Evaluating the performance of SHADE on
CEC 2013 benchmark problems
Ryoji Tanabe and Alex Fukunaga
Graduate School of Arts and Sciences
The University of Tokyo
Abstract—This paper evaluates the performance of SuccessHistory based Adaptive DE (SHADE) on the benchmark set for
the CEC2013 Competition on Real-Parameter Single Objective
Optimization. SHADE is an adaptive differential algorithm whichuses a history-based parameter adaptation scheme. Experimental
results on 28 problems from the CEC2013 benchmarks for 10,
30, and 50 dimensions are presented, including measurements ofalgorithmic complexity. In addition, we investigate the parameter
adaptation behavior of SHADE on these instances.

I.

I NTRODUCTION

The single-objective, global optimization problem is the
problem of finding a vector x = (x1 , ..., xD ) which minimizes
an objective function f (x), where D is the dimensionalityof the problem. In black-box optimization the task is solving
a global optimization problem without explicit knowledge ofthe form or structure of the objective function, i.e., f is a
“black box”. This problem occurs in many real-world problems(e.g. engineering optimization where the objective function iscomputed using a complex simulation), and has been studied
by many researchers.
Differential Evolution (DE) is a stochastic, black-box
search method that was originally designed for numerical
optimization problems [1]. Despite of its relative simplicity,
DE has been shown to be competitive with other more complex optimization algorithms, and has been applied to many
practical problems [2].
The search efficiency of DE depends largely on the mutation strategy, as well as the control parameters population size,
scaling factor F , crossover rate CR, and many previous researchers have shown that optimal settings for these parameters
are domain-dependent (c.f. [1], [3], [4], [5]). Thus, there has
been much recent work which seeks to automate the selectionof the mutation strategy, as well as the control parameter values[6], [2].
Success-History based Adaptive DE (SHADE), which wasintroduced in a companion paper [7], is a novel, adaptive
DE algorithms and an improved version of JADE [8] whichuses a different parameter adaptation mechanism based on ahistory. In SHADE, the mean values of successful parameter
CR, F for each generation are stored in a historical memory
MCR , MF . SHADE maintains a diverse set of parameters to
guide control parameter adaptation as search progresses formore robust search.
In our companion paper [7], SHADE was evaluated bycomparisons with CoDE [9], EPSDE [10], JADE [8] anddynNP-jDE [11] on 28 problems from the CEC2013 benchmark set [12], as well as CEC2005 benchmarks [13] and theset of classical 13 benchmark problems [14]. The experimental
results show that SHADE is competitive with previous, stateof the art DE algorithms on these benchmarks.
This paper presents an in-depth analysis of the performance
evaluation of SHADE on the CEC2013 benchmark set. Thisbenchmark set is improved version of the CEC2005 benchmark
set [13]. Experiments were performed for D = 10, 30, and 50
dimensional problems.
The rest of the paper is organized as follows. In Section
II, we describe the SHADE algorithm. Section III presents ourempirical evaluation of SHADE on the CEC2013 benchmarks,
which includes empirical measurements of the algorithm complexity (Section III-B), performance results on the benchmark
problems (Section III-C), and an analysis of the parameter
adaptation behavior of SHADE (Section III-D). Section IV
concludes the paper with a discussion and directions for future
work.
II.

S UCCESS -H ISTORY BASED A DAPTIVE DE

Success-History based Adaptive DE (SHADE) [7] is anadaptive DE, which uses a historical memory in order to adapt
the control parameters F, CR. The overall algorithm is shownin Algorithm 1. Below, we give an overview of the algorithm:
A DE population is represented as a set of real parameter
vectors xi = (x1 , ..., xD ), i = 1, ..., N , where N is the
population size. At the beginning of the search, the individual
vectors in population are initialized randomly.
SHADE maintains a historical memory with H entries forboth of the DE control parameters CR and F , MCR , MF . In
the beginning, the contents of MCR,i , MF,i (i = 1, ..., H) are
all initialized to 0.5.
In each generation, the DE control parameters F and CR
are generated based on the history based parameter adaptation,
trial vectors are generated, selection applied, and the historicalmemory (MCR , MF ) is updated. This process is repeated until
some termination criterion is achieved.
In each generation, the control parameters CRi and Fi
used by each individual xi are generated by first selecting an
index ri randomly from [1, H], and then applying the equations
below:
CRi = randni (MCR,ri , 0.1)
Fi = randci (MF,ri , 0.1)

(1)
(2)

Here, randni (µ, σ 2 ), randci (µ, σ 2 ) are values selected randomly from normal and Cauchy distributions with mean µ,

In Eq. (3), the index r1 is randomly selected from [1, N ]such that they differ from each other as well as i. x̃r2,G(x̃r2,G 6= xr1,G 6= xi,G ) in Eq. (3) is randomly selectedfrom the union of the population P and the external archive
A, P ∪ A. Parent vectors xi,G which were worse than thetrial vectors ui,G (and are therefore not selected for survivalin the standard DE, Eq. 7) are preserved in archive. The sizeof the archive is set to the same as that of the population,i.e., |A| = |P |. Whenever the size of the archive exceeds |A|,
randomly selected elements are deleted to make space for the
newly inserted elements.

Algorithm 1: SHADE// Initialization phase
1 G = 0;2 Initialize population P 0 = (x1,0 , ..., xN,0 ) randomly;
3 Set all values in MCR , MF to 0.5;
4 Archive A = ∅;5 Index counter k = 1;// Main loop
6 while The termination criteria are not met do
7
SCR = ∅, SF = ∅;8
for i = 1 to N do
9ri = Select from [1, H] randomly;
10
CRi,G = randni (MCR,ri , 0.1);
11
Fi,G = randci (MF,ri , 0.1);
12pi,G = rand[pmin , 0.2];
13
Generate trial vector ui,G by current-to-pbest/1/bin;
14
end
15
for i = 1 to N do
16
if f (ui,G ) ≤ f (xi,G ) then
17xi,G+1 = ui,G ;
18
else
19
xi,G+1 = xi,G ;
20
end
21
if f (ui,G ) < f (xi,G ) then22xi,G → A;
23
CRi,G → SCR , Fi,G → SF ;
24
end
25
end
26
Whenever the size of the archive exceeds |A|, randomly
selected individuals are deleted so that |A| ≤ |P |;27
if SCR 6= ∅ and SF 6= ∅ then28
Update MCR,k , MF,k based on SCR , SF ;29
k + +;
30
If k > H, k is set to 1;
31
end
32 endFor each dimension j, if the mutant vector element vj,i,G
is outside the boundaries [xmin, xmax], we applied the samejjcorrection performed in [8]: min
(xj + xj,i,G )/2 if vj,i,G < xmin
j(5)
vj,i,G =(xmax+ xj,i,G )/2 if vj,i,G > xmaxjj
After generating the mutant vector v i,G , it is crossed withthe parent xi,G in order to generate trial vector ui,G . Binomial
Crossover is used in SHADE and implemented as follows:
vj,i,G if rand[0, 1) ≤ CRi or j = jranduj,i,G =(6)xj,i,G otherwise
rand[0, 1) denotes a uniformly selected random number from[0, 1), and jrand is a decision variable index which is uniformly
randomly selected from [1, D]. CRi is the CR parameter usedby individual xi .

and variance σ 2 . In case a value for CRi outside of [0, 1] isgenerated, it is replaced by the limit value (0 or 1) closest to
the generated value. When Fi > 1, Fi is truncated to 1, andwhen Fi ≤ 0, Eq. (2) is repeatedly applied to try to generate
a valid value.
In each generation, a mutant vector v i,G is generated
from an existing population member xi,G by applying somemutation strategy. SHADE uses current-to-pbest/1 mutation
strategy and external archive the same as JADE [8].
v i,G = xi,G + Fi · (xpbest,G − xi,G )+ Fi · (xr1,G − x̃r2,G )(3)In Eq. (3), the individual xpbest,G is randomly selected fromthe top N × p (p ∈ [0, 1]) members in the G-th generation. Fi
is the F parameter used by individual xi . The greediness ofcurrent-to-pbest/1 depends on the control parameter p in orderto balance exploitation and exploration (small p behaves more
greedily).
In SHADE, each individual xi has an associated pi , which
is set according to the equation by generation:pi = rand[pmin , 0.2]

(4)where pmin is set such that when the pbest individual isselected, at least 2 individuals are selected, i.e., pmin = 2/N .
The maximum value 0.2 in Eq. (4) is the maximum value ofthe range for p suggested by Zhang and Sanderson [8].

After all of the trial vectors ui,G , 0 ≤ i ≤ N have been
generated, a selection process determines the survivors forthe next generation. The selection operator in standard DE
compares each individual xi,G against its corresponding trial
vector ui,G , keeping the better vector in the population.
ui,G if f (ui,G ) ≤ f (xi,G )(7)
xi,G+1 =xi,G otherwise
As with JADE, the CRi and Fi values used by successful
individuals in Eq. (7) are recorded in SCR and SF , and at theend of the generation, the contents of memory are updated asfollows:
meanW A (SCR ) if SCR 6= ∅(8)MCR,k,G+1 =MCR,k,Gotherwise
meanW L (SF ) if SF 6= ∅MF,k,G+1 =(9)MF,k,G
otherwise
An index k (1 ≤ k ≤ H) determines the position in
the memory to update. At the beginning of the search k is
initialized to 1. k is incremented whenever a new element
is inserted into the history. If k > H, k is set to 1. In
generation G, the k-th element in the memory is updated. In
the update equations (8) and (9), note that when all individuals
in generation G fail to generate a trial vector which is better
than the parent, i.e., SCR = SF = ∅, the memory is not
updated.
In the update equation (8), the weighted arithmetic mean
meanW A (SCR ) is computed according to Equation (10) by

Peng et al [15].

TABLE I: Algorithm Complexity|SCR |

meanW A (SCR ) =X

wk · SCR,k

(10)

T0
D = 10D = 30D = 50k=1∆fk
wk = P|S |CR
k=1 ∆fk(11)

where ∆fk = |f (uk,G ) − f (xk,G )| in Eq. (11).
The weighted Lehmer mean meanW L (SF ) in the update
equation (9) is computed using the formula below, and as with
meanW A (SCR ), the amount of improvement is used in order
to influence the parameter adaptation.
P|SF |
2
k=1 wk · SF,kmeanW L (SF ) = P|S |(12)F
k=1 wk · SF,k
III.

E XPERIMENTAL E VALUATION OF SHADE ON
CEC2013 C OMPETITION P ROBLEMS

We evaluated the performance of SHADE on the CEC2013
Special Session on Real-Parameter Single Objective Optimization benchmark suite [12]. This benchmark set is consist of 28
scalable test functions where the global optimum is shifted too, o = (o1 , ...oD ) (D is the number of dimensions).
These problems are based on an earlier set of benchmarks,
the CEC2005 benchmark set [13]. In addition to introducing
some entirely new functions, a major difference from the
CEC2005 benchmark set is that for the composite functions
that are derived by combining multiple functions, the basefunctions that are combined are selected more carefully, andan effort was made to make the regions around local optima
be continuous.
Functions F1 ∼ F5 are unimodal. F6 ∼ F20 are multimodal. F17 and F18 are double-funnel functions which havehigh deceptive structure [16]. F21 ∼ F28 are composite functions which combine multiple test problems into a complex
landscape. See [12] for details.
A. Algorithm Parameters
This section describes the parameter setting of SHADE and
its execution environment. The control parameters we set for
SHADE are:•Population size N = 100, and•memory size H = N = 100

These values were used for all experiments in this paper(same for D = 10, 30, 50).
The value N = 100 was chosen because it was used byseveral other authors in recent work [17], [8], so we have nottuned N . The memory size H = 100 was chosen because it
setting the historical memory size to be the equivalent to the
population size seemed like a natural choice. An evaluationof H ∈ {5, 10, 30, 50, 100, 200, 300, 400, 500} confirmed thatH = 100 resulted in good overall performance [7].
Although the standard DE parameters F (scaling factor)
and CR (crossover rate) are used internally by SHADE,
these parameters are automatically tuned by the history-based
adaptation mechanism and are not set by the user.

0.0009T1
0.5144
1.54792.6709T̂ 2
0.60891.71982.8802(T̂ 2 − T 1)/T 0
99.7084
181.4152
220.7530

B. Algorithm Complexity
This section reports “algorithm complexity” of our SHADE
code as defined in [12].
All experiments were executed on the following system:•

OS: Ubuntu 12.04 LTS•

CPU: core i7 (2.20GHz)•

RAM: 8GB•

Language: C++•

Compiler: g++ (gcc) with -O3 optimization flag

The algorithm complexity of SHADE was calculated according to the detailed instructions in [12]. Table I shows thecomputed algorithm complexity on 10, 30 and 50 dimensions.
As defined in [12], T 0 is the time calculated by running thefollowing test problem:
for i=1:1000000
x=0.55+(double)i; x=x+x; x=x./2; x=x*x;
x=sqrt(x); x=log(x); x=exp(x); y=x/x;
end
T 1 is the time to execute 200,000 evaluations of benchmark
function f14 by itself with D dimensions, and T 2 is the time
to execute SHADE with 200,000 evaluations of f14 in D
dimensions. T̂ 2 is the mean T 2 values of 5 runs. According
to Table I, both T 1 and T̂ 2 scaled linearly with the number ofdimensions, as shown by the linear growth of (T 2 − t1)/T 0.
C. Results
We followed the rules of the CEC2013 benchmark competition [12]. For all of the problems, the size of the search
space is [−100, 100]D . When the gap between the error valuesof the best solution found and the optimal solution was 10−8
or smaller, the error was treated as 0. The three different
dimensions D = 10, 30, 50 are tested. The maximum numberof objective function calls per run was D × 10, 000. The
number of runs per problem was 51, and we recorded best,
worst, median, mean and standard deviation of the error value
between the best fitness values found in each run and the true
optimal value. The results on D = 10, 30 and 50 dimensions
are shown in Table II, III and IV respectively.
We have also plotted average error (compared to the optimal solution) vs. the number of evaluations (for D = 10, 30, 50
dimensions) in Figures 1, 2 and 3. Note that the y-axis (average
error value) is logarithmic scale.
For functions F1 , F5 and F11 , SHADE found the optimal
solution in every single run for D = 10, D = 30, and D =50. This is most likely because these functions are separable,
which results in a easy search space for DE.
In several problems, including F17 , F26 , F28 (D =10, 30, 50), and F6 (D = 50), we noticed that SHADE
converged to the same local optimum in most every single
run. This may be because these functions have a particularly
deceptive landscape structure.
As shown in Tables II, III and IV and Figures 1, 2 and3, the search performance of SHADE usually degrades as thenumber of dimensions (D) increases.
However, there are interesting exceptions. For F6 and F21 ,the results on 30 dimensions is better than the results for
10 dimensions on these functions. Similarly, on F22 , SHADE
performed better in for D = 50 dimensions than for D = 30
dimensions. It has been noted by Whitley et al that for somescalable benchmark functions, finding the global optimum witha higher number of dimensions is easier than finding the global
optimum on a smaller number of dimensions [18]. It is possiblethat our anomalous results with F6 , F21 , and F22 are related
to these previous observations.
D. Behavior of Adaptive Parameters
This section investigates the behavior of the DE control
parameters that are adapted by SHADE. Figure 4 shows the
how the F (scaling factor) and CR (crossover rate) parametersthat are automatically tuned by SHADE change over time on
benchmark problems F1 , F4 , F11 , F14 , F24 , F27 (D = 30). The
plots start at the beginning of the search, and continue until
SHADE converges (search progress stops).
Each of these figures represents 1 run of SHADE on abenchmark problem. In each figure, the F and CR values forall elements in the historical memory are shown. In all cases,
we observe that the values in memory are clustered in a bandthat moves coherently as the search progresses.
The F and CR parameters converge to different sets of values, depending on the benchmark problem. This is consistent
with observation by previous researchers that optimal settings
for these control parameters are domain-dependent (c.f. [1],[3], [4], [5]).
One notable pattern can be observed. For the separable andmultimodal functions (F11 and F14 ), the control parameters
adapt so that F is high and CR converges to relatively low
values, while on the nonseparable functions F4 , F24 , F27 , thecontrol parameters adapt so that CR is very high and F has
a moderate value.
IV.

C ONCLUSIONSThis paper reported the performance evaluation of SuccessHistory based Adaptive DE (SHADE) [7] on the CEC2013
benchmark set. Results for D = 10, 30, 50, including algorithm
complexity measurements were reported. We also analyzed theparameter adaptation behavior of SHADE.
Although we have shown in a companion paper thatSHADE significantly outperforms previous DE variants [7],a detailed analysis of SHADE’s behavior on the CEC2013
benchmarks shows that for some problems, SHADE repeatedly
converges to one particular local optimum. Identifying theTABLE II: Results for D = 10
Func.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28

Best
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
9.5226e-05
2.0120e+01
1.1600e+00
0.0000e+00
0.0000e+00
1.2161e+00
1.0962e+00
0.0000e+00
1.9595e+02
3.6202e-01
1.0122e+01
1.2911e+01
2.4592e-01
1.4155e+00
4.0019e+02
2.4467e-06
1.1922e+02
1.0576e+02
2.0000e+02
1.0125e+02
3.0000e+02
3.0000e+02

Worst
0.0000e+00
0.0000e+00
6.3150e+00
0.0000e+00
0.0000e+00
9.8124e+00
2.4897e-02
2.0488e+01
4.5580e+00
3.1293e-02
0.0000e+00
5.2012e+00
8.6912e+00
6.2454e-02
6.5915e+02
1.2787e+00
1.0122e+01
2.0465e+01
4.4723e-01
2.9698e+00
4.0019e+02
2.0529e+01
7.8669e+02
2.0679e+02
2.0424e+02
2.0002e+02
3.0000e+02
3.0000e+02

Median0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
0.0000e+00
9.8124e+00
1.4915e-03
2.0369e+01
3.5258e+00
9.1545e-03
0.0000e+00
3.2817e+00
3.5469e+00
0.0000e+00
4.3993e+02
6.5561e-01
1.0122e+01
1.6775e+01
3.4914e-01
2.1117e+00
4.0019e+02
8.0390e-01
4.8381e+02
2.0000e+02
2.0000e+02
1.0619e+02
3.0000e+02
3.0000e+02

Mean
0.0000e+00
0.0000e+00
1.2663e-01
0.0000e+00
0.0000e+00
7.8884e+00
3.2593e-03
2.0353e+01
3.3895e+00
1.1987e-02
0.0000e+00
3.1413e+00
3.7708e+00
4.8984e-03
4.2075e+02
7.0799e-01
1.0122e+01
1.6878e+01
3.4352e-01
2.1572e+00
4.0019e+02
4.8396e+00
4.6061e+02
1.9344e+02
2.0015e+02
1.3333e+02
3.0000e+02
3.0000e+02

Std.
0.0000e+00
0.0000e+00
8.8399e-01
0.0000e+00
0.0000e+00
3.9346e+00
4.5447e-03
8.9488e-02
7.3507e-01
8.9885e-03
0.0000e+00
9.7343e-01
1.8530e+00
1.6958e-02
1.1444e+02
2.1188e-01
0.0000e+00
1.5363e+00
4.8987e-02
3.5157e-01
0.0000e+00
6.1972e+00
1.7839e+02
2.4638e+01
7.0243e-01
4.3581e+01
1.4604e-08
0.0000e+00

TABLE III: Results for D = 30
Func.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28

Best
0.0000e+00
1.2842e+03
0.0000e+00
5.7095e-07
0.0000e+00
0.0000e+00
4.2101e-01
2.0439e+01
2.2707e+01
1.7236e-02
0.0000e+00
1.5092e+01
2.3594e+01
0.0000e+00
2.5418e+03
9.7759e-02
3.0434e+01
5.9875e+01
9.6271e-01
9.3862e+00
2.0000e+02
1.2528e+01
2.4634e+03
2.0054e+02
2.0031e+02
2.0000e+02
3.0638e+02
3.0000e+02

Worst
0.0000e+00
4.1728e+04
1.5000e+03
1.2831e-03
0.0000e+00
2.6407e+01
2.6482e+01
2.1000e+01
3.0889e+01
1.8974e-01
0.0000e+00
3.2652e+01
8.8801e+01
1.0410e-01
3.9055e+03
1.2039e+00
3.0434e+01
8.4548e+01
1.5258e+00
1.2128e+01
4.4354e+02
1.1344e+02
4.3517e+03
2.2239e+02
2.9217e+02
3.0601e+02
8.6714e+02
3.0000e+02

Median0.0000e+00
6.7037e+03
1.5249e-04
7.4687e-05
0.0000e+00
0.0000e+00
2.5300e+00
2.0732e+01
2.7519e+01
7.1416e-02
0.0000e+00
2.2989e+01
5.0733e+01
2.0819e-02
3.2363e+03
9.3896e-01
3.0434e+01
7.2844e+01
1.3811e+00
1.0437e+01
3.0000e+02
1.0599e+02
3.5402e+03
2.0332e+02
2.6163e+02
2.0000e+02
3.5512e+02
3.0000e+02

Mean
0.0000e+00
9.0022e+03
4.0206e+01
1.9216e-04
0.0000e+00
5.9596e-01
4.6016e+00
2.0723e+01
2.7466e+01
7.6855e-02
0.0000e+00
2.3039e+01
5.0337e+01
3.1841e-02
3.2179e+03
9.1315e-01
3.0434e+01
7.2457e+01
1.3557e+00
1.0473e+01
3.0904e+02
9.8095e+01
3.5078e+03
2.0525e+02
2.5944e+02
2.0208e+02
3.8763e+02
3.0000e+02

Std.
0.0000e+00
7.4746e+03
2.1298e+02
3.0100e-04
0.0000e+00
3.7286e+00
5.3852e+00
1.7608e-01
1.7694e+00
3.5751e-02
0.0000e+00
3.7320e+00
1.3371e+01
2.3315e-02
2.6367e+02
1.8549e-01
3.8300e-14
5.5837e+00
1.2013e-01
6.0435e-01
5.6473e+01
2.5213e+01
4.1093e+02
5.2904e+00
1.9645e+01
1.4844e+01
1.0897e+02
0.0000e+00

cause of this phenomenon and improving SHADE to avoid
always being attracted to a particular local optimum is an areaof future work.
Finally, we noted that there are some problems which appear to be easier for SHADE when the number of dimensions
is smaller. Further understanding of this anomaly is an areafor future work.

TABLE IV: Results for D = 50
Func.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28

Best
0.0000e+00
7.1372e+03
1.9530e+00
3.0589e-05
0.0000e+00
4.5539e+00
8.9970e+00
2.0599e+01
5.0202e+01
7.3960e-03
0.0000e+00
3.7373e+01
1.1086e+02
0.0000e+00
5.7513e+03
7.3791e-01
5.0786e+01
1.0509e+02
1.8845e+00
1.7610e+01
2.0000e+02
9.0382e+00
5.8816e+03
2.1652e+02
2.7557e+02
2.0000e+02
5.2911e+02
4.0000e+02

Worst
0.0000e+00
5.3130e+04
9.9033e+06
6.8969e-03
0.0000e+00
4.9127e+01
4.8966e+01
2.1218e+01
5.8880e+01
1.8721e-01
0.0000e+00
8.6039e+01
1.8693e+02
7.4950e-02
7.6092e+03
1.7119e+00
5.0786e+01
1.6692e+02
3.2937e+00
2.0681e+01
1.1224e+03
5.4272e+01
9.0177e+03
2.5768e+02
3.7893e+02
4.1906e+02
1.7083e+03
3.3500e+03

R EFERENCES
[1]

[2]

[3]

[4]
[5]

[6]

[7]

[8]

[9]

[10]

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R. Mallipeddi, P. N. Suganthan, Q. K. Pan, and M. F. Tasgetiren,

Median0.0000e+00
2.6821e+04
2.9846e+05
1.1310e-03
0.0000e+00
4.3447e+01
2.2616e+01
2.0862e+01
5.5710e+01
6.6532e-02
0.0000e+00
5.7777e+01
1.4084e+02
2.4983e-02
6.8016e+03
1.2904e+00
5.0786e+01
1.3588e+02
2.6798e+00
1.9361e+01
1.1222e+03
1.1645e+01
7.6280e+03
2.3246e+02
3.4952e+02
2.0162e+02
8.4616e+02
4.0000e+02

Mean
0.0000e+00
2.6565e+04
8.7997e+05
1.6131e-03
0.0000e+00
4.2796e+01
2.3296e+01
2.0905e+01
5.5424e+01
7.3658e-02
0.0000e+00
5.8632e+01
1.4538e+02
3.4536e-02
6.8203e+03
1.2821e+00
5.0786e+01
1.3679e+02
2.6414e+00
1.9271e+01
8.4485e+02
1.3263e+01
7.6289e+03
2.3384e+02
3.3964e+02
2.5751e+02
9.3554e+02
4.5784e+02

Std.
0.0000e+00
1.1300e+04
1.9583e+06
1.4141e-03
0.0000e+00
5.5197e+00
9.3203e+00
1.6831e-01
1.9838e+00
3.6745e-02
0.0000e+00
1.1128e+01
1.9515e+01
1.9286e-02
4.4061e+02
2.0717e-01
4.2668e-14
1.2906e+01
2.8329e-01
7.6995e-01
3.6263e+02
7.1167e+00
6.5792e+02
1.0145e+01
3.0862e+01
8.0778e+01
3.0665e+02
4.1309e+02“Differential evolution algorithm with ensemble of parameters andmutation strategies,” Applied Soft Computing, vol. 11, no. 2, pp. 1679–
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[18] D. Whitley, K. Mathias, S. Rana, and J. Dzubera, “Evaluating evolutionary algorithms,” Artificial Intelligence, vol. 85, pp. 245–276, 1996.

1.0e+06

1.0e+10
D = 10D = 30D = 501.0e+00
1.0e-021.0e-04
1.0e-06

1.0e+06
1.0e+04
1.0e+02
1.0e+00
1.0e-021.0e-04
1.0e-06

1.0e-08

1.0e+101.0e+05
1.0e+00

1.0e-100100000 200000 300000 400000 500000
Number of evaluations

1.0e-050(a) F1

100000 200000 300000 400000 500000
Number of evaluations

0(b) F21.0e+08

1.0e+00
1.0e-021.0e-04
1.0e-06

1.0e+02
1.0e+00
1.0e-021.0e-04
1.0e-06

1.0e-101.0e-100100000 200000 300000 400000 500000
Number of evaluations

1.0e+01

100000 200000 300000 400000 500000
Number of evaluations

0(e) F51.0e+08

1.0e+02

1.0e+04
1.0e+03
1.0e+02
1.0e+01
1.0e+00

D = 10D = 30D = 50
Average error values

D = 10D = 30D = 50
Average error values

1.0e+05

100000 200000 300000 400000 500000
Number of evaluations(f) F6

1.0e+02
D = 10D = 30D = 501.0e+06

1.0e+021.0e-010(d) F41.0e+071.0e+03

1.0e+00

1.0e-08

1.0e-08

D = 10D = 30D = 501.0e+04
Average error values

1.0e+02

1.0e+05
D = 10D = 30D = 501.0e+04
Average error values

1.0e+04

100000 200000 300000 400000 500000
Number of evaluations(c) F31.0e+06
D = 10D = 30D = 501.0e+06
Average error values

1.0e+15

1.0e-08

1.0e-10

Average error values

D = 10D = 30D = 501.0e+20
Average error values

1.0e+02

1.0e+25
D = 10D = 30D = 501.0e+08
Average error values

Average error values

1.0e+04

1.0e+01

1.0e-01
1.0e-021.0e-03

1.0e+01
0

100000 200000 300000 400000 500000
Number of evaluations

1.0e+000(g) F70(h) F81.0e+05

1.0e+01
1.0e+00
1.0e-01

D = 10D = 30D = 50
Average error values

1.0e+02

1.0e+04
D = 10D = 30D = 501.0e+02
Average error values

1.0e+03

100000 200000 300000 400000 500000
Number of evaluations(i) F91.0e+04
D = 10D = 30D = 501.0e+04
Average error values

100000 200000 300000 400000 500000
Number of evaluations

1.0e+00
1.0e-021.0e-04
1.0e-06

1.0e+03

1.0e+02

1.0e+01

1.0e-08

1.0e-021.0e-100100000 200000 300000 400000 500000
Number of evaluations(j) F101.0e+00
0

100000 200000 300000 400000 500000
Number of evaluations(k) F11

Fig. 1: Convergence graph f1 ∼ f12

0

100000 200000 300000 400000 500000
Number of evaluations(l) F12

1.0e+04

1.0e+05

1.0e+02

1.0e+01

1.0e+03

D = 10D = 30D = 50
Average error values

1.0e+03

1.0e+05
D = 10D = 30D = 501.0e+04
Average error values

Average error values

D = 10D = 30D = 501.0e+02
1.0e+01
1.0e+00
1.0e-01

1.0e+04

1.0e+031.0e-02
1.0e+00

1.0e-03
0

100000 200000 300000 400000 500000
Number of evaluations

1.0e+020(a) F13

100000 200000 300000 400000 500000
Number of evaluations(b) F141.0e+01

1.0e+04

1.0e-01

D = 10D = 30D = 50
Average error values

D = 10D = 30D = 50
Average error values

Average error values

1.0e+00

1.0e+03

1.0e+02

1.0e+01
0

100000 200000 300000 400000 500000
Number of evaluations

1.0e+02

100000 200000 300000 400000 500000
Number of evaluations

0(e) F171.0e+08

1.0e+05

1.0e+04
1.0e+03
1.0e+02
1.0e+01

D = 10D = 30D = 50
Average error values

D = 10D = 30D = 50
Average error values

1.0e+05

100000 200000 300000 400000 500000
Number of evaluations(f) F18

1.0e+02
D = 10D = 30D = 501.0e+06

1.0e+03

1.0e+010(d) F16

1.0e+07

100000 200000 300000 400000 500000
Number of evaluations(c) F15

1.0e+04
D = 10D = 30D = 50

Average error values0

1.0e+01

1.0e+04

1.0e+03

1.0e+00
1.0e-01

1.0e+00
0

100000 200000 300000 400000 500000
Number of evaluations

1.0e+020(g) F190(h) F20

1.0e+05

1.0e+03
D = 10D = 30D = 501.0e+03

1.0e+02

D = 10D = 30D = 50
Average error values

1.0e+04

100000 200000 300000 400000 500000
Number of evaluations(i) F211.0e+05
D = 10D = 30D = 50
Average error values

Average error values

100000 200000 300000 400000 500000
Number of evaluations

1.0e+04

1.0e+03

1.0e+01

1.0e+00

1.0e+020

100000 200000 300000 400000 500000
Number of evaluations(j) F221.0e+020

100000 200000 300000 400000 500000
Number of evaluations(k) F23

Fig. 2: Convergence graph f13 ∼ f24

0

100000 200000 300000 400000 500000
Number of evaluations(l) F24

1.0e+03

1.0e+03

Average error values

D = 10D = 30D = 50

Average error values

D = 10D = 30D = 501.0e+02

1.0e+020

100000 200000 300000 400000 500000
Number of evaluations

0

100000 200000 300000 400000 500000
Number of evaluations(a) F25(b) F26

1.0e+04

1.0e+05
D = 10D = 30D = 50
Average error values

Average error values

D = 10D = 30D = 501.0e+03

1.0e+02

1.0e+04

1.0e+03

1.0e+020

100000 200000 300000 400000 500000
Number of evaluations

0

100000 200000 300000 400000 500000
Number of evaluations(c) F27(d) F28

Fig. 3: Convergence graph f25 ∼ f28

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

CR
F
0

10000
20000
Number of evaluations

30000

CR
F
0

(a) F1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

100000200000
Number of evaluations

300000CRF0(b) F4
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

CR
F

0

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

100000200000
Number of evaluations(d) F1430000040000
80000
120000
Number of evaluations(e) F24

150000(c) F11

CR
F
0

50000
100000
Number of evaluations

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

CR
F
0

60000
120000
Number of evaluations(f) F27

Fig. 4: Parameter adaptation process of SHADE on F1 , F4 , F11 , F14 , F24 , F27 .

180000