A Differential Evolution Algorithm for Continuous Function Optimization M. Fatih Tasgetiren
Yun-Chia Liang
Department of Management, Fatih University 34500 Buyukcekmece, Istanbul, Turkey
[email protected]
Department of Industrial Engineering and Management, Yuan Ze University No 135 Yuan-Tung Road, Chung-Li, Taoyuan County, Taiwan 320, R.O.C.
[email protected]
Gunes Gencyilmaz
Ipek Eker
Department of Management, Istanbul Kultur University Sirinevler, Istanbul, Turkey
[email protected]
Department of Management, Fatih University 34500 Buyukcekmece, Istanbul, Turkey
[email protected]
Abstract- This paper presents a differential evolution algorithm to solve continuous function optimization problems. The algorithm was tested using 14 newly proposed benchmark instances in Congress on Evolutionary Computation 2005. For these benchmark problems, the problem definition files, codes and evaluation criteria are available in http://www.ntu.edu.sg/home/EPNSugan. Since these benchmarks are newly designed, unfortunately there exist no results for comparison purposes. For this reason, the computational results are given along with the convergence graphs to be compared with the DE or nonDE algorithms during the conference and afterwards. I. INTRODUCTION Differential evolution (DE) is one of the latest evolutionary optimization methods proposed by Storn and Price 1997 in [13]. Like other evolutionary-type algorithms, DE is a population-based, stochastic global optimizer. In a DE algorithm, candidate solutions are represented as chromosomes based on floating-point numbers. The major difference between DE and genetic algorithm (GA) is that in DE some of the parents are generated through a mutation process before performing crossover operator whereas GA usually selects parents from current population, performs crossover, and then mutates the offspring. In the mutation process of a DE algorithm, the weighted difference between two randomly selected population members is added to a third member to generate a mutated solution. Then, the crossover operator is introduced to combine the mutated solution with the target solution so as to generate a trial solution. Then a selection operator is applied to compare the fitness function value of both competing solutions, namely, target
and trial solutions to determine who can survive for next generation. Since DE was first introduced to solve the Chebychev polynomial fitting problem by Storn and Price 1995 in [12], it has been successfully applied in a variety of applications including digital filter design by Storn 1996 in [10], neural network training by Masters and Land 1997 in [7], multisensor fusion by Joshi and Sanderson 1999 in [5], heat transfer by Babu and Sastry 1999 in [3], system design by Storn 1999 in [11], cancer diagnosis by Abbass 2002 in [1], and scheduling problems by Rüttgers 1997 in [9], and by Tasgetiren et al. 2004 in [14]. More introduction and literature surveys of DE can be found by Corne et al. 1999 in [4], by Lampinen 2001 in [6], by Babu and Onwubolu 2004 in [2], and by Price et al. 2005 in [8]. In addition, the advantages of DE such as simple concept, immediately accessible for practical applications, simple structure, ease of use, speed to get the solutions, and robustness has all led itself a good candidate to solve difficult nonlinear continuous functions. Therefore, this paper aims at employing DE to optimize a newly developed benchmark suite in Congress on Evolutionary Computation 2005. This paper is organized as follows. Section II gives the methodology of the proposed DE algorithm, and computational results of test problems are shown in Section III. Finally, Section IV summarizes the concluding remarks. II. DE ALGORITHM Currently, there exist several variants of DE. We follow the DE/rand/1/bin scheme of Storn and Price 1997 in [13]. The pseudo code of the DE algorithm is given in Figure 1.
Initialize parameters Initialize target population Evaluate Do { Obtain the mutant population Obtain the trial population Evaluate trial population Selection While (Termination)
construct the initial continuous parameter values of the individual uniformly:
xij0 = xmin + ( xmax − xmin )* r1
Figure 1. A Simple DE Algorithm.
The basic elements of the DE algorithm is summarized as follows: Target individual: X it denotes the ith individual in the population at generation t and is represented by n number of dimensions as X it = xit1 , xit2 ,.., xint , where xijt is the
[
]
optimized parameter value of the ith individual with respect to the jth dimension ( j = 1,2 ,..., n ). Mutant individual: Vi t denotes the ith individual in the population at generation t and is represented by n number of dimensions as Vi t = vit1 , vit2 ,.., vint , where vijt is the
[
]
optimized parameter value of the ith individual with respect to the jth dimension ( j = 1,2 ,..., n ). Trial individual: U it denotes the ith individual in the population at generation t and is represented by n number of dimensions as U it = u it1 ,u it2 ,..,u int , where u ijt is the
[
xijt = xmin + ( xmax − xmin )* r1
The population size is taken as 100. As the formulation of 14 functions suggests that the objective is to minimize the 14 continuous functions, the fitness value is the continuous function value of the individual X t . That is, f it X it . For simplicity, f it X it will be denoted as f i t .
( )
Target population: X is the set of NP individuals in the t population at generation t, i.e., X t = X 1t , X 2t ,..., X NP .
[
]
The complete computational procedure of the DE algorithm can be summarized as follows: Step 1: Initialization
Trial population: U t is the set of NP individuals in the t . population at generation t, i.e., U t = U 1t ,U 2t ,...,U NP
[
]
A population of individuals is constructed randomly for the DE algorithm. The continuous parameter values are established randomly. The following formula is used to
0 in
]
( )
t = t +1 For each target individual, X it , i = 1,2 ,..., NP , at generation t, a mutant individual, Vi t +1 = vit1+1 ,vit2+1 ,..,vint +1 , is determined such that:
[
]
Vi
where
t X best
t +1
=
t X best
(
+ F X bti − X cti
)
is the best individual so far in the
population and bi , and ci are two randomly choosen individuals from the population such that ( bi ≠ ci ).
F > 0 is a mutant factor which affects the differential variation between two individuals.
( )
A. Initial Population
}
X = x , x ,..., x . Evaluate each individual i in the population using the objective function f i0 X i0 for i = 1,2 ,..., NP . 0 i2
Step 3: Generate mutant population
F ∈ (0 ,2 ) is a real constant which
Mutant constant: affects the differential variation between two individuals. Crossover constant: CR ∈ (0 ,1) is a crossover constant which affects the diversity of population for the next generation. Fitness function: In a minimization problem, the objective function is the continuous function value denoted as f X it . Termination criterion: It is a condition that the search process will be terminated. It might be a maximum number of function evaluations or maximum CPU time to terminate the search.
[
{
0 i1
Step 2: Update generation counter
]
Set t=0, NP =100. Generate NP individuals randomly as explained before, X i0 ,i = 1,2 ,..., NP where 0 i
Mutant population: V t is the set of NP individuals in t the population at generation t, i.e., V t = V1t ,V2t ,...,VNP .
[
( )
B. Computational Procedure
]
optimized parameter value of the ith individual with respect to the jth dimension ( j = 1,2 ,..., n ). t
where x min and x max are given bounds of the continuous functions and r1 is a uniform random number between 0 and 1. During the reproduction of the DE algorithm, it is possible to extend the search outside of the initial range of the search space. For this reason, parameter values violating the initial range are restricted to the feasible range as follows:
Step 4: Generate trial population
Following the mutation phase, the crossover (recombination) operator is applied to obtain the trial population. For each mutant individual, t +1 t +1 t +1 t +1 Vi = vi 1 ,vi 2 ,.., vin , an integer random number
[
]
between 1 and n, i.e, Di ∈ (1,2 ,.., n ) , is chosen, and a
[
t +1 trial individual, U t +1 = U 1t +1 ,U 2t +1 ,...,U NP generated such that:
]
is
Step 5: Evaluate trial population
Evaluate the trial population using the objective function f it +1 U it +1 for i = 1,2 ,..., NP .
(
)
Step 6: Selection
To decide whether or not the trial individual U it +1 should be a member of the target population for the next generation, it is compared to its counterpart target individual X it at the previous generation. The selection is based on the survival of fitness among the trial population and target population such that:
(
) ( )
U t +1 ,if f U it +1 ≤ f X it X it +1 = i t X i , otherwise
( )
If the number of generation exceeds the maximum number of function evaluations, or maximum CPU time, then stop; otherwise go to step 2.
log(f-f*)
Step 7: Stopping criterion
p1 p2 p3 p4 p5 p6 p7
FES
Figure 2. Convergence Graph of DE for D=10 1.E+05 p8
1.E+04 log(f-f*)
p9
1.E+03
p10
1.E+02
p11
1.E+01
p12 p13
1.E+00
p14
1.E-01
10 00
In order to solve continuous function optimization problems, several optimization algorithms have been presented in the literature with their results based on a small subset of the standard test problems such as Sphere, Schwefel, Rosenbrock, Rastrigin, and so on. Often, confusing results limited to the test problems were reported in the literature in such a way that the same algorithm working for a set of functions may not work for some other set of functions. For these reasons, these algorithms should be evaluated more systematically by determining a common termination criterion, size of problems, initialization scheme, running time. The special session on real-parameter optimization in CEC2005 aimed at developing new benchmark functions to be publicly available to the researchers for evaluating their algorithms. The problem definition files, codes and evaluation criteria are made available in http://www.ntu.edu.sg/home/EPNSugan and
10 00
III. COMPUTATIONAL RESULTS
1.E+09 1.E+07 1.E+05 1.E+03 1.E+01 1.E-01 1.E-03 1.E-05 1.E-07 1.E-09 1.E-11 75 00 0 10 00 00
random number between 0 and 1. In other words, the trial individual is made up with some parameters of mutant individual, or at least one of the parameters randomly selected, and some other parameters of target individual.
50 00 0 75 00 0 10 00 00
constant in the range [0, 1], and rijt +1 is a uniform
25 00 0 50 00 0
generation U it , CR is a user-defined crossover
25 00 0
where the index D refers to a randomly chosen dimension (j=1,2,..,n), which is used to ensure that at least one parameter of each trial individual U it +1 differs from its counterpart in the previous
75 00 10 00 0
Otherwise
http://staffx.webstore.ntu.edu.sg/MySite/Public.aspx?acco untname=epnsugan. The traditional DE algorithm was coded in C and run on an Intel P4 1.33 GHz PC with 256MB memory. Regarding the DE parameters, mutation (MR) and crossover rates (CR) are taken as 0.9 respectively. The population size was 100. The maximum number of function evaluations is fixed at 10000*D where D is the size of dimension and varied from 10 to 50. The DE algorithm was run for the 14 benchmark functions recently developed. The performance evaluation of the DE algorithm is also conducted through the guidelines described in the webpage above. 25 replications are conducted for each benchmark function to record the error values, f (x ) − f x* , after 1e3 FES, 1e4 FES, 1e5 FES and at the termination. The error values achieved at different FES levels are given in details in Tables 2 to 4. The complexity of the DE algorithm is given in Table 1. The convergence graphs are also given in Figures 2 to 7. Since these benchmarks are newly designed, unfortunately there exist no results for comparison purposes. For this reason, the computational results are given along with the convergence graphs to be compared with the DE or non-DE algorithms during the conference and afterwards. The objective of this paper was initially to compare the results of the DE algorithm with those by the particle swarm optimization (PSO) algorithm. Due to the page restriction, the results for the PSO algorithm were given in a separate paper for CEC2005.
75 00 10 00 0
j = Di
50 00
rijt +1 ≤ CR or
50 00
v t +1 ,if u ijt +1 = ij t xij ,
FES
Figure 3. Convergence Graph of DE for D=10
P1 P2 P3 P4 P5 P6 P7
10 00 50 00 75 0 10 0 00 0 25 00 0 50 00 75 0 0 10 00 00 15 00 00 20 00 00 0 25 0 00 30 00 00 00
log(f-f*)
IV. CONCLUSIONS
1.E+12 1.E+10 1.E+08 1.E+06 1.E+04 1.E+02 1.E+00 1.E-02 1.E-04 1.E-06 1.E-08 1.E-10
FES
1.E+07 1.E+05 1.E+03 1.E+01 1.E-01 1.E-03 1.E-05 1.E-07 1.E-09 1.E-11
P8 P9 P10 P12 P13
[1] Abbass, H. A. (2002) “An Evolutionary Artificial Neural Networks Approach for Breast Cancer Diagnosis,” Artificial Intelligence in Medicine, vol. 25, pp. 265-281. [2] Babu, B. V. and Onwubolu, G. C. (eds.) (2004) New Optimization Techniques in Engineering, Springer Verlag.
Figure 5. Convergence Graph of DE for D=30 1.E+11 1.E+09 1.E+07 1.E+05 1.E+03 1.E+01 1.E-01 1.E-03 1.E-05 1.E-07 1.E-09 1.E-11
p1 p2 p3 p4 p5 p6 p7
10 0 50 0 0 75 0 10 00 0 25 00 0 50 00 0 75 00 10 000 0 15 000 0 20 000 0 25 000 00 30 00 0 35 000 0 40 000 0 45 000 0 50 000 00 00
log(f-f*)
BIBLIOGRAPHY
P14
FES
Figure 6. Convergence Graph of DE for D=50 1.E+07 p8
1.E+06
p9
1.E+05
p10
1.E+04
p11
1.E+03
p12
1.E+02
p13
1.E+01
p14
10 0 50 0 0 75 0 10 00 0 25 00 0 50 00 0 75 00 10 000 0 15 00 0 0 20 000 00 25 0 0 0 30 000 0 35 00 0 0 40 000 00 45 00 0 50 000 00 00
1.E+00
FES
Table 1. Complexity of the DE Algorithm D=30
[4] Corne, D., Dorigo, M., and Glover, F. (eds.) (1999) “Part Two: Differential Evolution,” New Ideas in Optimization, McGraw-Hill, pp. 77-158. [5] Joshi, R. and Sanderson, A. C. (1999) “Minimal Representation Multisensor Fusion Using Differential Evolution,” IEEE Transactions on Systems, Man, and Cybernetics, Part A, vol. 29, pp. 63-76.
[7] Masters, T. and Land, W. (1997) “A New Training algorithm for the General Regression Neural Network,” Proc. of the 1997 IEEE International Conference on Systems, Man, and Cybernetics, pp. 19901994. [8] Price, K., Storn, R., and Lampinen, J. (2005) Differential Evolution – A Practical Approach to Global Optimization, Springer-Verlag. [9] Rüttgers, M. (1997) “Design of a New Algorithm for Scheduling in Parallel Machine Shops,” Proc. of the Fifth European Congress on Intelligent Techniques and Soft Computing, pp. 2182-2187. [10] Storn, R. (1996) “Differential Evolution Design of an IIT-Filter with Requirements for Magnitude and Group Delay,” Proc. of IEEE International Conference on Evolutionary Computation, pp. 268-273. [11] Storn, R. (1999) “System Design by Constraint Adaptation and Differential Evolution,” IEEE Transactions on Evolutionary Computation, vol. 3, pp. 22-34.
Figure 7. Convergence Graph of DE for D=50
D=10
[3] Babu, B. V. and Sastry, K. K. N. (1999) “Estimation of Heat Transfer Parameters in a Trickle-Bed Reactor Using Differential Evolution and Orthogonal Collocation,” Computers and Chemical Engineering, vol. 23, pp. 327-339.
[6] Lampinen, J. (2001) “A Bibliography of Differential Evolution Algorithm,” Technical Report, Lappeenranta University of Technology, Department of Information Technology, Laboratory of Information Processing.
FES
log(f-f*)
However, some sophisticated DE algorithms are recently presented to claim that they produce better results than the traditional one on a subset of the standard test problems. For this reason, we present the results of the newly designed 14 benchmark problems for the traditional DE to be compared with the competing DE or non-DE algorithms during the conference and afterwards.
P11
10 00 50 00 75 00 10 00 25 0 00 50 0 00 75 0 0 10 00 00 15 00 00 20 00 00 25 00 00 30 00 00 00
log(f-f*)
Figure 4. Convergence Graph of DE for D=30
In this paper, a traditional version of the DE/rand/1/bin scheme of Storn and Price 1997 is presented to solve the benchmark problems designed for the special session on real-parameter optimization at CEC2005. Since these benchmark problems are newly designed for the special session, there are unfortunately no existing results for them to be compared.
D=50
T0
551
551
551
T1
1442
4526
7481
T2
7150
25094
49497
Complexity
10.36
37.33
76.25
[12] Storn, R. and Price, K. (1995) “Differential Evolution – a Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces,” Technical Report TR-95-012, ICSI, 1995. [13] Storn, R. and Price, K. (1997) “Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Space,” Journal of Global Optimization, vol. 11, pp. 341-359.
[14] Tasgetiren, M. F., Liang, Y.-C., Sevkli, M., and Gencyilmaz, G. (2004) “Differential Evolution Algorithm for Permutation Flowshop Sequencing Problem with Makespan Criterion,” Proc. of the Fourth
International Symposium on Intelligent Manufacturing Systems, Sakarya, Turkey, pp. 442-452.
Table2. Error Values Achieved at 1e3 FES, 1e4 FES, and 1e5 FES for the DE Algorithm, D=10 FES 1e3
1e4
1e5
1
2
3
4
5
6
7
1st
1.85424042E+03
3.01950854E+03
1.90610404E+07
5.26307402E+03
1.15818716E+03
5.02057882E+07
5.58453440E+01
7th
3.59666809E+03
5.04842971E+03
3.02341202E+07
7.50755258E+03
1.72110031E+03
1.11428265E+08
1.04098324E+02
13th
4.29586080E+03
6.09159746E+03
4.51606799E+07
8.48740167E+03
2.26923753E+03
1.93328016E+08
1.16677973E+02
19th
5.00819927E+03
7.77913908E+03
6.24066195E+07
9.40238053E+03
2.83958633E+03
2.92389769E+08
1.47885612E+02
25th
6.60095704E+03
9.57528322E+03
9.00850862E+07
1.35383508E+04
5.75893494E+03
5.35614353E+08
2.48522549E+02
Mean
4.23316050E+03
6.29731897E+03
4.63044434E+07
8.64123168E+03
2.44959097E+03
2.18674934E+08
1.23054697E+02
Std D.
1.25635031E+03
1.73963030E+03
1.95727850E+07
1.90329611E+03
1.09489663E+03
1.28320965E+08
4.04756254E+01
1st
3.09619028E+00
6.46119682E+01
1.13580243E+05
1.34154008E+02
3.03966000E-03
3.28694019E+03
1.04623495E+00
7th
1.44382561E+01
1.26492739E+02
6.82541627E+05
1.83790359E+02
8.20477000E-03
7.95176922E+03
1.24115951E+00
13th
1.87168552E+01
1.86523583E+02
9.17870169E+05
2.58114014E+02
1.61936800E-02
1.27184306E+04
1.39226109E+00
19th
2.16113562E+01
2.21808864E+02
1.10021861E+06
3.61044020E+02
2.02880100E-02
2.18604190E+04
1.46351830E+00
25th
3.38956911E+01
6.97320132E+02
2.52788530E+06
4.37839128E+02
1.05464340E-01
4.11937938E+04
1.77392536E+00
Mean
1.88166548E+01
1.94334532E+02
1.00104904E+06
2.67162061E+02
1.88126044E-02
1.55603166E+04
1.37673183E+00
Std D.
6.52481764E+00
1.16797831E+02
5.43199732E+05
9.98846625E+01
1.98725906E-02
1.01945994E+04
1.74155175E-01
1st
0.00000000E+00
0.00000000E+00
4.10000000E-07
0.00000000E+00
0.00000000E+00
0.00000000E+00
5.08014300E-02
7th
0.00000000E+00
0.00000000E+00
2.02000000E-06
0.00000000E+00
0.00000000E+00
0.00000000E+00
1.03748080E-01
13th
0.00000000E+00
0.00000000E+00
7.09000000E-06
0.00000000E+00
0.00000000E+00
0.00000000E+00
1.50056560E-01
19th
0.00000000E+00
0.00000000E+00
6.23900000E-05
0.00000000E+00
0.00000000E+00
0.00000000E+00
2.07026720E-01
25th
0.00000000E+00
0.00000000E+00
3.83810000E-04
0.00000000E+00
0.00000000E+00
3.98657911E+00
2.58674320E-01
Mean
0.00000000E+00
0.00000000E+00
5.46328000E-05
0.00000000E+00
0.00000000E+00
6.37852662E-01
1.50829436E-01
Std D.
0.00000000E+00
0.00000000E+00
1.01191111E-04
0.00000000E+00
0.00000000E+00
1.49164132E+00
6.40648056E-02
8
9
10
11
12
13
14
1st
2.04152779E+01
6.17860059E+01
6.21051071E+01
9.68018798E+00
1.57965846E+04
7.07424982E+01
3.97908718E+00
7th
2.06656385E+01
7.06628237E+01
8.11574636E+01
1.14143431E+01
4.72954195E+04
3.73212875E+02
4.31507750E+00
13th
2.07916277E+01
7.49742879E+01
9.09541402E+01
1.16785718E+01
5.14369354E+04
6.16887210E+02
4.37865709E+00
19th
2.08319215E+01
8.13895175E+01
9.64873632E+01
1.22943012E+01
6.01442038E+04
1.20720050E+03
4.41748757E+00
25th
2.09122801E+01
8.93127813E+01
1.02934924E+02
1.30496441E+01
8.13749078E+04
2.72286933E+03
4.51682169E+00
Mean
2.07451368E+01
7.57053848E+01
8.74951410E+01
1.16930681E+01
5.28402633E+04
9.12927940E+02
4.35079277E+00
Std D.
1.31186065E-01
7.77409520E+00
1.07633555E+01
8.09602684E-01
1.44527417E+04
6.91357389E+02
1.18453228E-01
1st
2.03215622E+01
2.02641419E+01
3.34415225E+01
7.73432393E+00
7.49916286E+02
2.32086074E+00
3.58983468E+00
FES 1e3
1e4
1e5
7th
2.05162497E+01
3.29482195E+01
4.42874150E+01
9.11806837E+00
1.42067605E+03
3.89394678E+00
3.93831782E+00
13th
2.05685793E+01
3.74767314E+01
5.22460038E+01
9.96182919E+00
2.23430323E+03
5.13629510E+00
4.03646983E+00
19th
2.05919487E+01
4.16617905E+01
5.45106038E+01
1.03578021E+01
3.35669953E+03
5.86466685E+00
4.08288292E+00
25th
2.06642183E+01
4.52071867E+01
6.31434797E+01
1.11137930E+01
8.52399814E+03
7.46223545E+00
4.32128800E+00
Mean
2.05474174E+01
3.58942657E+01
5.00817797E+01
9.78996188E+00
2.91231444E+03
4.99484267E+00
3.99999366E+00
Std D.
7.82285870E-02
7.37227742E+00
7.99405348E+00
8.82700003E-01
2.08499103E+03
1.35426725E+00
1.82656711E-01
1st
2.01870148E+01
1.98991811E+00
7.79444897E+00
7.93340000E-04
0.00000000E+00
3.81196130E-01
1.00236119E+00
7th
2.02972996E+01
2.98487717E+00
1.14446694E+01
2.43548000E-03
0.00000000E+00
5.33666510E-01
3.45017003E+00
13th
2.03631326E+01
4.97479529E+00
1.52434616E+01
5.85140000E-03
2.00000000E-08
7.16250120E-01
3.58983468E+00
19th
2.03872002E+01
6.96470836E+00
1.89896977E+01
1.50192166E+00
1.00030465E+01
9.19177750E-01
3.75965729E+00
25th
2.05101926E+01
8.95462648E+00
3.63416127E+01
3.00220739E+00
1.34735272E+03
1.56296326E+00
3.92518546E+00
Mean
2.03384214E+01
5.09418876E+00
1.65176635E+01
7.09221810E-01
5.89551096E+01
7.66957323E-01
3.44691730E+00
Std D.
7.50173269E-02
2.09729376E+00
6.98306214E+00
9.72805229E-01
2.68501349E+02
3.34237256E-01
5.73178240E-01
Table3. Error Values Achieved at 1e3 FES, 1e4 FES, 1e5 FES, and at Termination for the DE Algorithm, D=30 FES 1e3
1e4
1e5
Trm
1st
1e4
1e5
2
3
4
5
6
7
6.34975237E+04
4.19985770E+08
6.74118174E+04
1.84970168E+04
1.13667763E+10
1.09899550E+03
7th
4.97148721E+04
7.33956224E+04
8.27104662E+08
8.60348323E+04
2.21714648E+04
1.87083958E+10
1.34025439E+03
13th
5.45560568E+04
8.31322837E+04
1.04649460E+09
1.03477350E+05
2.39905575E+04
2.26590666E+10
1.44404952E+03
19th
6.04220211E+04
9.42251907E+04
1.20096710E+09
1.13989836E+05
2.56208746E+04
3.12495693E+10
1.57084249E+03
25th
6.54650364E+04
1.22545015E+05
1.34340411E+09
1.48995303E+05
2.85591517E+04
4.41783843E+10
1.90925497E+03
Mean
5.44088225E+04
8.59289785E+04
1.00926361E+09
9.98428668E+04
2.39569787E+04
2.60006776E+10
1.45691929E+03
Std D.
7.20866400E+03
1.74239230E+04
2.39042404E+08
1.86456304E+04
2.63150656E+03
9.42864407E+09
1.95041627E+02
1st
7.37040994E+03
2.50504951E+04
1.43161289E+08
3.43310772E+04
7.59216958E+03
8.00479416E+08
1.91022251E+02
7th
1.17345687E+04
4.66857531E+04
2.87888290E+08
4.82081398E+04
9.70474400E+03
1.21020594E+09
3.46103054E+02
13th
1.28904650E+04
5.00936429E+04
3.13126040E+08
5.11596337E+04
1.02530421E+04
1.49847820E+09
4.21541562E+02
19th
1.47663491E+04
5.40391673E+04
3.84733178E+08
5.52669554E+04
1.15159792E+04
1.78986602E+09
5.45858281E+02
25th
1.81025020E+04
6.45799939E+04
5.20868453E+08
6.71006318E+04
1.31914581E+04
4.59953781E+09
7.43593610E+02
Mean
1.27943297E+04
4.91996632E+04
3.26878253E+08
5.08241261E+04
1.04777358E+04
1.65636481E+09
4.44258650E+02 1.41394243E+02
Std D.
2.62878783E+03
9.25481151E+03
8.82835547E+07
8.59486266E+03
1.34589043E+03
8.34415627E+08
1st
8.83870000E-04
1.75557267E+02
1.84216727E+06
3.61277298E+02
1.33227709E+01
3.78990126E+01
1.85171710E-01
7th
5.89966000E-03
3.60806197E+02
3.24895247E+06
9.14245748E+02
7.03625752E+02
6.80511767E+01
3.63209960E-01
13th
9.70001000E-03
4.29233149E+02
4.18139444E+06
1.27675113E+03
2.82632665E+03
1.23096747E+02
5.09019030E-01
19th
1.15735200E-02
6.19900836E+02
5.42529640E+06
1.85841654E+03
3.25818339E+03
1.99468029E+02
7.65749370E-01
25th
2.69409200E-02
8.46556874E+02
9.51084012E+06
3.56927869E+03
6.61988660E+03
1.62801948E+03
9.68941790E-01
Mean
9.85309680E-03
4.82823155E+02
4.64793259E+06
1.45544748E+03
2.56001668E+03
2.31274685E+02
5.40389176E-01
Std D.
6.55475777E-03
1.89155714E+02
2.02291314E+06
8.24725636E+02
1.93256670E+03
3.50876203E+02
2.32871580E-01
1st
0.00000000E+00
1.00082200E-02
2.52866376E+05
4.44201410E-01
1.08400000E-05
7.91051720E-01
0.00000000E+00
7th
0.00000000E+00
2.69799400E-02
3.63513263E+05
1.56984803E+00
9.65160000E-03
6.02897119E+00
0.00000000E+00
13th
0.00000000E+00
4.70619200E-02
6.71937721E+05
2.26302961E+00
2.19951281E+03
1.01826559E+01
1.00000000E-08
19th
0.00000000E+00
8.74604400E-02
9.74699986E+05
4.37444810E+00
2.91858327E+03
1.26779155E+01
5.39371200E-02
25th
0.00000000E+00
1.58316260E-01
1.45497078E+06
1.58945239E+01
5.93939047E+03
1.80863107E+01
2.89676720E-01
Mean
0.00000000E+00
6.18957020E-02
7.34622326E+05
4.06987050E+00
1.89052136E+03
9.45833547E+00
4.16066952E-02
Std D.
0.00000000E+00
4.21460779E-02
3.83248088E+05
4.20637434E+00
1.88806962E+03
5.37511000E+00
6.68922481E-02
8
9
10
11
12
13
14
1st
2.10371002E+01
3.51505207E+02
3.86807775E+02
4.30149677E+01
1.21493413E+06
7.96154487E+04
1.36177043E+01
FES 1e3
1 3.88764767E+04
7th
2.11788301E+01
3.81350091E+02
4.53468021E+02
4.48355068E+01
1.35209016E+06
1.57959003E+05
1.41171735E+01
13th
2.12374808E+01
4.08660945E+02
4.74525568E+02
4.54247189E+01
1.49367677E+06
2.18094934E+05
1.41871422E+01
19th
2.12775388E+01
4.20830954E+02
4.93344100E+02
4.61805178E+01
1.66345136E+06
3.04821261E+05
1.42366729E+01
25th
2.13001739E+01
4.74661471E+02
5.22231496E+02
4.80001430E+01
2.02807803E+06
4.92660893E+05
1.44037304E+01
Mean
2.12194622E+01
4.04523863E+02
4.69684588E+02
4.55424008E+01
1.51555047E+06
2.28310623E+05
1.41609786E+01
Std D.
6.80105871E-02
2.94053422E+01
3.16318829E+01
1.28294042E+00
2.01894868E+05
1.06163834E+05
1.48381425E-01
1st
2.09519581E+01
2.33280451E+02
2.84955398E+02
4.06642918E+01
4.32073230E+05
1.46392849E+03
1.36177043E+01
7th
2.10682700E+01
2.59948835E+02
3.18477988E+02
4.25439523E+01
6.48983418E+05
2.67124771E+03
1.38430561E+01
13th
2.11070513E+01
2.80316075E+02
3.42834521E+02
4.30718489E+01
6.71627208E+05
3.83519302E+03
1.39494154E+01
19th
2.11398472E+01
2.91904233E+02
3.51147017E+02
4.35359055E+01
7.95225929E+05
5.17356358E+03
1.40067600E+01
25th
2.11875304E+01
3.13804233E+02
3.81146888E+02
4.43857469E+01
1.05816248E+06
7.88898250E+03
1.41079504E+01
Mean
2.10956296E+01
2.76173604E+02
3.34932913E+02
4.28563828E+01
7.10077488E+05
4.00501625E+03
1.39080641E+01
Std D.
5.95784469E-02
2.17571138E+01
2.61488639E+01
9.64590167E-01
1.36520733E+05
1.86824600E+03
1.32340585E-01
1st
2.08069979E+01
2.19159473E+01
3.00718954E+01
1.75624575E+01
0.00000000E+00
2.44370195E+00
1.30431085E+01
7th
2.09823691E+01
3.28623317E+01
6.86396655E+01
3.83301139E+01
0.00000000E+00
4.40461100E+00
1.35254311E+01
13th
2.10002068E+01
4.09024374E+01
1.14042907E+02
3.96573441E+01
0.00000000E+00
5.26838403E+00
1.36171782E+01
19th
2.10285615E+01
4.77875505E+01
1.95522412E+02
4.06290325E+01
0.00000000E+00
9.04497730E+00
1.36717312E+01
25th
2.10878291E+01
6.07722612E+01
2.28968907E+02
4.27992327E+01
0.00000000E+00
1.55170251E+01
1.37656063E+01
Mean
2.09985640E+01
4.09120166E+01
1.26305732E+02
3.82544031E+01
0.00000000E+00
6.64955422E+00
1.35706703E+01
Trm
Std D.
5.40782888E-02
1.12044936E+01
6.52609716E+01
5.23821753E+00
0.00000000E+00
3.19828604E+00
1.60775657E-01
1st
2.08069979E+01
2.18890942E+01
2.34153005E+01
5.37357858E+00
0.00000000E+00
1.97830146E+00
1.29577683E+01
7th
2.09519581E+01
3.28336035E+01
4.49232272E+01
8.52890898E+00
0.00000000E+00
2.68040834E+00
1.32525942E+01
13th
2.09704502E+01
4.07932709E+01
5.00901443E+01
1.09034039E+01
0.00000000E+00
2.95264959E+00
1.33952849E+01
19th
2.09983757E+01
4.77579490E+01
5.49341958E+01
1.56893411E+01
0.00000000E+00
4.04031404E+00
1.35254311E+01
25th
2.10301674E+01
6.06924067E+01
7.95864803E+01
1.87345068E+01
0.00000000E+00
5.77030220E+00
1.36598673E+01
Mean
2.09600061E+01
4.08728614E+01
5.02669506E+01
1.17989926E+01
0.00000000E+00
3.49834965E+00
1.33659445E+01
Std D.
5.14799419E-02
1.12049301E+01
1.33350662E+01
4.16056629E+00
0.00000000E+00
1.17382890E+00
1.99833636E-01
Table4. Error Values Achieved at 1e3 FES, 1e4 FES, 1e5 FES and at Termination for the DE Algorithm, D=50 FES 1e3
1e4
1e5
Trm
1st
1e4
2
3
4
5
6
7
1.58504520E+05
2.07214795E+09
2.01007726E+05
3.60707613E+04
6.24046572E+10
2.70669600E+03
7th
1.24213485E+05
2.15679197E+05
3.34100009E+09
2.49302705E+05
4.19096551E+04
7.05401502E+10
3.24870319E+03
13th
1.32941680E+05
2.51151610E+05
4.15514697E+09
2.69409198E+05
4.37841233E+04
7.54364456E+10
3.42658116E+03
19th
1.42118402E+05
2.72773513E+05
4.62200572E+09
3.15075628E+05
4.60201977E+04
9.19462520E+10
3.55218199E+03
25th
1.55556907E+05
3.38457623E+05
5.17138758E+09
3.67871127E+05
4.96219684E+04
1.23847914E+11
3.87866585E+03
Mean
1.32369395E+05
2.47596798E+05
4.02173553E+09
2.80243469E+05
4.39235557E+04
8.17463411E+10
3.40662939E+03
Std D.
1.30691091E+04
4.56184816E+04
8.47471367E+08
4.69921636E+04
3.58509430E+03
1.70616903E+10
2.88949417E+02
1st
3.98083164E+04
1.08122547E+05
7.87258061E+08
1.24568107E+05
2.00127999E+04
6.18604131E+09
1.06680008E+03
7th
4.43059347E+04
1.36973281E+05
1.07956226E+09
1.53487315E+05
2.50087236E+04
9.91086638E+09
1.34261794E+03
13th
4.65329206E+04
1.49203539E+05
1.17726243E+09
1.61513846E+05
2.60129082E+04
1.20288326E+10
1.50405398E+03
19th
5.08940804E+04
1.56631020E+05
1.46263328E+09
1.79026478E+05
2.73081642E+04
1.53635072E+10
1.67262880E+03
25th
6.81645472E+04
1.93489815E+05
1.82079096E+09
2.08216994E+05
3.01435042E+04
1.82118509E+10
2.08144012E+03
Mean
4.79235965E+04
1.47231543E+05
1.26727971E+09
1.63019317E+05
2.60249258E+04
1.23105024E+10
1.51560346E+03
Std D.
6.01149768E+03
2.03000801E+04
2.65941467E+08
1.99856704E+04
2.39857300E+03
3.46090616E+09
2.61940464E+02
1st
3.91216224E+00
1.05025114E+04
9.72749744E+06
1.36500475E+04
4.43185685E+03
7.58732668E+03
1.89022668E+00
7th
5.40716704E+00
1.82657894E+04
2.42321479E+07
3.05232757E+04
7.12124427E+03
2.55062364E+04
2.40128959E+00
13th
7.11906131E+00
1.99615128E+04
3.64854720E+07
3.30408587E+04
9.76171684E+03
3.59316291E+04
3.05294171E+00
19th
9.80515483E+00
2.11415867E+04
4.16240778E+07
3.99110229E+04
1.03183423E+04
8.55642772E+04
3.79685390E+00
25th
2.13349717E+01
3.28744039E+04
7.62817030E+07
5.55904245E+04
1.24092126E+04
4.16202397E+05
7.23378121E+00
Mean
8.63916087E+00
2.00253237E+04
3.73886638E+07
3.53287453E+04
9.03571259E+03
6.28620419E+04
3.50588763E+00
Std D.
4.72937274E+00
4.71459125E+03
1.68235583E+07
9.48968364E+03
2.21534260E+03
8.09669778E+04
1.51754487E+00
1st
0.00000000E+00
2.67055935E+01
9.83716510E+05
6.42149636E+02
5.38846427E+02
2.13773028E+01
1.00000000E-07
7th
0.00000000E+00
5.11243860E+01
1.49749327E+06
1.61208917E+03
3.66470924E+03
4.01438958E+01
6.60000000E-07
13th
0.00000000E+00
5.74726101E+01
2.10192373E+06
2.26645561E+03
5.61799320E+03
8.45228419E+01
2.66000000E-06
19th
0.00000000E+00
7.93419114E+01
2.56510263E+06
3.58109587E+03
7.33205088E+03
9.32595445E+01
1.66393000E-02
25th
0.00000000E+00
1.40927436E+02
4.52990884E+06
1.04787718E+04
1.04085511E+04
3.88045474E+02
4.86659500E-02
Mean
0.00000000E+00
6.70520822E+01
2.18206322E+06
3.00509038E+03
5.49714668E+03
9.13380346E+01
1.07342384E-02
Std D.
0.00000000E+00
2.92808581E+01
9.15385365E+05
2.18710616E+03
2.49396728E+03
8.49118047E+01
1.58607108E-02
8
9
10
11
12
13
14
1st
2.11749540E+01
7.39051239E+02
7.59303768E+02
7.53533902E+01
5.39738149E+06
3.04673585E+05
2.36851054E+01
FES 1e3
1 1.09919212E+05
7th
2.13132765E+01
7.74119475E+02
8.65019525E+02
7.97840852E+01
6.42234538E+06
9.05402186E+05
2.38685505E+01
13th
2.13442214E+01
7.99985607E+02
8.88168100E+02
8.14681320E+01
6.91973597E+06
1.06372977E+06
2.39822883E+01
19th
2.13675710E+01
8.31189543E+02
9.14221109E+02
8.26105502E+01
7.23733190E+06
1.34206302E+06
2.40376980E+01
25th
2.13887244E+01
8.72940430E+02
9.52398094E+02
8.52553619E+01
8.34791399E+06
1.82215415E+06
2.41530379E+01
Mean
2.13327267E+01
8.03629808E+02
8.81998588E+02
8.12443558E+01
6.86580761E+06
1.07914542E+06
2.39449478E+01
Std D.
4.94162947E-02
3.42247412E+01
4.49513140E+01
2.23696850E+00
7.18679955E+05
3.54193134E+05
1.40011212E-01
1st
2.11554297E+01
5.37857250E+02
5.77084747E+02
7.40360183E+01
2.74711467E+06
1.98472893E+04
2.33464592E+01
7th
2.12198224E+01
5.93322275E+02
6.24282945E+02
7.67951283E+01
3.11405142E+06
3.29730821E+04
2.36234624E+01
13th
2.12656486E+01
6.18468455E+02
6.48086187E+02
7.78536807E+01
3.39803563E+06
5.31763874E+04
2.37075318E+01
19th
2.12914574E+01
6.26707452E+02
6.60825244E+02
7.85654017E+01
3.69400897E+06
7.25397169E+04
2.37937725E+01
1e5
Trm
25th
2.13342762E+01
6.50200676E+02
7.11173536E+02
7.98898988E+01
4.19301734E+06
1.11900986E+05
2.39553168E+01
Mean
2.12562759E+01
6.09454445E+02
6.47202996E+02
7.76676600E+01
3.41839270E+06
5.45567981E+04
2.36952123E+01
Std D.
5.06525509E-02
2.99157879E+01
3.30762417E+01
1.39660852E+00
3.99080725E+05
2.55809289E+04
1.59300841E-01
1st
2.10798778E+01
7.02114354E+01
3.75240293E+02
7.17453236E+01
1.24834475E+04
2.46533857E+01
2.29296696E+01
7th
2.11601110E+01
8.80808147E+01
4.02222311E+02
7.41917063E+01
6.09260384E+04
3.06321105E+01
2.33553904E+01
13th
2.11857992E+01
9.69368956E+01
4.17645544E+02
7.48307488E+01
8.93235585E+04
3.50081067E+01
2.34080481E+01
19th
2.12054600E+01
1.16184212E+02
4.35727985E+02
7.62535794E+01
1.30699201E+05
3.75568992E+01
2.35047972E+01
25th
2.12310934E+01
1.46614282E+02
4.84708024E+02
7.72227109E+01
1.99061994E+05
4.37736162E+01
2.36513447E+01
Mean
2.11807660E+01
1.01463342E+02
4.20275090E+02
7.50219491E+01
9.48682960E+04
3.44406311E+01
2.33992457E+01
Std D.
3.33359464E-02
2.02831078E+01
2.47437627E+01
1.35698901E+00
4.33802472E+04
4.87465995E+00
1.52067649E-01
1st
2.10276245E+01
6.66621661E+01
5.59456683E+01
2.44855365E+01
6.30067003E+02
5.19513613E+00
2.26742575E+01
7th
2.11233371E+01
7.95965935E+01
8.37425395E+01
3.74255920E+01
4.08554888E+03
7.35487789E+00
2.30477618E+01
13th
2.11432427E+01
8.65612814E+01
9.01579521E+01
5.70049321E+01
8.10494593E+03
8.78430931E+00
2.32035459E+01
19th
2.11633732E+01
9.65108165E+01
9.74666693E+01
7.11363402E+01
1.60856239E+04
1.15155361E+01
2.32610502E+01
25th
2.11989202E+01
1.42278822E+02
1.33334823E+02
7.44238325E+01
3.08874945E+04
1.59468035E+01
2.34931830E+01
Mean
2.11374162E+01
9.27697931E+01
9.14884221E+01
5.32926347E+01
1.06185337E+04
9.58925425E+00
2.31630192E+01
Std D.
3.69486980E-02
2.10558604E+01
1.68416709E+01
1.83730917E+01
8.53809015E+03
3.20361245E+00
1.74935016E-01
