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Differential evolution algorithm with ensemble of populations for global numerical optimization

OPSEARCH 46(2):184–213

R. Mallipeddi, P. N. Suganthan

School of Electrical and Electronic Engineering Nanyang Technological University, Singapore 639798

mallipeddi@https://www.doczj.com/doc/e61544147.html,.sg, epnsugan@https://www.doczj.com/doc/e61544147.html,.sg

Abstract

Differential evolution (DE) is an ef ? cient and powerful population-based stochastic search technique for solving global optimization problems over continuous space, which has been widely applied in many scienti ? c and engineering ? elds. However, the success of DE to handle a speci ? c problem crucially depends on the proper choice of various parameters including the size of the population. Employing the trial and error scheme to search for the most suitable parameter settings requires high computational costs. In this paper, we propose a DE algorithm with an ensemble of parallel populations in which the number of function evaluations allocated to each population is self-adapted by learning from their previous experiences in generating superior solutions. Consequently, a more suitable population size takes most of the function evaluations adaptively to match different phases of the search process/evolution. Although the evolutionary algorithms have been investigated for about ? ve decades, to our best of knowledge so far no effective population adaptation scheme has been proposed. The performance of the DE algorithm with an ensemble of parallel populations is extensively evaluated on a suite of 14 bound-constrained numerical optimization problems and compares favorably with the conventional DE with different single population sizes.

Keywords

Differential evolution, Global numerical optimization, Self-adaptation, Ensemble of populations

1. Introduction

Differential evolution (DE) algorithm, proposed by Storn and Price [1], has been sucessfully applied in diverse ? elds such as mechanical engineering [2], [3], communication [4] and pattern recognition [5]. In DE, there exist

Paper accepted on February 6, 2009 ?

Operational Research Society of India

OPSEARCH 46(2):184–213185 many trial vector generation strategies out of which a few may be suitable for solving a particular problem. Moreover, three crucial control parameters involved in DE, i.e. population size NP, scaling factor F and crossover rate CR, may signi? cantly in? uence the optimization performance of the DE. Therefore, to successfully solve a speci? c optimization problem at hand, it is generally required to perform a time-consuming trial-and-error search for the most appropriate population size, strategy and to tune its associated parameter values. However, such a trial-and-error searching process requires high computational costs. Morover, as evolution proceeds, the population of DE may cover different regions in the search space, within which certain population size associated with speci? c strategy and parameter settings may be more effective than others. Therefore, it is desirable to adaptively determine an appropirate population size, strategy and its associated parameter values at different stages of evolution/search process. In this report, we propose a DE algorithm with an ensemble of parallel populations to avoid the trial-and-error procedure and the number of function evaluations given to a population size are gradually self-adapted by learning from their previous experiences in generating promising solutions. Consequently, a more suitable population size along with its parameter settings can be determined adaptively to match different search / evolution phases. Speci? cally, at each generation, the number of function evaluations given to a speci? c population size depends on its success in the previous generations. Although the evolutionary algorithms have been investigated for about ? ve decades, to our best of knowledge so far no effective population adaptation scheme has been proposed.

The remainder of this document is organized as follows. The conventional DE and related work are reviewed in Sections 2 and 3, respectively. Section 4 describes the proposed DE algorithm with an ensemble of parallel populations. Experimental results to demonstrate the performance of DE algorithm with an ensemble of parallel populations in comparison with the conventional single population DE over a suite of 14 bound constrained numerical optimization problems are presented in Section 5. Section 6 concludes the document. 2. The DE algorithm

Differential Evolution (DE) is a parallel direct search method which utilizes NP D-dimensional parameter vectors, so-called individuals, which encode the candidate solutions, i.e. . The initial population should cover the entire search space as much as possible by uniformly randomizing the initial individuals within the search space constrained by the prescribed minimum and maximum parameter bounds

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and For example, the initial value of the j th parameter in the i th individual at the generation G=0 is generated by:

(1) where rand (0,1) represents a uniformly distributed random variable within the range [0,1].

2.1 Mutation operation

After initialization, DE employs the mutation operation to produce a mutant

vector V

i,G with respect to each individual X

i,G

, so-called target vector, in

the current population. For each target vector X

i,G at the generation G, its

associated mutant vector can be generated via certain mutation strategy. The most commonly used mutation strategies are

“DE/rand/1”: (2)“DE/best/1”: (3)“DE/current-to-best/ 2”: (4)“DE/best/2”: (5)“DE/rand/2”: (6)“DE/rand-to-best/2”: (7) The indices are mutually exclusive integers randomly generated within the range [1, NP], which are also different from the index i. These indices are randomly generated once for each mutant vector. The scaling factor F is a positive control parameter for scaling the difference

vector. X

best,G is the best individual vector with the best ? tness value in the

population at generation G.

2.2 Crossover operation

After the mutation phase, crossover operation is applied to each pair of the

target vector X

i,G and its corresponding mutant vector V

i,G

to generate a trial

vector: . In the basic version, DE employs the binomial (uniform) crossover de? ned as follows:

(8) In equation (8), the crossover rate CR is a user-speci? ed constant within the range [0,1), which controls the fraction of parameter values copied from the

OPSEARCH 46(2):184–213187 mutant vector. is a randomly chosen integer in the range [1, D]. The binomial crossover operator copies the j th parameter of the mutant vector

i,G to the corresponding element in the trial vector U

i,G

if or

j = jrand. Otherwise it is copied from the corresponding target vector X

i,G

2.3 Selection operation

The objective function value of each trial vector is compared to that of its corresponding target vector in the current population. If the trial vector has less or equal objective function value than the corresponding target vector, the trial vector will replace the target vector and enter the population of the next generation. Otherwise, the target vector will remain in the population for the next generation. The selection operation can be expressed as follows:

(9) The above 3 steps are repeated generation after generation until some speci? c termination criteria are satis? ed.

The algorithmic description of DE is summarized in Table 1.

Table 1: The standard DE algorithm

Step 1 Set the generation Counter , and randomly initialize 3 populations of NP individuals with

uniformly distributed in the range , where and

Step 2 WHILE stopping criterion is not satis? ed

Step

2.1

Mutation step

/*Generate a mutated vector for each target vector */

FOR

i = 1 to NP

Generate a mutated vector corresponding to the

target

vector via one of the equations (2)-(6).

END

FOR

Step 2.2 Crossover step

/*Generate a trial vector for each target vector */

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a) Binomial crossover

FOR

i = 1 to NP

FOR j = 1to D

END

FOR

END

FOR

b) Exponential crossover

FOR

i = 1 to NP

, L=0, U

i,G = X

i,G

, , L=L+1

WHILE

( rand[0,1)

END

FOR

Step

2.3

Selection step

/*Selection*/

FOR

i =1 to NP

Evaluate the trial vector

IF ≤, THEN ,

IF , THEN ,

END

FOR

Step 2.4 Increment the generation count G = G + 1

Step 3 END WHILE

*Acute brackets

3. Brief literature review on DE

The performance of the conventional DE algorithm highly depends on the population size, the chosen trial vector generation strategy and associated parameter values used. Inappropriate choice of strategies and parameters may lead to premature convergence or stagnation, which have been extensively demonstrated in [6–10]. In the past decade, DE researchers have suggested many empirical guidelines for choosing trial vector generation strategies and their associated control parameter settings. Storn and Price [11] suggested that a reasonable value for NP should be between 5D and 10D, and a good initial choice of F was 0.5. The effective range of F values was suggested between 0.4 and 1. The ? rst reasonable attempt of choosing CR value can be

OPSEARCH 46(2):184–213189 0.1. However, since the large CR value can speed up convergence, the value of 0.9 for CR may also be a good initial choice if the problem is near uni-modal or fast convergence is desired.

It was recommended in [12] to use the trial vector generation strategy DE/ current-to-rand/1 and parameter settting . If the DE converges prematurely, one should increase the value of NP and F or decrease the value of K. If the population stagnates, one should increase the value of NP or F, or randomly choose K within the range [0, 1]. If none of the above con? guration works, one may try the strategy DE/rand/1/bin along with a small CR value. In [8] the experimental results showed that the searching capability and convergence speed are very sensitive to the choice of control parameters NP, F and CR. They recommended that the population size NP is between 3D and 8D, the scaling factor F equals 0.6 and the crossover rate CR is between [0.3, 0.9].

In [13] it was suggested to use F values between [0.4, 0.95] with F = 0.9 being a good initial choice. The CR values should lie in [0, 0.2] when the function is separable while in [0.9, 1] when the function’s parameters are dependent. However, when solving a real engineering problem, the characteristics of the problem are usually unknown. Consequently, it is dif? cult to choose the most appropriate CR value in advance.

Some adaptation techniques to avoid manual tuning of the control parameters are summarized in [14]. This document proposes a novel approach to adapting the DE population size by employing an ensemble of populations.

4. DE algorithm with ensemble of populations

The trail and error search in conventional DE obviously requires a huge amount of computational costs. Moreover, during different stages of evolution, different population sizes coupled with speci? c trial vector generation strategy and control parameter values can be more effective than others. Motivated by these observations, we develop a DE algorithm with ensemble of populations, in which the number of function evaluations given to a speci? c population size is self-adapted according to their previous performance generating superior solutions. The DE algorithm with ensemble of populations is described below.

4.1 Selection of mutation strategy and control parameters

The proposed DE algorithm with ensemble of populations is tested with the following two mutation strategies:

190

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The ‘DE/best/1/bin’ strategy is a degenerated case of the “DE/rand-to-best/1/bin” strategy with F equal to 1 and relies on the best solution found so far. This strategy usually has fast convergence speed and performs well when solving unimodal problems and may lead to a premature convergence when solving multimodal problems.

‘DE/rand-to-best/2/bin’ is a two difference vectors based strategy and ●

enables the algorithm to solve rotated problems more effectively. Two difference vectors based strategies may result in better perturbation than one difference vector based strategies.

The two trial vector generation strategies used are as follows.

DE/best/1/bin:

DE/rand-to-best/2/bin:

Regarding the other two parameters, CR is sensitive to problems with different characteristics such as the uni-modality and multi-modality.

F is closely related to the convergence speed. Based on the discussions in Section 3, the values of CR and F are choosen to be 0.3 and 0.5.

4.2 Population adaptation

The effect of population size parameter on the performance of DE has not been tackled and is taken to be constant value depending on the dimension of the problem. If the population size is small, the algorithm may converge fast; but the probability of premature convergence and stagnation are higher. The stagnation occurs virtually without any obvious reasons. When stagnation occurs the population remains diverse, but the optimization process does not progress anymore [7]. In premature convergence, the population loses its diversity. Instead, taking a large population requires more computational effort. It was proposed in [13] that separable and unimodal functions require the smallest population sizes, while parameter-linked, multi-modal problems require the largest populations.

OPSEARCH 46(2):184–213191 In the proposed ensemble of populations algorithm, 3 different population sizes are employed which are 2D, 4D and 8D where D is dimensionality of the problem. The off-springs of one population not only compete with their parents but also with the nearest member in the other populations. The number of function evaluations given to each of these populations depends on the success of the population in the previous generation. The success of the population is measured based on the following two criteria:

●

Number of replacements made by the offspring produced by the population.

●

Improvement in the best function value

50% of the function values are divided based on the ? rst criteria, while the remaining 50% are split based on the second criteria among the three populations. In every generation even if the population is unsuccessful, 5% of the function evaluations are given to the population so that its performance can be estimated in the next generation. The algorithmic description of the DE with population ensemble is presented in Table 2.

Table 2: The standard DE algorithm

Step 1 Set the generation number , and randomly initialize a population of NP (p1=2D, p2= 4D, p3=8D) individuals with

uniformly distributed in the range , where and

Step 2 Evaluate each of the population and ? nd the best individual and its function value. Initialize the number of function evaluations (Np1, Np2, Np3) to be given to each of the populations (p1, p2, p3) in the next generation to be equal to the size of the maximum population, i.e. p3.

Step 3 WHILE stopping criterion is not satis? ed

Xp1 = Best value so far of population size p1

Xp2= Best value so far of population size p2

Xp3 = Best value so far of population size p3

Rp1 = 0; % To count the number of successful offspring produced by population p1 Rp2 = 0; % To count the number of successful offspring produced by population p2 Rp3 = 0; % To count the number of successful offspring produced by population p3 FOR NP = [p1, p2, p3]

R = Rp1 or Rp2 or Rp3 (based on NP)

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Step

3.1

Mutation step

/*Generate a mutated vector for each target vector */

FOR

i = 1 to NP

Generate a mutated vector corresponding to the

target

vector via one of the equations (2)-(6).

END

FOR

Step 3.2 Crossover step

/*Generate a trial vector for each target vector */

a) Binomial crossover

FOR i = 1 to NP

FOR j = 1to D

END

FOR

END

FOR

b) Exponential crossover

FOR

i = 1 to NP

, L=0, U

i,G = X

i,G

, , L=L+1

WHILE

( rand[0,1)

END

FOR

Step

3.3

Selection step

/*Selection*/

FOR

i =1 to NP

Evaluate the trial vector

IF ≤, THEN , , R=R+1 IF , THEN ,

END

FOR

Step 3.4 Nearest Neighbor Replacement

FOR

NP1 = [p1, p2, p3]

IF NP1 ≠ NP

Find the nearest neighbor to each in the NP1 population.

OPSEARCH 46(2):184–213193 IF , THEN , R=R+1

END

END IF

FOR

END

FOR

END

Yp1 = Best value so far of population size p1

Yp2 = Best value so far of population size p2

Yp3 = Best value so far of population size p3

AIp1 = Yp1 – Xp1; % Improvement in Objective Value due to population p1 AIp2 = Yp2 – Xp2; % Improvement in Objective Value due to population p2 AIp3 = Yp3 – Xp3; % Improvement in Objective Value due to population p3

x1=AIp1/Np1; y1=AIp2/Np2; z1=AIp3/Np3;

c1=(x1+y1+z1) +10^ (-200);

x2=Rp1/Np1; y2=Rp2/Np2; z2=Rp3/Np3;

c2=(x2+y2+z2) +10^ (-200);

P = 5%of (Np1+Np2+Np3);

Q = (Np1+Np2+Np3) – 3*P;

Np1= P +round (Q/c1*x1); % Number of function evaluations for population p1 in next generation

Np2= P +round (Q/c1*y1); % Number of function evaluations for population p2 in next generation

Np3= P +round (Q/c1*z1); % Number of function evaluations for population p3 in next generation

Step 3.5 Increment the generation count G = G+1

Step 4 END WHILE

5. Numerical experiments and results

5.1 Test functions

Among the 14 functions f1 - f4 are unimodal and functions f5 - f 14 are multimodal. These 14 test functions (f1 - f 14) are dimension-wise scalable [15].

1) Shifted Sphere function

2) Shifted Schwefel’s problem 1.2

3) Rosenbrock’s function

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4) Shifted Schwefel’s problem 1.2 with noise in ? tness

5) Shifted Ackley’s function

6) Shifted rotated Ackley’s function

7) Shifted Griewank’s function

8) Shifted rotated Griewank’s function

9) Shifted Rastrigin’s function

10) Shifted rotated Rastrigin’s function

11) Shifted non-continuous Rastrigin’s function

12) Schwefel’s function

13) Composition function 1 (CF1) in [16]

14) Composition function 6 (CF6) in [16]

Table 3: Global optimum, search ranges and initialization ranges of the test

1 is the shifted vector for the ? rst basic function in the Composition function.

OPSEARCH 46(2):184–213195 5.2 Comparison between DE with ensemble of populations and

conventional DE

Experiments were conducted on a suite of 14 numerical functions to evaluate the conventional DE algorithm with different population sizes and the DE with ensemble of populations. For functions f1 - f14, both 10-dimensional (10-D) and 30-dimensional (30-D) functions were tested. The maximum number of function evaluations (FEs) is set to 200,000 when solving 10-D problems, and 600,000 when solving 30-D counterpart. All experiments were run 25 times, independently. For the study the following population sizes are used:

Concentional DE with NP=2D, 4D, 6D, 8D, 10D

●

Multi-population with 3 different population sizes (2D, 4D, 8D)

●

The 2 strategies used in the simulations are

‘DE/best/1/bin’

●

‘DE/rand-to-best/2/bin’

●

The control parameters are set to F=0.5 and CR=0.3 in our experiments [8,9]. 5.3 Experimental results and discussions

Tables 4–7 report the mean and standard deviation of function values and success rates. The best results are typed in bold. If the algorithm can result in a function value no worse than the pre-speci? ed optimal value of +1e-5 with the number of FEs less than the pre-speci? ed maximum number is considered as success. The success rate is the number of successful runsdivided by the total number of runs. Fig 1 and Fig 2 illustrate the convergence characteristics for some of the test functions in terms of the best ? tness value of the median run of each algorithm for functions f1 - f14 with D=10. In order to compare the convergence speed, we report the average number of function evaluations (NFE) required to ? nd the global optima when an algorithm solves the problem with 100% success rate.

1) Comparing DE with ensemble of populations and conventional DE

For the 10-dimensional problems f

1, f

, f

the single population cases with

conventional DE and the Multi-population DE give 100% success rate. In case

of functions f

3, f

– f

and f

– f

we can observe that the ensemble algorithm

shows improved performance in terms of success rate as well as the mean

and standard deviation of the function values. In case of f

14 the success rate

of multi-population algorithm is less than some of the single population cases, but the improvement in terms of mean and standard deviation of

function values can be observed. In case of f

8 and f

the single population

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196cases as well as the multi-population failed to achieve any success, but the mean and the standard deviation of the Multi-population is comparable to that of the single population cases.

For the 30-dimensional problems f 1, f 5, f 6 the single population cases with conventional DE and the multi-population DE give 100% success rate. In case of functions f 2, f 7, f 8 and f 13 we can observe that the multi-population algorithm shows improved performance in terms of success rate as well as the mean and standard deviation of the function values. In case of f 12 and f 14 the success rate of the multi-population algorithm is less than some of the single population cases, but the mean and the standard deviation values of the function values are comparable to that of the single population cases. In case of functions f 3, f 4, f 9, f 10 and f 11 the single population as well as the multi-population failed to achieve success, but the mean and the standard deviation of the multi-population are improved than the single population cases.In STRATEGY-2 also similar performance can observed as in STRATEGY-1 for 10-dimensional problems and 30-dimensional problems. Instead in case of 30-dimensional problems much better performance is observed with STRATEGY-2. We can observe from the results that in case of problems f 8 and f 9 the multi-population algorithm is able to some success even when none of the single population cases is successful.

In this part, we intend to compare the overall performances of the conventional DE algorithm with various populations and the multi-population DE on all functions by plotting empirical distribution of normalized success performance [32]. The success performance is de ? ned as: Success performance (SP) = mean (FEs for successful runs)*(# of total runs) / (# of successful runs).

We ? rst calculated the success performance of conventional DE algorithm with various populations and the multi-population DE algorithm on each test function, and then normalized the SP by dividing all SP’s by the SP of the best algorithm on the respective function. Results of all functions are used where at least one algorithm was successful at least once. Therefore, for 10-dimensional f 1-f 14 problems, we exclude f 8 and f 10 for STRATEGY-1 and , , and for STRATEGY-2. For 30-dimensional problems, we exclude f 3, f 8 , f 10 and f 14 for STRATEGY-1 and f 3, f 4 , f 10, f 11 and f 14 for STRATEGY-2. Small values of SP and large values of the empirical distribution in graphs are preferable. The ? rst one that reaches the top of the graph will be regarded as the best algorithm.

From Fig. 3 we can observe that the multi-population performs better than conventional DE with different population sizes.

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T a b l e 4(a ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n

f 1. S p h e r e

f 2. S c h w e f e l ’s P r o b l e m 1.2

f 3. R o s e n b r o c k

M e a n S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e M e a n S t d S u c c e s s r a t e

2D 00100%00100%4.921713.28830%4D 00100%00100%2.27991.56670%6D 00100%00100%2.03521.67920%8D 00100%3.8370E -296.3781E -29100%2.47901.63924%10D 00100%

4.7908E -24

8.1329E -24100%2.92791.40670%

M u l t i -p o p u l a t i o n

00100%

00100%1.19481.3939

28%

T a b l e 4(b ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n f 4. S c h w e f e l ’s P r o b l e m 1.2 w i t h n o i s e

f 5. A c k l e y

f 6. R o t a t e d A c k l e y

M e a n S t d

S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

2D 00100%0.04620.231096%0.04620.231096%4D 00100%00100%00100%6D 2.6396E -294.3245E -29100%00100%00100%8D 1.3034E -222.2394E -22100%00100%00100%10d 9.7199E -182.5121E -17100%00100%00100%M u l t i -p o p u l a t i o n

2.7703E -271.3518E -26

100%

00100%

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198

T a b l e 4(c ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n

f 7. G r i e w a n k

f 8. R o t a t e d G r i e w a n k

f 9. R a s t r i

g i n

M e a n S t d

S u c c e s s r a t e M e a n

S t d S u c c e s s r a t e M e a n S t d

S u c c e s s r a t e 2D 0.05340.02964%0.27360.42090%3.02472.35944%4D 0.02800.01750%0.05630.03570%0.55721.077772%6D 0.03410.02594%0.04970.02380%0.55720.708256%8D 0.02510.022220%0.05300.02550%0.07960.275592%10D 0.01610.014520%0.07070.02990%0.03980.199096%M u l t i -p o p u l a t i o n

0.00450.0079

72%0.05940.03360%00100%T a b l e 4(d ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n

f 10. R o t a t e d R a s t r i

g i n

f 11. N o n -c o n t i n u o u s R a s t r i

g i n

f 12. S c h w e f e l

M e a n S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

2D 7.57264.11580%2.72001.54164%163.44543146.341628%4D 6.65582.63790%0.64000.637744%104.225792.503336%6D 6.38902.18920%0.72000.678240%18.950155.971888%8D 7.21552.35220%0.40000.707172%23.687748.352280%10D 6.98831.70290%0.08000.276992%4.737523.687796%M u l t i -p o p u l a t i o n

7.82352.61010%

0.04000.200096%0

100%

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199

T a b l e 4(e ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n f 13. C o m p o s i t i o n F u n c t i o n 1

f 14. C o m p o s i t i o n F u n c t i o n 6

M e a n

S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e

2D 2854.160376%54.933277.34028%4D 2452.281380%24.321443.409460%6D 3255.677672%8.054727.673688%8D 3247.609568%20.275140.688260%10D 5258.594752%16.201637.331568%M u l t i -p o p u l a t i o n

2096%4.286819.958960%T a b l e 5(a ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -2

P o p u l a t i o n

f 1. S p h e r e

F 2. S c h w e f e l ’s P r o b l e m 1.2

f 3. R o s e n b r o c k

M e a n S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

2D 00100%00100%0.12660.39114%4D 00100%8.3391E -151.0907E -15100%0.21120.44320%6D 00100%3.3256E -093.4301E -09100%0.46500.64960%8D 00100%2.2027E -061.3866E -06100%1.01490.82720%10D 00100%1.4431E -047.5235E -030%1.45511.10620%M u l t i -p o p u l a t i o n

100%

3.3140E -106.7241E -10100%0.2255

0.2012

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200

T a b l e 5(b ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -2

P o p u l a t i o n

f 4. S c h w e f e l ’s P r o b l e m 1.2 w i t h n o i s e

f 5. A c k l e y

f 6. R o t a t e d A c k l e y

M e a n S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e M e a n S t d S u c c e s s r a t e 2D 3.3692E -199.9584E -19100%00100%1.4211E -067.1054E -16100%

4D 6.3203E -095.7332E -09100%00100%00100%6D 2.5330E -052.1156E -0516%00100%00100%8D 0.00158.9346E -040%00100%00100%10D 0.02160.01120%00100%00100%M u l t i -p o p u l a t i o n

1.7270E -054.5310E -05

68%

00100%00100%T a b l e 5(c ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -2

P o p u l a t i o n

f 7. G r i e w a n k

f 8. R o t a t e d G r i e w a n k

f 9. R a s t r i

g i n

M e a n S t d

S u c c e s s r a t e M e a n

S t d

S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

2D 00100%0.06290.02160%00100%4D 00100%0.08080.01920%00100%6D 00100%0.10500.02860%00100%8D 00100%0.10790.03160%00100%10D 00100%0.11730.03180%00100%M u l t i -p o p u l a t i o n

00100%

0.10320.02550%0

100%

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201

T a b l e 5(d ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -2

P o p u l a t i o n

f 10. R o t a t e d R a s t r i

g i n

f 11. N o n -c o n t i n u o u s R a s t r i

g i n

f 12. S c h w e f e l

M e a n S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e M e a n S t d S u c c e s s r a t e

2D 9.17642.24100%00100%00100%4D 9.63771.75320%00100%00100%6D 11.53092.33990%00100%00100%8D 12.81662.59010%00100%00100%10D 12.89472.10640%00100%00100%M u l t i -p o p u l a t i o n

10.35602.6553

00100%00100%

T a b l e 5(e ): R e s u l t s f o r 10-D p r o b l e m s w i t h s t r a t e g y -2

P o p u l a t i o n f 13. C o m p o s i t i o n f u n c t i o n 1

f 14. C o m p o s i t i o n f u n c t i o n 6

M e a n

S t d

S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

2D 0.23261.041988%5.619419.69750%4D 9.0500E -124.5250E -11100%1.35480.95740%6D 2.1598E -147.4764E -14100%1.65551.06070%8D 3.6437E -081.8211E -07100%2.16500.99310%10D 1.9466E -079.7238E -07100%2.63541.03670%M u l t i -p o p u l a t i o n 1.0744E -07

3.3864E -07100%

2.28611.11440%

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202

T a b l e 6(a ): R e s u l t s f o r 30-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n

f 1. S p h e r e

f 2. S c h w e f e l ’s P r o b l e m 1.2

f 3. R o s e n b r o c k

M e a n S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e M e a n S t d S u c c e s s r a t e 2D 00100%6.4603E -089.5040E -08100%24.013921.57930%4D 00100%0.07600.05980%29.528921.36880%6D 00100%3.80992.70840%32.101123.75530%8D 00100%27.115016.73950%26.734415.56320%10D 00100%70.905923.62660%26.813615.89200%M u l t i -p o p u l a t i o n

00100%

6.5517E -052.6677E -0468%15.792318.74050%T a b l e 6(b ): R e s u l t s f o r 30-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n

f 4. S c h w e f e l ’s P r o b l e m 1.2 w i t h n o i s e

f 5. A c k l e y

f 6. R o t a t e d A c k l e y

M e a n

S t d

S u c c e s s r a t e

M e a n

S t d S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

2D 0.13100.39410%5.6843E -151.7764E -15100%5.4001E -151.8115E -15100%4D 17.119712.77590%5.5422E -151.7999E -15100%5.5422E -151.7999E -15100%6D 120.554570.31750%5.6843E -151.7764E -15100%6.1107E -151.6281E -15100%8D 333.2408165.95590%6.2528E -151.5486E -15100%6.1107E -151.6281E -15100%10d 590.1934283.51670%6.1107E -151.6281E -15100%5.8265E -151.7405E -15100%M u l t i -p o p u l a t i o n

2.30182.3663

3.6948E -157.1054E -15100%

4.1211E -15

1.3293E -15

100%

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203

T a b l e 6(c ): R e s u l t s f o r 30-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n

f 7. G r i e w a n k

f 8. R o t a t e d G r i e w a n k

f 9. R a s t r i

g i n

M e a n S t d

S u c c e s s r a t e M e a n

S t d S u c c e s s r a t e M e a n S t d

S u c c e s s r a t e

2D 0.00380.006068%0.01130.013620%13.01415.42620%4D 0.00270.004772%0.00380.005564%6.44732.18970%6D 0.00220.004176%0.00460.005856%4.41762.48870%8D 0.00360.007376%0.00540.007152%4.65642.79530%10D 0.00170.003580%0.00470.007968%4.17881.54670%M u l t i -p o p u l a t i o n 8.8752E -040.0025

88%0.00220.003868%6.68612.95040%

T a b l e 6(d ): R e s u l t s f o r 30-D p r o b l e m s w i t h s t r a t e g y -1

P o p u l a t i o n

f 10. R o t a t e d R a s t r i

g i n

f 11. N o n -c o n t i n u o u s R a s t r i

g i n

f 12. S c h w e f e l

M e a n

S t d

S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

M e a n

S t d

S u c c e s s r a t e

2D 120.231319.90910%15.36004.92340%514.0227262.83520%4D 140.692215.00610%9.72002.62230%203.7139200.880236%6D 147.384712.03060%8.68002.32240%269.2499186.997012%8D 158.396416.20860%7.91173.47530%198.9764135.558916%10D 162.471310.36260%10.74145.98950%156.3386173.395236%M u l t i -p o p u l a t i o n 94.6520

15.58260%8.48002.51860%241.6142

162.106316%