2013 Sixth International Conference on Advanced Computational Intelligence October 19-21,2013, Hangzhou, China

Differential Evolution Based on Population Reduction with

Minimum Distance

Ming Yang, Jing Guan*, Zhihua Cai, and Changhe Li

Abstract- In Differential Evolution (DE), there are many adaptive DE algorithms proposed for parameter adaptation. However, they mainly focus on tuning the mutation factor F and the crossover probability CR. The adaptation of population size N P has not been widely studied in the literature of DE. Reducing population size without jeopardizing the performance of an algorithm could save computational resources and hence accelerate it's convergence speed. This is beneficial to algorithms for optimization problems which need expensive evaluations. In this paper, we propose an improved population reduction method for DE, called dynNPMinD-DE, by considering the difference between individuals. When the reduction criterion is satisfied, dynNPMinD-DE selects the best individual and pairs of individuals with minimal-step difference vectors to form a new population. dynNPMinD-DE is tested on a set of 13 scalable benchmark functions in the number of dimensions of D=30 and D=50, respectively. The results show that dynNPMinD­DE outperforms the other peer DE algorithms in terms of both solution accuracy and convergence speed on most test functions.

I. INTRODUCTION

Differential Evolution (DE), introduced by Price and Storn [1], is a simple yet powerful evolutionary algorithm (EA) for global optimization. Nowadays, DE has become one of the most frequently used EAs for solving the global optimiza­tion problems [2], mainly because it has good convergence properties and it is principally easy to understand.

DE creates new candidate solutions by combining the parent individual and different information between several other individuals of the same population. DE has three control parameters: amplification factor of the difference vector-F, crossover control parameter-CR, and popula­tion size-N P. The control parameters involved in DE are highly dependent on the problems to be solved, and the original DE algorithm keeps all the three control parameters fixed during the optimization process [3]. For a specific task, it may have to spend a huge amount of time to try and fine­tune the corresponding parameters.

For adapting control parameters F and C R, some adaptive and self-adaptive DE algorithms were developed to solve general problems more efficiently, such as SaDE [4], FADE [5], DESAP [6], jDE [7], jDE-2 [8] and JADE [9]. Although

Ming Yang, Zhihua Cai, and Changhe Li are with the School of Computer Science, China University of Geosciences, Wuhan, China (e-mail: yangming0702@gmail.com, zhcai@cug.edu.cn, changhe.lw@gmail.com).

ling Guan is with the China Ship Development and Design Center, Wuhan, China (email: g_jing0414@163.com).

The work was supported by the fund of the National Natural Science Foundation of China (No.61305086, No.61075063) and the Special Fund for Basic Scientific Research of Central Colleges, China University of Geosciences (Wuhan) (No.CUGLl00230 & No.G1323521289).

* Author for correspondence.

978-1-4673-6343-3/13/$31.00 ©2013 IEEE 96

adaptive DE algorithms have been proposed, they mainly focus on tuning the mutation factors F and crossover prob­abilities CR. The adaptation of population size parameter N P has not been widely studied in the scope of DE. If an algorithm decreases the number of evaluations for individuals by reducing population size, but not jeopardizes its own performance, then it could save computational resources to solve the optimization problems which need expensive evaluations. This paper is to study the effect of population reduction on the classic DE/rand/1/bin and develop an im­proved population reduction method.

In the DESAP algorithm [6], population size evolves with individuals in the evolution process and it may increase or decrease. If population size decreases, DESAP selects the best new-size individuals to form a new population; if population size increases, all current individuals are chosen into the new population as well as the remaining individuals by cloning the best individual from the current population. This new population generation method is also used in [10] and the population size is adjusted according to the population diversity. In [11], Brest et al. proposed a novel population size reduction method based on jDE [7], called dynNP-DE. In the beginning of evolutionary process, the population size is large, and later it decreases by half at each predefined control point. The individuals of new half population are better ones, based on their fitness values, of individuals from the first half of the current population and corresponding individuals from the second half. At the beginning of the evolutionary optimization process, the population size is large and its diversity is high. Then the population size gradually descends with evolution process, and hence, the algorithm can spend much more time to improve the best individual using the saved resources.

In this paper, we will further and systematically study the effect of population reduction on DE/rand/llbin based on the works in [11], [12].

II. DE WITH AN IMPROVED POPULATION SIZE

REDUCTION METHOD

In this section, we will introduce an improved popu­lation reduction method for DE, called dynNPMinD-DE. In dynNPMinD-DE, the population size adaptation strategy adopts the strategy in [11], which defined the time to decrease population. When population reduction criterion is met, dynNPMinD-DE employs the proposed population reduction method in this paper.

A. Population Size Adaptation Strategy

Suppose that N Pp (p = 0, 1,2, ... ,pmax) is the pop­ulation size at reduction point p where pmax denotes the times of reductions need to be performed except N Po, which is the initial population size. genp denotes the number of generations with population size N Pp, which is calculated as:

-l maxnfeval J genp -

NP + rp, pmax· p

(1)

where Ip ;::: 0 is a small non-negative integer value, which is greater than zero when maxnfeval is not divisable by pmax. The reduction is performed in the following genera-tions:

GR E {genl,genl + gen2, ... , P:�-

l

genp} . (2)

At each reduction, dynNPMinD-DE reduces population size by half.

TABLE I

TYPICAL RUN, WHEN MAXNFEVAL=100000 AND PMAX=4

P 1 2 3 4

NPp 200 100 50 25

genp 125 250 500 1000

NPp x genp 25000 25000 25000 25000

It should be noticed that the population size must be at least 4 for the original DE/rand/1/bin. Table I illustrates the case of each reduction point with maxnfeval=lOOOOO and pmax=4.

B. An Improved Population Reduction Method

In DE/rand/l/bin strategy, for each target vector Xi,G, a mutation vector Vi is generated according to

with randomly chosen indexes '1, r2, r3 E [1, N P], '1 i­

r2 i- r3 i- i. F E [0,2] and is a real number, which controls the amplification of the difference vector (xr2 ,G - xr3 ,G) .

The dynNP-DE algorithm selects better individuals into new population. This selection operation does not consider the steps of difference vectors. However, the steps of differ­ence vectors between individuals may not be suitable after selection. Individuals may not learn useful information from difference vectors, so algorithms will be likely to stagnate. In order to make individuals learn useful information from difference vectors and alleviate the stagnation problem for DE algorithms, in this section, we propose an improved selection method, considering the steps of difference vector (xr2,G - xr3 ,G) in mutation, and embed it into classic DE/rand/1/bin strategy. When population reduction takes place, we select the individuals with minimal-step difference vectors into the new population. In order to guarantee the algorithm to keep getting better results, we also select the best individual into the new population. We call this variant DE as dynNPMinD-DE algorithm.

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Algorithm 1 An improved population reduction method 1: Compute dij,C for each pair individuals:

o dij,c = L I Xi,k,C - Xj,k,C I ,

k=l

where X�i,G, Xj,G E Pc and i < j;

(3)

2: PC+l = {Xbest,G}, where Xbest,Cis the best individual; 3: while I PC+ll < N PC+l do 4: Find individuals X= and Xn with the minimal dij,c, in the condition that

Xm) X'n E Pc, but Xm and Xn tJ. PC+l simultaneously; 5: 6: 7: 8: 9:

10:

if Xm � PC+l then PC+l = PC+l U{x=};

end if if IpC+11 < N PC+1 and Xn � PC+1 then

PC+l = PC+l U{xn};

end if 11: end while

• X2

• Xs

Fig, 1. Illustration of Algorithm 1, where Xl is the best individual and d4,5 and d5,7 are the minimum and the second-minimum distances, respectively,

Algorithm 1 illustrates the population reduction procedure with minimum step difference in the selection operation. PG denotes the population set at the G-th generation, PG =

{XI,G, X2,G, . . . , XNPc,G}. In Algorithm 1, firstly, the best individual is selected into the new population in order to make the algorithm evolve forward. Then the population re­duction method chooses a pair individuals with the minimum distance, the second-minimum distance, and so on, till half­population individuals are selected. As a result, individuals that are close to each other will survive after each reduction. Therefore, the mutation step at average level will decrease with the evolution process. It should be noticed that if an individual exists in the new population, it will not be selected into the new population again.

Fig. 1 illustrates how to select individuals to form a new population when population reduction criterion is met. In Fig. 1, there are eight individuals in the current population, PG = {Xl, X2, ... , xs}. The new population size will be N PG+l = 4. That is, we select four individuals from PG to form the new population PG+I. Firstly, we select the best individual {xd into PG+I, then PG+l = {xd. Individuals X4 and X5 are also selected into the new population, be­cause the minimum distance is d4,5. After that, the second­minimum distance is d5,7, so individuals X5 and X7 will be considered. Due to X5 already exists in PG+I, X5 can not be selected again. Only X7 is selected into PG+l ' After this step, Algorithm 1 will terminate as I PG+l I = N PG+I = 4. Eventually, the new population is PG+I = {XI,X4,X5,X7}.

Algorithm 2 dynNPMinD-DE algorithm 1: Generate uniform randomly the initial population Po;

2: Evaluate the fitness for each individual in Po;

3: G = 0; 4: while the stop criterion is not satisfied do 5: for each individual Xi E Pc do 6: Select uniform randomly Tl # T2 # T3 # i; 7: jrand=rndint(I,D); 8: for j = 1 to D do 9: if rnd(O, 1) < C R or j == j.rand then

10: Ui,j = Xr1,j + F , (X'r2,j - Xr3 ,j); 11: else 12: Ui,j = Xi,j; 13: end if 14: end for 15: if Ui,j exceeds boundary constraints then 16: { 2· Xlow,j - Ui,j Ui,j = 2. Xupp,j - Ui,j

if Ui,j < Xlow,j if Ui,j > Xupp,j

where Xlow,j and Xupp,j are respectively the predefined lower and upper bounds;

17: end if 18: Evaluate the offspring Ui; 19: end for 20: for each individual Xi E Pc do 21: if Ui is not worse than Xi then 22: Xi = Ui; 23: end if 24: end for 25: if G = G R then 26: Reduce the population using Algorithm I and generate new population

PC+l;

27: end if 28: G = G + 1; 29: end while

Fig, 2 shows an simulation of population reduction process for the 2-dimensional Schwefel function using Algorithm L From the graphs in Figure 2, it can be seen that individuals far away from each other are removed in the old population. Only individuals that are close enough will survive into next generation. As a result, the mutation step will decrease at average leveL This is beneficial for individuals to focus on exploitation in a local area, where an optimum is possibly located, due to small mutation step, That is, reducing mu­tation step would accelerated the exploitation progress and hence speed up convergence for the algorithm. This can be seen in Figure 2(b), 2( c) and 2( d). The individuals evolves with small difference vectors and exploit the global optimum in the late period.

The pseudo-code of dynNPMinD-DE is presented in Algo­rithm 2. The dynNPMinD-DE algorithm uses the population reduction method in Algorithm 1. For the initial population, it should be to set to be a large value in order to achieve large population diversity in the beginning of evolutionary process. After some generations, when population undergoes reduction process, the population is reduced by half using Algorithm 1. The best individual and individuals with mini­mum step difference vectors are chosen to form a new popu­lation. Then the algorithm is able to decrease the number of evaluations of individuals and focus on exploitation progress. These can speed up the convergence of the algorithm.

Compared with original DE/randlllbin algorithm, dynN­PMinD-DE only needs to perform additional population reduction procedure in O(N P�). However, the population reduction is only performed for several times (predefined

98

constant in our present work) in the whole run rather than every generation. Compared with dynNP-DE, for each reduc­tion, dynNPMinD-DE performs distance comparison among individuals in O(N P'J;), which is higher than the time cost by dynNP-DE.

III. EXPERIMENTAL STUDY

In this section, dynNPMinD-DE is applied to minimize a set of l3 scalable benchmark functions in dimensions D=30 and D=50, repectively. These functions are Sphere(h),

Schwefel's Problem 2.22(12), Schwefel's Problem 1. 2(13),

Schwefel's Problem 2.21(f4)' Rosenbrock(f5), Step(f6), Quartic(17), Schwefel's Problem 2. 26(fs), Rastrigin(fg),

Ackley(ho), Griewank(fu), Penalized Function1(fd and Penalized Function2(f13)' A more detailed description of all functions is given in [l3]. h-14 are continuous unimodal functions. 15 is the Rosenbrock function which is unimodal for D=2 and D=3 but may have multiple minima in high dimension cases [14]. 16 is a discontinuous step function, and 17 is a noisy quartic function. Is-h3 are multimodal functions and the number of local minima increases exponen­tially with the problem dimension [13]. For Is, when D=30, Imin=-12569.5, and when D=50, Imin=-20949.L And the global optimal function values are 0 for the other functions in the paper.

dynNPMinD-DE is compared with the classic DE/rand­Illbin to study the effect of dynamic poplualtion on DE. It is also compared with dynNP-DE [11] to test the effectiveness of the proposed population reduction method.

Parameters F and CR were set to be 0.5 and 0.9, respec­tively, which were recommended in [1], and N P=200 for DE/randlllbin. For dynNP-DE and dynNPMinD-DE, the par­ticular parameters were set to be pmax=4 and N Pinit=200 as recommended in [11]. The algorithm stop criterion is the number of function evaluations which are pre-defined (see Eq.(l)). The number of function evaluations for each test function is shown in Table II in the two dimensional cases.

A. Experimental Results

Table II summarizes the average mean and standard de­viation results of 50 independent runs for each algorithm on functions h -h3 with D=30 and D=50, respectively. For each function, the corresponding best results of the three peers algorithms are shown in bold font. Fig. 3 shows the convergence graphs of function Is in 30 and 50 dimensions. The convergence graphs of the other functions are not provided due to space limitation in this paper. In Fig. 3, the curves record the mean error E of the best fitness 1 (x) to the function value of the global optimum over 50 independent runs:

where N is the run times, whose value was 50 in the paper and j is the number of fitness evalutations.

The comparisons in Table II show that dynNPMinD­DE performs significantly better than DE/rand/l/bin for all

TABLE II

AVERAGE RESULTS OF DYNNPMIND-DE, DYNNP-DE, AND DE/RANDIl/BIN ON ALL FUNCTIONS IN 30 AND 50 DIMENSIONS OVER 50 INDEPENDENT

RUNS, WHERE "MEAN" AND "STD DEV" INDICATE THE MEAN BEST RESULTS AND CORRESPONDING STANDARD DEVIATION VALUES.

D-30 D-50

F Func. dynNPMinD-DE d.vnNP-DE DEirand/llbill Fum:. d.vnNPMiIlD-DE dynNP-DE DEirandillbin

Eva/. Mean (Sid Dev) Mean (Sid Dev) Mean (Sid Dev) Eva/. Mean (Sid Dev) Mean (SId Dev) Mean (SId Dev) /1 100000 1.727E·OIO (4.399E·OIO) 2. I 89E-01O (7.783E-01O) 1.720E+OOO (5.58IE-OOI)j 200000 3.531E-0I2 (6.540E-OI2) 2.430E-011 (6.966E-OII) 4.705E-OOI (1.633E-OOI)t h 150000 9.373E-OIO (2.833E-009) 9.767E-OIO (2.317E-009) 2.315E-OOI (7.443E-002)j 200000 3.980E-008 (6.S14E-008) 1.214E-007 (3.487E-007) 1.860E+OOO (4.987E-OOI)j h 300000 1.872E-004 (9.962E-004) 2.846E-004 (8.454E-004) 3.60 I E+OO I (1.309E+OO l)j 800000 3.379E-003 (7.722E-003) 3.63IE-003 (9.421 E-003) 3.986E+OO2 (1.310E+OO2)j f, 100000 1.904E+OOO (I.338E+OOO) 1.95 I E+OOO (I. I 29E+OOO) 9.072E+OOO (1.305E+OOO)t 100000 1.304E+OOI (3.083E+OOO) 1.337E+OOI (3.709E+OOO) 3.284E+OOI (3.068E+OOO)j fs 30000 7.738E+OOI (6.219E+OOI) 8.690E+OOI (1.302E+OO2) 5.882E+OO5 (1.906E+OO5)t 100000 1.013E+OO2 (1.885E+OO2) 7.406E+OOI (3.7S8E+OOI) 1.760E+OO4 (7.I44E+OO3)j f6 50000 O.OOOE+OOO (O.OOOE+OOO) O.OOOE+OOO (O.OOOE+OOO) 1.923E+OO2 (4.955E+OOI)t 200000 O.OOOE+OOO (O.OOOE+OOO) O.OOOE+OOO (O.OOOE+OOO) O.OOOE+OOO (O.OOOE+OOO) h I()()(){)(){)() 3.436E-OOS (I.32SE-OOS) 3.640E-005 (1.355E-005) 3.414E-004 (9.827E-005)j I()()(){)(){)() 8.073E-OOS (3.028E-OOS) 8.456E-005 (2.637E-005) 8.250E-004 (1.935E-004)t f8 200000 -1.189E+OO4 (3.603E+OO2) -1.1 58E+OO4 (5.446E+OO2lt -5.460E+OO3 (2.779E+OO2lt 250000 -1.843E+004 (7.S28E+OO2) -1.753E+004 (8.082E+OO2)t -7.000E+OO3 (2.744E+002)t h 200000 2.I90E+OOI (6.986E+OOO) 2.222E+OOI (7.432E+OOO) 1.92IE+OO2 (9.147E+OOO)t 200000 4.466E+OOI (1.396E+OOI) 4.592E+OOI (1.216E+OOI) 3.913E+OO2 (1.53IE+OOI)j /10 200000 9.784E-0I3 (1.900E-OI2) 2.358E-OI2 (3.976E-0I2) 2.994E-003 (5.99IE-004)j 200000 4.742E-007 (1.0S4E-006) 5.554E-007 (8.030E-007) 2.805E-OOI (7.194E-002)j /11 70000 1.973E-003 (4.252E-003) 1.580E-003 (3.S47E-003) 1.324E+OOO (7.948E-002)j 200000 7.684E-OI2 (1.789E-OII) 4.923E-012 (1.296E-OI\) 4.477E-OOI (1.007E-OOI)t 112 150000 5.132E-OI7 (4.859E-OI7) 3.828E-OI7 (1.844E-OI7) 2.553E-003 (1.656E-003)j 200000 4.729E-OI3 (1.134E-012) 7.720E-OI3 (2.21IE-OI2) 9. I 62E-002 (5.670E-002)t /13 150000 1.791E-016 (3.989E-OI6) 8.994E-OI7 (1.36SE-OI6) 1.652E-002 (1.089E-002)j 200000 4.792E-004 (2.18SE-003) 6.592E-004 (2.636E-003) 4.772E-OOI (1.76IE-OOI)t t dynNPMmD-DE pertorms slgmficantly belief than the algonthm al a 0.05 level of slgmficance by palfcd samples Wilcoxon sIgned rank test : dynNPMinD-DE performs significantly worse than the algorithm al a 0.05 level of significance by paired samples Wilcoxon signed rank test

I ) g

, � O�

-200 -100 0 lOa Xl

(a) population (200 individuals) before the I-st reduction

600

400

200

-200

-400

-600

(c) population (100 individuals) before the 2-nd reduction at the l7-ve

generation

600

400

200

-200

-400

-600

(b) population (100 individuals) after the I-st reduction at the 6-th

generation

600

400

200

-200

-400

-600

(d) population (50 individuals) after the 2-nd reduction at the 18-th

generation

Fig. 2. An example of population reductions by Algorithm 1 for the Schwefel function in 2-dimension where the black points are individuals in the fitness landscape.

functions. And dynNPMinD-DE also achieves better results than dynNP-DE on most functions. Especially, dynNPMinD­DE performs significantly better than dynNP-DE on is in both D=30 and D=50 dimensions. In addition, there is no functions where dynNPMinD-DE performs significantly worse than dynNP-DE. From the results, it can be seen that

99

dynNPMinD-DE is able to solve not only low dimensional problems but also higher dimensional problems, e.g., 50 dimensions.

From Fig. 3, it can be seen that the convergence of both dynNPMinD-DE and dynNP-DE become faster and faster than DE/rand/1/bin with each population reduction.

10'

0.5 Function Evaluations

(a) !s, D = 30

_______ DE/rand/l/bin ---&- dynNP-DE -- dynNPMinD-DE

1.5

lO

',----,----------.----7---+-="'D"'ElO',,="""'I1"'''''=, =il

---6-dynNP-DE -- dynNPMinD-DE

1030"--------::0-=.62=-5 -----:-1.=25:-------,'-= .• =75------='2 .5

Function Evaluations x 105

(b) Is, D = 50

Fig. 3. Convergence graph for function /s in dimensions D = 30 and D = 50. The horizontal axis is the number of function evaluations, and the vertical axis is the mean error of best fitness over 50 independent runs. The dotted vertical lines are the moments where population reduction is performed.

> Cl

!

-1O',----

-,-------

---.----7=�c==c===il

I ----+---- DEJrand/.l/bin

---6-dynNP-DE - dynNPMlnD-DE

_105 0 0.5 , 1.5

Function Evaluations Xl05

(a) best fitness

10'

10'

10°

10-2

10-4 � DEJrand/1/bin

---&- dynNP-DE

-- dynNPMinD-DE

10-6

0 0.5 1.5 Function Evaluations

(C) AvgDV x 10

5

10' ,--------.-----,------,-----,

10-5

0 0.5

0.7

0.6

0.5

0.4

"' '"

0.3

0.2

0.'

0.5

______ DE/rand/1Jbin

---&- dynNP-DE

-- dynNPMinD-DE

Function Evaluations

(b) AvgDist

, Function Evaluations

(d) SR

1.5

----+---- DE/rand/libin ---e--- dynNP-DE -- dynNPMinD-DE

Fig. 4. One run results on function /s (D = 30). The horizontal axis is the number of function evaluations, and the vertical axes are (a) best fitness, (b) AvgDist, (c) AvgDV, and (d) SR, respectively. The dotted vertical lines are the moments where population reduction is performed.

Interestingly, the convergence of dynNPMinD-DE is even faster than dynNP-DE on is (D=30 and D=50). This is because dynNPMinD-DE is able to spend much more energy to improve the best individual than dynNP-DE when the population size becomes smaller as evolution progresses.

The results of Table II and Fig. 3 show that our idea works by considering the distance between individuals when performing population reduction procedure.

100

B. The Analysis of Experimental Results

In this section, we will analysis the effect of the pop­ulation reduction on DE/rand/llbin and give the explana­tions why dynNPMinD-DE is better than dynNP-DE and DE/rand/llbin.

We define AvgDist as the average distance between individuals of population Pc:

. 2 """' AvgDzst = NPc' (NPc - 1) � dij,C' t<J

where Xi,C, Xj,C E Pc, and AvgDV as the average step of difference vectors involved in mutation (step 10 in Algorithm 2):

n

AvgDV = � L I X�2,j - X�3,j l , i=l

where n is the times of performing mutation in one genera­tion. In a generation, the success rate of trial vectors entering the next generation is SR:

ns SR= NPc' where ns is number of trial vectors successfully entering the next generation.

In Fig. 4, compared with the other two algorithms, dynNPMinD-DE can obtain smaller value of AvgDist after the second population reduction, which results in smaller value of AvgDV (step of difference vectors) in mutation. The success rate S R also gets higher when step of difference vectors in mutation gets smaller. Therefore, dynNPMinD­DE is able to generate much better trial vectors and get better fitness than other algorithms. It also validates that small step of difference vectors in mutation is beneficial to generating better trial vectors in the late of evolution. If an algorithm makes steps of difference vectors smaller as evolution progress by population reduction, it can reduce the number of evaluations for individuals and accelerate the algorithm's convergence speed.

IV. CONCLUSIONS

Reduction population size without jeopardizing an algo­rithm's performance significantly, it could reduce the number of evaluations for individuals and accelerate the algorithm's convergence speed. This is beneficial to solve the optimiza­tion problems which need expensive evaluations.

In the late of evolution, small step of difference vectors in mutation is helpful to generate good trial vectors. Based on this motivation, this paper proposes an dynNPMinD-DE algorithm based on population reduction. In dynNPMinD­DE, it will select the best individual and the individuals with minimal-step difference vectors to form a new population when population reduction takes place. dynNPMinD-DE was tested on a set of 13 scalable benchmark functions in dimen­sions D=30 and D=50 in this paper. From the results, two conclusions can be drawn. First, dynNPMinD-DE is able to obtain good trial vectors by considering the distance between individuals; Second, the performance of dynNPMinD-DE is

much better than the other two peer algorithms in terms of both solution accuracy and convergence speed. on the problems tested in the paper.

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