47e+00 4.25e+00 6.99e+00 1.76e+02 5.11e+00
+/=/−−−−++++−
rf4Mean 0.00e+00 0.00e+00 0.00e+00 0.00e+00 4.63e+00 9.83e+00 2.87e+02 0.00e+00
Std. 0.00e+00 0.00e+00 0.00e+00 0.00e+00 7.68e+00 1.16e+01 3.13e+02 0.00e+00
+/=/−====+++−
Tabl e 9 Statistical results on all test functions and real-world problems
Func. Metric SADE JADE GDE MGBDE SinDE C-ABC CCPSO2 GPDE
30-D+/=/−21/5/4 22/5/3 23/4/3 24/2/4 15/7/8 23/6/1 22/4/4 −
50-D+/=/−22/3/5 21/4/5 23/4/3 22/1/7 14/12/4 19/10/1 22/4/4 −
100-D+/=/−16/4/10 14/8/8 22/5/3 18/2/10 20/7/3 20/7/3 24/1/5 −
Real-word +/=/−1/2/1 2/1/1 2/1/1 3/1/0 3/1/0 3/1/0 4/0/0 −
Tot a l +/=/ −60/14/20 59/18/17 70/14/10 67/6/21 52/27/15 65/24/5 72/9/13 −
123

G. Sun et al.
0 0.5 1 1.5 22.5 3
x 105
103
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f1
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3
x 105
10−20
10−15
10−10
10−5
100
105
1010
Function Evaluations
Fitness Error Value
Function f2
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 22.5 3
x 105
10−25
10−20
10−15
10−10
10−5
100
105
Function Evaluations
Fitness Error Value
Function f3
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3
x 105
10−4
10−3
10−2
10−1
100
101
102
103
104
105
Function Evaluations
Fitness Error Value
Function f4
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 22.5 3
x 105
10−2
10−1
100
101
Function Evaluations
Fitness Error Value
Function f12
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3
x 105
10−1
100
101
102
Function Evaluations
Fitness Error Value
Function f13
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
Fig. 1 Convergence graphs (mean curves) for eight algorithms on functions f1,f2,f3,f4,f12 and f13 with D=30 over 50 independent runs
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
00.5 11.5 22.5 3
x 105
101
102
103
104
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f18
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
00.5 11.5 22.5 3
x 105
100
101
102
103
104
Function Evaluations
Fitness Error Value
Fu n ct io n f19
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3
x 105
101
102
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f20
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
00.5 11.5 22.5 3
x 105
102
103
104
105
106
107
108
109
Function Evaluations
Fitness Error Value
Fu n ct io n f21
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3
x 105
102
103
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f29
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
00.5 11.5 22.5 3
x 105
102
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Fu n ct io n f30
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
Fig. 2 Convergence graphs (mean curves) for eight algorithms on functions f18,f19,f20,f21 ,f29 and f30 with D=30 over 50 independent runs
123

G. Sun et al.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
104
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f1
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
00.5 1 1.5 2 2.5 33.5 44.5 5
x 105
10−10
10−5
100
105
1010
Function Evaluations
Fitness Error Value
Function f2
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
10−4
10−2
100
102
104
106
Function Evaluations
Fitness Error Value
Function f3
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
00.5 11.5 22.5 33.5 4 4.5 5
x 105
101
102
103
104
105
106
Function Evaluations
Fitness Error Value
Function f4
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
10−2
10−1
100
101
Function Evaluations
Fitness Error Value
Function f12
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
00.5 11.5 22.5 33.5 4 4.5 5
x 105
10−1
100
101
102
Function Evaluations
Fitness Error Value
Function f13
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
Fig. 3 Convergence graphs (mean curves) for eight algorithms on functions f1,f2,f3,f4,f12 and f13 with D=50 over 50 independent runs
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
101
102
103
104
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f18
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 22.5 33.5 44.5 5
x 105
100
101
102
103
104
Function Evaluations
Fitness Error Value
Function f19
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
102
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f20
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
103
104
105
106
107
108
109
Function Evaluations
Fitness Error Value
Function f21
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
102
103
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f29
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f30
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
Fig. 4 Convergence graphs (mean curves) for eight algorithms on functions f18,f19,f20,f21 ,f29 and f30 with D=50 over 50 independent runs
123

G. Sun et al.
0 1 2 3 4 5 6 7 8 9 10
x 105
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f1
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
10−10
10−5
100
105
1010
Function Evaluations
Fitness Error Value
Function f2
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
10−2
10−1
100
101
102
103
104
105
106
107
Function Evaluations
Fitness Error Value
Function f3
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
102
103
104
105
106
Function Evaluations
Fitness Error Value
Function f4
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
10−2
10−1
100
101
Function Evaluations
Fitness Error Value
Function f12
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
10−1
100
101
102
Function Evaluations
Fitness Error Value
Function f13
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
Fig. 5 Convergence graphs (mean curves) for eight algorithms on functions f1,f2,f3,f4,f12 and f13 with D=100 over 50 independent runs
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
0 1 2 3 4 5 6 7 8 9 10
x 105
102
103
104
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f18
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
101
102
103
104
Function Evaluations
Fitness Error Value
Fu n ct i o n f19
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
102
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f20
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Fu n ct i o n f21
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
103
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f29
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
0 1 2 3 4 5 6 7 8 9 10
x 105
103
104
105
106
107
108
109
Function Evaluations
Fitness Error Value
Fu n ct i o n f30
SADE
JADE
GDE
MGBDE
SinDE
C−ABC
CCPSO2
GPDE
Fig. 6 Convergence graphs (mean curves) for eight algorithms on functions f18 ,f19,f20 ,f21,f29 and f30 with D=100 over 50 independent
runs
123

G. Sun et al.
Tabl e 1 0 Parameter configurations of different GPDEs
Par. GPDE-C1 GPDE-C2 GPDE-C3 GPDE-C4 GPDE-C5 GPDE-C6 GPDE-C7 GPDE-C8 GPDE-C9
FR 0.03 0.05 0.1 0.03 0.05 0.10.03 0.05 0.1
V0.03 0.03 0.03 0.05 0.05 0.05 0.10.10.1
tested 50 independent runs for every function and the mean
results are used in the comparison, and the maximum allow-
able generations are set to 10000 for all the test functions and
real-world problems.
5.3 Comparative results
To evaluate the performances of the participant algorithms
and provide a comprehensive comparison, we, respectively,
report the mean (denoted by “Mean”) fitness error value
fxbest−fx∗, the corresponding standard deviation
(Std.) and the statistical conclusion of comparative results
based on 50 independent runs, where xbest is the best obtained
solution and x∗is the known optimal solution. In addition, the
statistical conclusions of the comparative results are based
on the paired Wilcoxon rank sum test which is conducted at
0.05 significance level to assess the significance of the perfor-
mance difference between GPDE and each competitor. We
mark the three kinds of statistical significance cases with “+,”
“=” and “−” to indicate that GPDE is significantly better
than, similar to, or worse than the corresponding competitor,
respectively. The comparative results of test functions with
different dimensions are summarized in Tables 2,3,4,5,6,7
and 8, and the best results are indicated with boldface font to
highlight the best algorithm for each test function. Moreover,
the numbers of the three cases (+/=/−)obtained by the
compared results are summarized in Table 9.
Table 9shows that GPDE obtains the best overall perfor-
mance among the eight compared algorithms. In details, for
the 94 functions, GPDE performs better than SADE, JADE,
GDE, MGBDE, SinDE, C-ABC and CCPSO2 on 60, 59, 70,
67, 52, 65 and 72 functions, and only loses in 20, 17, 10, 21,
15, 5 and 13 functions, respectively. Moreover, GPDE out-
performs its competitors on every adopted dimensions of test
functions and has no worst “Mean” in 94 functions, which
means that GPDE is robust, and thus it is a reliable algorithm
for handling various problems with different dimensions.
In addition, to observe the convergence characteristics of
the compared algorithms, we select 36 functions with dif-
ferent dimensions and plot their convergence graphs based
on the mean values over 50 runs in Figs. 1,2,3,4,5and
6. Obviously, GPDE has a wonderful convergence rate. As a
conclusion, the experimental results and convergence graphs
have demonstrated that GPDE performs significantly better
than the other seven compared algorithms.
5.4 Robustness analysis of control parameters
Generally speaking, the involved control parameters may
have important effects on the algorithmic performance, but it
is good news for the users if the control parameters is robust.
To verify the robustness of control parameters in GPDE, we
compare various GPDEs with different parameter configura-
tions, which are listed in Table 10. Note that we only evaluate
the robustness of periodic adjustment parameter FR and the
variance Vof crossover rate, because population size NP
usually has no obvious effect on the performance of DE.
Since most of the results obtained by different GPDEs
are very close to each other, we only select 36 test functions
with different dimensions whose results have relatively clear
differences to reveal the robustness of control parameters
FR and Vvia convergence graphs. The convergence graphs
of 36 selected functions obtained by GPDEs with different
parameter configurations are plotted in Figs. 7,8,9,10,11
and 12.
Three discoveries can be found out from Figs. 7,8,9,10,
11 and 12. Above all, the results of GPDEs with different
parameter configurations have no obvious fluctuation, which
implies that the control parameters of GPDE are robust. Sec-
ondly, the variance Vof crossover rate has a slightly bigger
influence on the performance of GPDE than periodic adjust-
ment parameter FR, because Vhas more direct effect on the
population diversity than FR. At last, under a prescribed limit,
a bigger value of Vleads to a better result, because a big-
ger value of Vis often corresponding to a better population
diversity.
6 Conclusions
Differential evolution is an excellent evolutionary algo-
rithm for global numerical optimization, but it is not com-
pletely free from the problems of premature convergence and
stagnation. In order to alleviate these problems and enhance
DE, we propose a new variant called GPDE. In GPDE, a novel
Gaussian mutation operator who, respectively, takes the posi-
tion of the best individual among three randomly selected
individuals and the distance between the other two as the
mean and standard deviation, and a modified common muta-
tion operator is applied to cooperatively generate the mutant
vectors. Moreover, scaling factor adopts a cosine function to
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
0 0.5 1 1.5 2 2.5 3
x 105
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Fu n ct i o n f1
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
00.5 11.5 22.5 3
x 105
10−20
10−15
10−10
10−5
100
105
1010
Function Evaluations
Fitness Error Value
Function f2
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3
x 105
10−25
10−20
10−15
10−10
10−5
100
105
Function Evaluations
Fitness Error Value
Fu n ct i o n f3
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
00.5 11.5 22.5 3
x 105
10−2
10−1
100
101
102
103
104
105
Function Evaluations
Fitness Error Value
Function f4
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3
x 105
10−1
100
101
Function Evaluations
Fitness Error Value
Function f12
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
00.5 11.5 22.5 3
x 105
10−1
100
101
102
Function Evaluations
Fitness Error Value
Function f13
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
Fig. 7 Convergence graphs (mean curves) for the GPDE with different parameter configurations on functions f1,f2,f3,f4,f12 and f13 with
D=30 over 50 independent runs
123

G. Sun et al.
0 0.5 1 1.5 2 2.5 3
x 105
101
102
103
104
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f18
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3
x 105
100
101
102
103
104
Function Evaluations
Fitness Error Value
Function f19
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3
x 105
101
102
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f20
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3
x 105
103
104
105
106
107
108
109
Function Evaluations
Fitness Error Value
Function f21
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3
x 105
102
103
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f29
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3
x 105
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f30
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
Fig. 8 Convergence graphs (mean curves) for the GPDE with different parameter configurations on functions f18,f19,f20,f21 ,f29 and f30 with
D=30 over 50 independent runs
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f1
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
100
102
104
106
108
1010
1012
Function Evaluations
Fitness Error Value
Fu n ct i o n f2
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
10−1
100
101
102
103
104
105
106
107
Function Evaluations
Fitness Error Value
Function f3
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
101
102
103
104
105
106
Function Evaluations
Fitness Error Value
Fu n ct i o n f4
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
10−1
100
101
102
Function Evaluations
Fitness Error Value
Function f12
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
10−1
100
101
102
Function Evaluations
Fitness Error Value
Fu n ct i o n f13
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
Fig. 9 Convergence graphs (mean curves) for the GPDE with different parameter configurations on functions f1,f2,f3,f4,f12 and f13 with
D=50 over 50 independent runs
123

G. Sun et al.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
101
102
103
104
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f18
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
00.5 11.5 22.5 33.5 44.5 5
x 105
100
101
102
103
104
Function Evaluations
Fitness Error Value
Function f19
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
101
102
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f20
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
00.5 11.5 22.5 33.5 44.5 5
x 105
104
105
106
107
108
109
Function Evaluations
Fitness Error Value
Function f21
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
102
103
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f29
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
00.5 11.5 22.5 33.5 44.5 5
x 105
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f30
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
Fig. 10 Convergence graphs (mean curves) for the GPDE with different parameter configurations on functions f18 ,f19,f20,f21 ,f29 and f30 with
D=50 over 50 independent runs
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
0 1 2 3 4 5 6 7 8 9 10
x 105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f1
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
102
104
106
108
1010
1012
Function Evaluations
Fitness Error Value
Function f2
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
102
103
104
105
106
107
Function Evaluations
Fitness Error Value
Function f3
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
102
103
104
105
106
Function Evaluations
Fitness Error Value
Fu n ct i o n f4
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
10−1
100
101
Function Evaluations
Fitness Error Value
Function f12
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
10−1
100
101
102
Function Evaluations
Fitness Error Value
Function f13
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
Fig. 11 Convergence graphs (mean curves) for the GPDE with different parameter configurations on functions f1,f2,f3,f4,f12 and f13 with
D=100 over 50 independent runs
123

G. Sun et al.
0 1 2 3 4 5 6 7 8 9 10
x 105
102
103
104
105
106
107
108
109
1010
1011
Function Evaluations
Fitness Error Value
Function f18
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
101
102
103
104
105
Function Evaluations
Fitness Error Value
Function f19
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
102
103
104
105
106
107
108
Function Evaluations
Fitness Error Value
Function f20
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f21
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
103
104
105
106
107
108
109
1010
Function Evaluations
Fitness Error Value
Function f29
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
0 1 2 3 4 5 6 7 8 9 10
x 105
103
104
105
106
107
108
109
Function Evaluations
Fitness Error Value
Function f30
GPDE−C1
GPDE−C2
GPDE−C3
GPDE−C4
GPDE−C5
GPDE−C6
GPDE−C7
GPDE−C8
GPDE−C9
Fig. 12 Convergence graphs (mean curves) for the GPDE with different parameter configurations on functions f18 ,f19,f20,f21 ,f29 and f30 with
D=100 over 50 independent runs
123

Differential evolution with Gaussian mutation and dynamic parameter adjustment
adjust its value periodically, which has potential advantage
in balancing the exploration and exploitation abilities, and
crossover rate employs a Gaussian function to produce its
value dynamically, which can adjust the population diver-
sity. The test suite of IEEE CEC-2014 which contains 30 test
function and four real-world problems, and seven remarkable
meta-heuristic algorithms are used to evaluate the perfor-
mance of GPDE, and the obtained results show that GPDE
performs much better than the other seven compared DE
algorithms. In addition, the parameter analysis indicates that
the control parameters involved in GPDE are robust.
Acknowledgements The authors wish to thank the anonymous review-
ers, whose valuable comments lead to an improved version of the paper.
This work was supported by the National Natural Science Foundation
of China under Grant Nos. 71701187, 71771166 and 71471126, and
Research Project of Zhejiang Education Department under Grant No.
Y201738184, and High Performance Computing Center of Tianjin Uni-
versity, China.
Compliance with ethical standards
Conflict of interest All the authors declares that they have no conflict
of interest.
Human and animal rights This article does not contain any studies
with human participants or animals performed by any of the authors.
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