Replicator Dynamic Inspired Differential Evolution Algorithm for

Global Optimization

Shichen Liu, Yan Xiong, Qiwei Lu and Wenchao Huang

Department of Computer Science and Technology, University of Science and Technology of China,

Hefei, Anhui 230027, China

Keywords: Differential Evolution, Numerical Optimization, Parameter Adaptation, Self-adaptation, Replicator

Dynamic, Natural Computation.

Abstract: Differential Evolution (DE) has been shown to be a simple yet efficient evolutionary algorithm for solving

optimization problems in continuous search domain. However the performance of the DE algorithm, to a

great extent, depends on the selection of control parameters. In this paper, we propose a Replicator Dynamic

Inspired DE algorithm (RDIDE), in which replicator dynamic, a deterministic monotone game dynamic

generally used in evolutionary game theory, is introduced to the crossover operator. A new population is

generated for an applicable probability distribution of the value of Cr, with which the parameter is evolving

as the algorithm goes on and the evolution is rather succinct as well. Therefore, the end-users do not need to

find a suitable parameter combination and can solve their problems more simply with our algorithm.

Different from the rest of DE algorithms, by replicator dynamic, we obtain an advisable probability

distribution of the parameter instead of a certain value of the parameter. Experiment based on a suite of 10

bound-constrained numerical optimization problems demonstrates that our algorithm has highly competitive

performance with respect to several conventional DE and parameter adaptive DE variants. Statistics of the

experiment also show that our evolution of the parameter is rational and necessary.

1 INTRODUCTION

The evolutionary algorithms are heuristic search

algorithms which have been developed for over 50

years (Friedberg, 1958); (Box, 1957); (Holland,

1962); (Fogel, 1962). There are three main aspects

in EAs, i.e., genetic algorithms, evolutionary

programming and evolutionary strategies. They are

now generally used to solve optimization problems

in continuous search space. Differential Evolution

(DE) algorithm, proposed by Storn and Price, is one

of the state-of-the-art evolutionary algorithms (Storn

and Price, 1995). DE algorithm is a simple yet

efficient population-based stochastic method for

global optimization problems, and it has been

successfully applied to a whole host of engineering

problems such as aerodynamic design (Rogalsky et

al., 1999), digital filters design (Storn, 1996); (Storn,

2005), power system optimization

(Lakshminarasimman and Subramanian, 2008), etc.

Generally, just as other evolutionary algorithms,

there are three main operations in DE, i.e., mutation,

crossover and selection. In these operations three

crucial control parameters are required to be

specified. They are the population size NP, scale

factor F and the crossover rate Cr. These parameters

significantly affect the optimization performance of

the DE. In this regard, although the use of

evolutionary algorithms to solve problems of design

and optimization is varied, different end-users

confront the same problem that they have to find a

suitable parameter combination that matches the

evolutionary algorithms before actual design or

optimization can begin (Lobo and Goldberg, 2001);

(Harik and Lobo, 1999). Hence, it's reasonable and

necessary to turn parameters setting into a part of the

algorithm itself instead of leaving it as a problem to

the end-users.

In the past, on the choice of parameters of all

sorts of EAs, researchers always try to find a best

definite value for a parameter. To achieve this,

literature either uses a trial-and-error searching

process, or gets the parameter adapted or self-

adapted. However, in most cases, one cannot find a

best value for parameter configuration to optimize

the performance, and even whether there exists such

133

Liu S., Xiong Y., Lu Q. and huang W..

Replicator Dynamic Inspired Differential Evolution Algorithm for Global Optimization.

DOI: 10.5220/0004053401330143

In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 133-143

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

a best value is doubtful. More than one value of the

parameter may be appropriate: a value may win in a

run while a different value may perform better in

another run. Furthermore, there are also many cases

that some individuals of the population use a

parameter while the others use a different one obtain

better results than that the whole population uses a

definite parameter. So, unlike the previous studies,

which focus on finding a certain value for a

parameter, we focus on the probability distribution

of all possible or suitable values of the parameter, a

definite value is just a special form of the

distribution.

Based on the above observation, in this paper, we

propose a Replicator Dynamic Inspired DE

algorithm (RDIDE), in which crossover rate Cr is

configured using replicator dynamic, a deterministic

monotone game dynamic generally used in

evolutionary game theory. Since the probability

distribution of the crossover rate Cr is self-adapted

in our algorithm, the end-users can be able to simply

run the algorithm as a black-box without

consideration of the parameters, which may greatly

improve the working efficiency of the end-users.

To sum up, this paper makes the following

contributions:

 We propose a new self-adaptive DE algorithm,

with which the users can solve their problems more

simply, with a higher success rate and a quicker

convergence speed.

 Replicator dynamic is introduced to the

parameter setting of the DE algorithm. We no more

discuss about a proper parameter, but about an

advisable probability distribution of the parameter.

 In the dynamic of the distribution, we design a

new mechanism for believable success rate based on

principle of statistics.

The remainder of this paper is organized as follows.

In section 2, the conventional DE is reviewed.

Section 3 describes the proposed RDIDE and the use

of replicator dynamic. A suite of 10 bound-

constrained numerical optimization problems is set

to evaluate the performance of the algorithm in

section 4. Finally, section 5 summarizes the main

conclusions arising from this work.

2 DIFFERENTIAL EVOLUTION

ALGORITHM

Without loss of generality, in this paper, DE is

aiming to minimize an objective function. Let S be

the search space of the optimization problem,

RS 

. The population of DE includes NP individuals and

each of them is a D-dimensional solution particle. At

any certain generation G, the individuals are of the

form

NP,1,2,i ,

321

 )

i,D



X 



where i indicates the index of the particle. The

particles develop from one generation to another

constraint by the search space. At each generation,

every particle goes through the operations of

mutation, crossover and selection, and a trial particle

will be generated for each target particle. The

evolution processes as follows.

2.1 Initialization

DE algorithm starts with an initial population

}XX,X,X{





, these particles are expected to be

initialized filling the entire search space as much as

possible. For this purpose, generally, the initial

population is generated within the boundary

constraints at random

D,1,2,j ,10

 )X(X),N(XX

ji,j

(1)

where

and

are the lower and upper boundary

of j-th component respectively, and

),N( 10

denotes

a uniformly distributed random value within the

range [0, 1].

2.2 Mutation

A mutant vector

)y,y,y(yY

i,D

321





is generated

for every associated target vector



in this

operation at each generation G. Several mutation

methods could be used to generate



, and a

conventional one is like this:

NP,2, 1,i,

321

 )XX(FXY



(2)

where index

iii

randrr

321

are random integers from

the range [1, NP], mutually different, and each is

different from the base index i. F is a scaling factor

for differential vectors.

2.3 Crossover

Crossover operation comes after mutation. The trial

vector

),,,,(

,3,2,1,

uuuuU 



is generated from the

combination of its parent and mutant vector:















otherwise,x

)jor(jCR)[0,1)if(Uniform,y

ji,

randji,

ji,

(3)

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

134

where

rand

is a random index chosen from [1, D] to

ensure at least one component is different from



and



, and the parameter Cr is within the range [0,

1], indicating the crossover rate of the generation. If

any component of the trial vectors is beyond the

search space, they will be reinitialized randomly and

uniformly within the search space.

2.4 Selection

In this phase, we determine which vector is going

into the next generation and which should be

deleted. The procedure is done following rule for the

function minimization:













otherwiseU

UfXfifX

))()((,



(4)

Every trial vector is only compared with its target

vector, and the one with better fitness is kept. Hence,

all the individuals of the next generation are going to

get better or remain the same, thus the whole

population evolves.

3 REPLICATOR DYNAMIC

INSPIRED DE ALGORITHM

Being a crucial factor of the DE algorithm, control

parameters selection determines the performance of

the algorithm directly. Hence, a good deal of

research on the parameters selection of DE has been

done. Storn (1995) suggested that F within the range

[0.5,1], Cr in [0.8,1] and NP = 5D or 10D. Gämperle

et al. (2002) suggested that NP be between 3D and

8D, F= 0.6, and Cr between [0.3,0.9]. At the same

time, several adaptive and self-adaptive mechanisms

have been proposed to dynamically change the value

of the parameters. Zaharie (2003) used a

multipopulation method for the parameter adaptation

(ADE). Omran et al. (2005) proposed a mechanism

to self-adapt the scaling factor F (SDE). Later on,

Brest et al. (2006) encoded F and Cr into individuals

and modulate them by two parameters. In the same

year, Teo (2006) proposed a DE algorithm with a

dynamic population sizing strategy based on self-

adaptation (DESAP). Lately, Qin et al. (2009)

proposed SaDE, in which both generation strategy

and the parameters are adapted.

In our paper, we focus on the adaptation of Cr

during the evolution, as Cr is an especially

significant parameter. The suitable choice of Cr can

lead to good result while an improper one may result

in the failure of the algorithm (Price et al., 2005).

3.1 Inspired by Replicator Dynamic

The main idea of this paper is to self-adapt the

probability distribution of the crossover rate, so that

the parameter could be more suitable to various

kinds of problems. At the same time, different

distributions of Cr may perform better at different

generations for a certain problem, so the distribution

of Cr is expected to be fit for every moment of the

evolution as well. To achieve this, a mechanism of

multiple evolutions is proposed: the first evolution

refers to DE algorithm itself, and the second one

means that the probability distribution of Cr value is

evolving independently with the idea of evolutionary

game theory.

We build a candidate set (CRSet), containing

several possible values of Cr. Whenever the

crossover operation is executed, each individual

choose one value from the set via a particular

probability distribution. The value of Cr is a real

number within the range [0, 1], and the set is

expected to cover the range uniformly. In our

proposal, we let



CRSet

,,,{

321

CRCRCR },

CRCR

where

is set to (0.2×i-0.1). For each

, a

is assigned to indicate the probability to choose it,

the distribution of



. At each generation,

every individual choose a Cr from the CRSet via the

distribution of

, and the distribution



is evolving

according to the fitness of each

of the current

and previous generations with replicator dynamic.

Probability distribution to choose values for Cr is

very similar to mixed strategy equilibrium of a game

theory, and a definite value of Cr corresponds with a

pure strategy. Our attention is on the dynamically

changing of the distribution, thus a method of

evolutionary game theory is introduced. We assume

that a new population of plentiful individuals is

generated to seek a reasonable probability

distribution for

with the idea of evolution. Any

individual in the population is called replicator,

choosing a certain value in the CRSet and passing its

choice to the descendants without modification. Let





be the number of individuals choosing

time point t, then the total population size is

 

tntN





, and the proportion of individuals to

choose













tNtntpcr

/

. The population

state is the distribution of



tpcr

, i.e.,







tpcrtP





















),,,

5432

tpcrtpcrtpcrtpcr

, Let



and



be the

ReplicatorDynamicInspiredDifferentialEvolutionAlgorithmforGlobalOptimization

135

background per capita birth and death rates in the

population. Then the rate of change of the number of

individuals choosing

(



) and rate of change of

total population (



) can be described as follows:





iii

nCRfitnessn





 )(



(5)



















)(

CRfitnessN

nCRfitness

NnnN



(6)

where









)()(

pcrCRfitnessCRfitness

is the

average fitness.

Since













tNtntpcr

/

, we take derivative to

both sides:

iii

pcrNnrcpN 





. So





iii

pcrCRfitnessCRfitnessrcp  )()(



(7)

(7) gives the replicator dynamic that will be used to

adjust distribution of

pcr

, increasing rate of the

proportion of the individual choosing

independent of the background per capita birth rate(



), death rate(



) and the size(



) of the

population. In another word, the evolution of



only dependent on the fitness of each Cr, which is

very simple to execute. We let the possibility of one

particle (an individual in DE) to choose

equal

the proportion of individuals (in Cr evolution) to

choose

, i.e.,

crii

PPpcrP



 ,

, to ensure that the

proportion of individuals in DE to choose different

Cr is approximate to



)(

CRfitness

can be

indicated by the success rate of the trial vectors

generated by





CrT

and successfully entering the

next generation





CrW

. So, (7) changes into the form

below:

CrT

CRSuccRateCRfitness

CrW

)()( 

(8)





pcrCRSuccRateCRSuccRateP

 )()(



(9)



51,2i ,)()(

，，





i

PCRSuccRateCRSuccRate

(10)

The distribution



changes by (8), (9) and (10)

succinctly, thus the evolution of Cr is achieved.

3.2 Design of Believable Success Rate

When we use the replicator dynamic for the

evolution of Cr, there is still a problem in (8), that if

the total quantity of individuals using

is not

enough, the corresponding fitness is trustless.

So we have to determine the minimum of

CrT

to ensure the trustiness of

)(

CRfitness

with high

confidence level and narrow confidence interval.

From de Moivre–Laplace theorem, we learn that if

CrT

is big, approximately







 



1 0,N~

1 CRfitnessCRfitnessCrT

CRfitnessCrTCrW





where





1 0,N

denotes the standard normal

distribution and indexes are omitted. Thus we have:



 



























2/2/

CRfitnessCRfitnessCrT

CRfitnessCrTCrW

(11)

Equation can be transformed into another form:











1)( BCRfitnessAP

(12)

The solutions of the equation









 



CRfitnessCRfitnessCrT

CRfitnessCrTCrW







are A and B:



()

ˆˆ

() 1 ()

fitness CR

CrT

CrT u

fitness CR fitness CR

CrT









































(13)

where

CrTCrWCRssefitn /)(



, denotes the value of

fitness that will be used in the algorithm，



is a

corresponding constant to





1 0,N

, A takes

negative sign and B takes positive sign.









is the confidence level and



BA,

is the

confidence interval for

)(CRfitness

. Since

1)(

0  CRssefitn



4/1)(

1)(

 CRssefitnCRssefitn

. When we assume



4/1)(

1)(

 CRssefitnCRssefitn

the width of





BA,

equals to

2/2/



uCrTu 

. Let

05.0





and

2.0widththe

, we have

96.1





and

the minimum of CrT is 93. This is to say, CrT must

be at least 93,

)(

CRssefitn

is trusted.

However, CrT could not be big enough in one

generation, so two extra methods are introduced to

RDIDE. First we build memories to store the

numbers of individuals choosing

and those

successfully entering the next generation within the

last M generations. With this method,



MGCrTCrWCrTCrWCRssefitn

MGg

iiii







 11

//)(

where G is the current generation, and respectively,

CrT

and

CrW

denote the number of vectors

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

136

choosing

in generation g and the number of

those successfully entering the next generation.

During the first M generations, we simply let the

value of

be 0.2 (i = 1, 2, 3, 4, 5), and add

CrT

and

CrW

to the memories. In the following

generations,

is dynamically changing with (9),

while

CrT

and

CrW

replace

CrT



and

CrW



Besides, we assign a constant (

min

) to constrain the

minimum of

, when

min

and

is going to

decrease, the value of

remains the same. The

expected value of CrT,

MPNPCrTE )(

. In this

paper,

)(CrTE

is expected to be equal or greater than

100 to ensure the trustiness of

)(CRfitness

, and we

achieve this by assuming a small P such as 0.1, and





NPPM //100

3.3 The Algorithmic Description

The algorithmic description of the RDIDE is

presented in Table 1.

Table 1: Algorithmic description of the RDIDE.

Step 1: Initialization

Set the generation counter G=0.

Initialize a population of NP individuals according to (1). Evaluate the

population. Store with best fitness as and its fitness .

Initialize the distribution of , , and establish two

memories, , (i=1 to M, j=1 to 5).

Step 2: Evolution

WHILE Termination Criterion is not satisfied

Step 2.1 renovate and

Replace and by and

Set ,

Step 2.2 Mutation, Crossover and Selection

FOR i=1 to NP

Choose a from CRSet due to

FOR j=1 to D

END FOR

WHILE the variable is outside the search region

Regenerate

END WHILE

Evaluate the trial vector.

END IF

ELSE

END IF

END FOR

Step 2.3 Dynamic Change of Distribution

IF G>M

ReplicatorDynamicInspiredDifferentialEvolutionAlgorithmforGlobalOptimization

137

Table 2: Benchmark functions.

Test problems

min

global optimum S







100,100













100,100

















11003

iii

xxxF

0 (1,1,…,1)



100,100















1,04.014

NzF









100,100





































202cos

exp

2.0exp205





32,32





































202cos

exp

2.0exp206





32,32























1cos

4000



500,500























1cos

4000



500,500









zzF

102cos109





5,5









mmF

102cos1010





5,5







zMmoooooxz

 ,,,

,,2,1

;o: the shifted global optimum, M: orthogonal rotation matrix.

F6:cond(M)=1; F8:cond(M)=3; F10:cond(M)=2

4 EXPERIMENTS AND RESULTS

4.1 Test Problems and Experimental

Conditions

In this paper, in order to assure a fair comparison,

the experimental conditions, the parameters setting

and the benchmark problems are the same to SaDE.

10 benchmark problems (F1-F10) were set to

evaluate the performance of our algorithm. Six

functions (F1, F2, F4, F5, F7, F9) are shifted and

three (F6, F8, F10) are further rotated. Among these

functions, F1-F4 are unimodal functions and F5-F10

are multimodal functions. All the functions are listed

in Table 2.

In our experiment, RDIDE is compared with 5

conventional DE and 4 adaptive DE variants. In

order to ensure reliability, the statistics of the

experiment with these 9 DE algorithms are results

found in literature (Qin et al. 2009). The conditions

of the experiment are as follows:

1) Population size NP=50, Scaling factor F=0.5;

2) Dimension D=10/30 for all problems:

FEs=100 000 with 10-D problems,

FEs=300 000 with 30-D problems.

3) Parameters for RDIDE,

1.0,20

min

 PM

4) Comparison DE, 5 conventional DE:

DE/rand/1(F=0.9,Cr=0.1),

DE/rand/1(F=0.9,Cr=0.9),

DE/rand/1(F=0.5,Cr=0.3),

DE/rand-to-best/1(F=0.5,Cr=0.3),

DE/rand-to-best/2(F=0.5,Cr=0.3).

4 adaptive DE variants:

SaDE, ADE, SDE, jDE.

5) All experiments were run 50 times,

independently.

4.2 Results and Analysis

1) In this section, we compare RDIDE with the 9

other DE. Two groups of comparison are conducted

to show the highly competitive performance of

RDIDE. In the first comparison, we concentrate on

the mean and standard deviation of the functions as

well as the success rates. The success rate refers to

the proportion that the success runs divided by the

total runs. The success of a run means that it results

in a value no worse than the pre-specified optimal

value, i.e.,

min



f

with the number of FEs less

than the pre-specified maximum number in this run.

In the second comparison, we focus on the average

number of function evaluations (NFE) required to

find the optima, as it’s a direct reflection of the

convergence speed. Table 3 and Table 4 report the

statistics of the first comparison, and Table 5 shows

the results of the second comparison. All best results

are typed in bold.

From the results of the first comparison, we can

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Table 3: Results for 10-D problems.

Algorithm

D=10

F1 F2 F3 F4

Mean Std SRate Mean Std SRate Mean Std SRate Mean Std SRate

CDE-1 0 0 100% 8.89E-01 4.96E-01 0% 9.01E-01 7.94E-01 0% 2.41E+01 1.28E+01 100%

CDE-2 4.95E-13 5..27E-13 100% 1.44E-05 1.13E-05 43% 7.11E-03 2.74E-02 0% 2.42E-04 1.38E-04 100%

CDE-3 0 0 100% 9.63E-09 5.99E-09 100% 1.76E+00 1.54E+00 0% 5.42E-06 4.44E-06 83%

CDE-4 0 0 100% 0 0 100% 2.57E+00 1.86E+00 0% 0 0 100%

CDE-5 0 0 100% 9.45E-13 9.90E-13 100% 2.37E+00 2.23E+00 0% 1.04E-08 1.20E-08 100%

SaDE 0 0 100% 0 0 100% 0 0 100% 0 0 100%

ADE 0 0 100% 1.44E-04 2.48E-04 3% 1.56E+00 2.64E+00 0% 7.00E-02 5.84E-02 0%

SDE 0 0 100% 0 0 100% 2.05E+00 1.68E+00 0% 0 0 100%

DE 0 0 100% 0 0 100% 1.34E-13 7.32E-13 100% 0 0 100%

RDIDE 0 0 100% 0 0 100% 5.78E-07 2.05E-06 100% 0 0 100%

Algorithm

D=10

F5 F6 F7 F8

Mean Std SRate Mean Std SRate Mean Std SRate Mean Std SRate

CDE-1 0 0 100% 3.81E-05 1.30E-05 90% 0 0 100% 1.22E-01 2.77E-02 0%

CDE-2 4.59E-07 2.41E-07 100% 6.86E-07 3.89E-07 100% 3.05E-01 2.02E-01 0% 2.41E-01 2.00E-01 0%

CDE-3 0 0 100% 3.32E-15 9.01E-16 100% 0 0 100% 1.60E-01 3.75E-02 0%

CDE-4 4.97E-15 1.77E-15 100% 4.26E-15 1.45E-15 100% 4.67E-03 8.13E-03 70% 2.91E-01 3.14E-01 0%

CDE-5 3.55E-15 1.87E-15 100% 3.55E-15 0 100% 0 0 100% 1.44E-01 3.97E-02 0%

SaDE 0 0 100% 0 0 100% 0 0 100% 1.37E-02 1.18E-02 20%

ADE 0 0 100% 0 0 100% 2.55E-07 1.40E-06 100% 7.93E-02 4.24E-02 0%

SDE 0 0 100% 0 0 100% 7.39E-03 7.59E-03 40% 3.81E-02 3.06E-02 0%

DE 0 0 100% 0 0 100% 5.75E-04 2.21E-03 93% 2.26E-02 1.77E-02 7%

RDIDE 0 0 100% 0 0 100% 0 0 100% 0 0 100%

Algorithm

D=10

F9 F10

Index of test functions with 100% success rate

Mean Std SRate Mean Std SRate

CDE-1 0 0 100% 1.33E+01 3.00E+00 0% 1, 4, 5, 7, 9

CDE-2 8.71E+00 5.53E+00 0% 1.63E+01 2.02E-01 0% 1, 4, 5, 6

CDE-3 0 0 100% 1.65E+01 2.99E+00 0% 1, 2, 5, 6, 7, 9

CDE-4 6.63E-02 2.52E-01 93% 1.00E+01 2.32E+00 0% 1, 2, 4, 5, 6

CDE-5 0 0 100% 1.63E+01 3.36E+00 0% 1, 2, 4, 5, 6, 7, 9

SaDE 0 0 100% 3.80E+00 1.35E+00 0% 1, 2, 3, 4, 5, 6, 7, 9

ADE 0 0 100% 9.41E+00 2.20E+00 0% 1, 5, 6, 7, 9

SDE 6.96E-01 8.72E-01 50% 7.79E+00 3.18E+00 0% 1, 2, 4, 5, 6, 9

DE 0 0 100% 5.78E+00 3.18E+00 0% 1, 2, 3, 4, 5, 6, 9

RDIDE 0 0 100% 0 0 100% 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

observe that, for 10-D problems, RDIDE can find

the global optimal value for all test functions with

100% success rate while other DE algorithms can

achieve 4-8 functions only. It outperforms most of

other algorithms, and is only with a little worse

mean value in F3 compared with SaDE and jDE. For

30-D problems, except a success rate of 90% in F3,

success rates of all other functions reach 100%. At

the same time, jDE succeeds in 6 functions, which is

the second best of all algorithms while CDE-2,

CDE-4 and ADE fail in all functions. The mean

value of RDIDE in F5, F6 and F9 is a little worse

than some other algorithms, yet despite all this, the

corresponding mean values are 5.91E-15, 4.25E-15

and 3.30E-14 which are still very close to the optima.

And in F4, F8 and F10, the results of RDIDE

surpass the other algorithms completely. For both

10-D and 30-D problems, F8 and F10 are so difficult

that most algorithms fail to find the global optima

while RDIDE achieves with 100% success rate.

From Table 5, we can observe that the

convergence speed of RDIDE is outstanding as well.

For 10-D problems it holds 4 best NFE values while

for 30-D problems it holds 6 best NFE values. In

contrast, CDE-4 for 10-D problems and SaDE for

30-D problems, which are the second fastest from

the result, holds only 2 best NFE values and 4 best

NFE values respectively.

2) The main idea of the proposed algorithm is to

self-adapt the crossover rate, which is reflected by

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Table 4: Results for 30-D problems.

Algorithm

D=30

F1 F2 F3 F4

Mean Std SRate Mean Std SRate Mean Std SRate Mean Std SRate

CDE-1 0 0 100% 3.62E+03 8.22E+02 0% 3.39E+01 1.51E+01 0% 1.24E+04 2.15E+03 0%

CDE-2 4.50E-02 6.15E-02 0% 1.73E+03 1.40E+03 0% 1.04E+02 6.25E+01 0% 8.98E+03 5.74E+03 0%

CDE-3 0 0 100% 1.38E+03 2.53E+02 0% 2.14E+01 1.98E+00 0% 5.19E+03 1.24E+03 0%

CDE-4 1.90E-07 1.04E-06 97% 1.92E+01 2.27E+01 0% 6.37E+01 4.01E+01 0% 2.36E+00 5.47E+00 0%

CDE-5 0 0 100% 1.04E+02 8.25E+01 0% 2.08E+01 1.19E+01 0% 1.51E+03 2.07E+02 0%

SaDE 0 0 100% 0 0 100% 3.99E-01 1.22E+00 90% 3.37E+00 1.37E+01 0%

ADE 0 0 100% 3.04E+02 6.86E+01 0% 4.69E+01 2.64E+01 0% 6.75E+04 1.02E+04 0%

SDE 4.56E-01 2.08E+00 50% 1.58E+00 4.48E+00 0% 7.73E+03 3.27E+04 0% 4.67E+02 5.09E+02 0%

DE 0 0 100% 8.91E-11 1.27E-10 100% 5.57E-01 1.38E-00 40% 2.15E-01 4.91E-01 0%

RDIDE 0 0 100% 0 0 100% 2.90E-01 2.04E+00 90% 3.85E-09 1.50E-08 100%

Algorithm

D=30

F5 F6 F7 F8

Mean Std SRate Mean Std SRate Mean Std SRate Mean Std SRate

CDE-1 0 0 100% 3.81E-05 1.30E-05 90% 0 0 100% 9.12E-02 3.08E-02 0%

CDE-2 3.86E-02 2.18E-02 0% 7.64E-02 5.11E-02 0% 1.52E-01 1.15E-01 0% 9.01E-01 1.40E-01 0%

CDE-3 4.03E-15 1.23E-15 100% 3.67E-15 6.49E-16 100% 0 0 100% 2.24E-05 1.19E-04 93%

CDE-4 3.10E-02 1.76E-01 93% 4.82E-03 2.64E-02 97% 1.08E+01 1.00E+01 0% 1.82E+02 5.47E+01 0%

CDE-5 7.58E-15 1.80E-15 100% 7.34E-15 1.30E-15 100% 0 0 100% 3.97E-03 1.85E-02 37%

SaDE 0 0 100% 0 0 100% 2.38E-03 5.03E-03 80% 8.54E-03 0.09E-03 40%

ADE 0 0 100% 0 0 100% 0 0 100% 2.93E-03 5.65E-03 10%

SDE 2.19E-01 3.87E-01 40% 1.01E-01 3.04E-01 63% 1.59E+00 2.23E+00 13% 1.39E+00 4.24E+00 13%

DE 0 0 100% 0 0 100% 0 0 100% 5.17E-03 6.64E-03 57%

RDIDE 5.91E-15 1.79E-15 100% 4.25E-15 8.52E-16 100% 0 0 100% 0 0 100%

Algorithm

D=30

F9 F10

Index of test functions with 100% success rate

Mean Std SRate Mean Std SRate

CDE-1 0 0 100% 1.68E+02 1.43E+01 0% 1, 5, 7, 9

CDE-2 8.54E+01 3.30E+01 0% 2.45E+02 2.20E+01 0%

one

CDE-3 3.10E+01 3.24E+00 100% 1.87E+02 1.09E+01 0% 1, 5, 6, 7, 9

CDE-4 9.58E+00 3.88E+00 93% 1.44E+02 2.09E+01 0%

one

CDE-5 4.03E+01 3.73E+00 100% 1.88E+02 7.15E+00 0% 1, 5, 6, 7, 9

SaDE 0 0 100% 1.67E+01 5.26E+00 0% 1, 2, 5, 6, 9

ADE 2.32E-01 5.01E-01 100% 1.21E+02 1.28E+01 0%

one

SDE 1.09E+01 4.23E+00 50% 3.63E+01 6.78E+00 0% 1, 5, 6, 7, 9

DE 0 0 100% 3.65E+01 8.29E+00 0% 1, 2, 5, 6, 7, 9

RDIDE 3.30E-14 2.83E-14 100% 3.07E-14 2.86E-14 100% 1, 2, 4, 5, 6, 7, 8, 9, 10

the dynamically change of the distribution of the

probabilities to choose different Cr value. So we

discuss about the property of RDIDE via the

changes of the distributions in this section.

Figure 1

illustrates changes of



in RDIDE for all functions

with both D=10 and D=30. In the Figure, x-axis

represents different values of Cr, y-axis represents

generations of the algorithm and z-axis represents

the probabilities to choose different values of Cr.

From the figure, it can be observed that the

distribution is evolving as the DE algorithm goes on.

Different values of Cr are suitable for different

problems, and generally, a proper choice of Cr value

is 0.1 and 0.9. Besides, even for a certain problem,

the proper probability distribution of Cr value may

change with the process of the algorithm. And we

discover that this kind of change is regular, as for

each problem, experiment was run 50 times

independently, and the corresponding changes of the

distribution are exceedingly similar. In F1, F7 and

F8, Cr should be constant 0.1, and in F2, Cr should

be constant 0.9. However in all other test problems,

the distribution should be changing as the algorithm

goes on, e. g., in F3 with D=30, the value of Cr

should be 0.1 with high probability at the beginning

of evolution, then it should change to 0.9 and be

back to 0.1 finally; in F5 with both D=10 and D=30,

the change of the distribution is complex at the

beginning, each value of Cr dominates for a short

time and 0.9 turns into the best choice finally; case

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Table 5: Comparison of NFE.

10D CDE-1

NFE SRate

CDE-2

NFE SRate

CDE-3

NFE SRate

CDE-4

NFE SRate

CDE-5

NFE SRate

SaDE

NFE SRate

RDIDE

NFE SRate

F1 16770 100% 53298 100% 10291 100% 6318 100% 10058 100% 8357 100% 10370 100%

F2 -- 0% -- 43% 72436 100% 23383 100% 53658 100% 14867 100% 14847 100%

F3 -- 0% -- 0% -- 0% -- 0% -- 0% 42446 100% 64525 100%

F4 -- 0% -- 0% -- 83% 30925 0% 71278 100% 15754 100% 9433 100%

F5 25335 100% 82919 100% 15157 100% 9436 100% 15045 100% 12123 100% 12729 100%

F6 -- 90% 85272 100% 16682 100% 9923 100% 16980 100% 12244 100% 15794 100%

F7 41247 100% -- 0% 29961 100% -- 70% 59205 100% 35393 100% 54942 100%

F8 -- 0% -- 0% -- 0% -- 0% -- 0% -- 0% 54561 100%

F9 19200 100% -- 0% 23155 100% -- 93% 30621 100% 23799 100% 26007 100%

F10 -- 0% -- 0% -- 0% -- 0% -- 0% -- 0% 24288 100%

30D

F1 66339 100% -- 0% 34687 100% -- 97% 31470 100% 20184 100% 32346 100%

F2 -- 0% -- 0% -- 0% -- 0% -- 0% 118743 100% 117799 100%

F3 -- 0% -- 0% -- 0% -- 0% -- 0% -- 90% -- 90%

F4 -- 0% -- 0% -- 0% -- 0% -- 0% -- 0% 191469 100%

F5 92941 100% -- 0% 49822 100% -- 93% 45948 100% 26953 100% 28594 100%

F6 -- 0% -- 0% 55108 100% -- 97% 49961 100% 33014 100% 46740 100%

F7 80741 100% -- 0% 39436 100% -- 0% 41314 100% -- 80% 39056 100%

F8 -- 0% -- 0% -- 0% -- 0% -- 0% -- 0% 45708 100%

F9 90391 100% -- 0% -- 0% -- 0% -- 0% 58732 100% 147483 100%

F10 -- 0% -- 0% -- 0% -- 0% -- 0% -- 0% 134735 100%

in F6 is similar to F5, yet 0.1 takes a more

dominating place initially. From later period in F9

and F10 with D=10, we see that probability of 0.1

and probability of 0.9 are equal, neither of the value

can surpass the other one.

So we conclude that an appropriate probability

distribution of the value of Cr is not only related to

the problem and the algorithm, but also the stage of

the evolution as well. Thus assuming a constant

value of Cr in conventional DE is not befitting, and

so does using a trial-and-error process to find the

parameter combination. Based on the analysis above,

RDIDE, which uses the probability distribution

instead of a definite value while the distribution is

self-adapted, is more rational for global optimization.

5 CONCLUSIONS

In this paper, to make DE algorithm more practical

to various kinds of optimization, we proposed a

RDIDE algorithm, in which replicator dynamic is

introduced to the crossover operator. With this

method, the end-users can simply run the algorithm

without considering the setting of the parameters.

The algorithm involves multiple evolutions: the first

evolution refers to DE algorithm, and the second one

means that the parameter Cr is evolving

independently with replicator dynamic. A new

population is assumed to find an advisable

probability distribution of Cr, and an extra technique

is designed for a believable success rate. The final

process according to the evolution is rather succinct.

We then compare RDIDE with 9 other DE

algorithms over a suite of 10 bound-constrained

numerical optimization problems and RDIDE

produced highly competitive results in both success

rate and the convergence speed. Furthermore, the

statistics of the experiment show that a good choice

of Cr not only rests with different problems but also

with different stages of the detailed evolution

process. Finally we conclude that RDIDE is a more

effective and simple DE algorithm to obtain the

global optima with a higher success rate and a

quicker convergence speed.

ACKNOWLEDGEMENTS

This work is supported by National Natural Science

Foundation of China under grant No.61170233,

No.60970128, post-doctoral foundation

No.2011M501397 and youth foundation of USTC.

We thank four anonymous referees for their precious

comments to improve this paper.

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141

Figure 1: Dynamic change of distributions of



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