TWO ELITIST VARIANTS OF DIFFERENTIAL

ANT-STIGMERGY ALGORITHM

Adrian Emanoil Şerbencu, Viorel Minzu, Adriana Şerbencu and Daniela Cernega

Control Systems and Industrial Informatics Department, Computer Science Faculty

"Dunarea de Jos" University from Galati, 800201, Str. Domneasca 111, Galati, Romania

Keywords: Optimization, Ant colony optimization, DASA algorithm, Stigmergy.

Abstract: This paper deals with the analysis for two types of elitist variant proposed for the DASA algorithm. It is

usual for the genetic algorithms to keep the best solution found in the population used from next generation.

Another way to insert elitist behaviour in algorithms that construct solution is to use the most attractive

components in order to obtain god quality solution, and may be the optimal ones. Based on particularities of

ant colony based metaheuristics these two types of elitist behaviour were successfully applied to DASA

algorithm. In this paper the efficiency of the proposed elitist variants of DASA algorithms is analyzed using

experimental results. The analysis is applied to six benchmark functions from the class of high-dimensional

real-parameter optimization problems.

1 INTRODUCTION

When a metaheuristic proves to be effective for a

class of optimisation problems, one of the logical

steps forward is to be adapted to solve other class of

problems. Ant Colony Optimization (ACO) is a

metaheuristic which has passed such a process. ACO

was originally developed for combinatorial

optimization problems such as Travel Salesman

Problem (TSP) (M. Dorigo et al., 1996) or Single

Machine Total Weighted Tardiness problem (Bauer

et al. 1999). After that, several variants of algorithms

which use a pheromone mediated communication

have been proposed to solve real parameter

optimization problems. The continuous ant colony

optimization (CACO) (Bilchev and Parmee, 1995)

was the first proposed adaptation of Ant System

metaheuristic to continuous search space. CACO

initialize the population of ants with the same

solution (nest) and generate random directions

which will be followed by ants in their search. If an

ant improves the fitness function, the used direction

is updated. The API algorithm was proposed in

2000(Monmarche et al, 2000). Here, all ants start

from the nest and each of them search independently

for solution. This algorithm also uses a recruitment

strategy to refine the search. In 2004, Socha

proposes ACOԹ (Socha, 2004) that uses a

population including the n best solutions found so

far by ants, to probabilistically sample the search

space. Finally, Korosec proposed in 2006 the

differential ant-stigmergy algorithm (DASA)

(Korosec, 2006). This algorithm uses one solution

which is improved iteratively. In DASA, the ants do

not operate on the search space, but on the space of

differences that will modify the current solution.

This algorithm was successfully applied to high-

dimensional benchmark functions.

In this paper, two elitist variants of DASA

algorithm are proposed. Keeping the best found so

far solution in the population is usual in genetic

algorithms implementations. Using the most

attractive components in construction of the solution

is another way to insert elitist behaviour in

algorithms. This type of elitist behaviour was

successfully applied in ant based metaheuristics. The

motivation for appealing to elitism is, as usual, the

desire to increase the speed of convergence towards

promising areas of search space. Based on the

special properties of the DASA algorithm, both

approaches were investigated.

The second section of this paper presents the two

elitist proposed variants of the DASA algorithm.

The presentation starting point is the basic form of

the DASA algorithm and the improvement of the

optimisation strategies are formulated as two elitist

derivate behaviours.

The third section of the paper named

Experimental results is structured in four subsections

136

Emanoil ¸Serbencu A., Minzu V., ¸Serbencu A. and Cernega D..

TWO ELITIST VARIANTS OF DIFFERENTIAL ANT-STIGMERGY ALGORITHM.

DOI: 10.5220/0003539301360141

In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 136-141

ISBN: 978-989-8425-74-4

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

presenting the experimental environment and the

benchmark functions, the algorithm parameter

settings, the testing procedure and the obtained

results. The last section is dedicated to conclusion

and future work.

2 PROPOSED VARIANTS

OF DIFFERENTIAL

ANT

-STIGMERGY

ALGORITHM

2.1 Basic Form of DASA

Differential ant-stigmergy algorithm (DASA) uses a

fine-grained discrete form of the continuous spaces

(Korosec, 2006). For each direction of the search

space, the difference, which can be applied to the

current solution, may take a value from a finite set of

discrete values. Each discrete value of the difference

is attached to a node in a graph. In metaheuristics

based on ant metaphor, nodes in the graph are

associated with pheromone values, which will

measure their attractiveness. In DASA algorithm,

the nodes form level j of the construction graph

corresponds to j direction (dimension) of the search

space. To each level in the graph is associated a

pheromone distribution function, which correspond

to a Cauchy Probability Density Function (PDF)

()

⎛⎞

−

⎛⎞

⋅⋅ +

⎜⎟

⎝⎠

(1)

where l is the location offset and s=s

global

- s

local

is a

scale factor.

An ant constructs a path in the graph by sampling

the PDFs. The constructed path corresponds to a

difference vector

{

}

δδδ

,...,...,

=Δ

which specify the amplitude of the move in the

search space. Adding to temporary solution the

vector Δ with the values corresponding to path

constructed by an ant weighted with a random

values, generates the solution X={x

,…. x

}

that ant with

jjjj

+= '

(2)

where - x

is the j component of ant solution;

- x’

is the j component of temporary best solution;

- the weight ω

is a random integer number draw

from {1, 2,…, (b-1)};

is the sampled offset step.

Table 1: Percent of optimum selection.

Number of choice

10.000

100.000

function optimum selection [%]

f1 11.34 16.37

f2 11.75 4.37

f3 11.26 4.21

f4 10.98 9.79

f5 11.76 6.77

f6 12.06 11.80

mean 11.52 8.88

After an improvement of the current solution, the

pheromone is redistributed by centering the Cauchy

PDFs on the differences that generated the

improvement. In each of algorithm iteration, the

parameters s

global

and s

local

are updated with the aim

to balance between exploration of the search space

and exploitation of a promising area.

Experimental results with the percent in which the

ants chose the node corresponding to the maximum

of pheromone are presented in table 1. It may be

noted that ants, by sampling the Cauchy distribution,

select the node corresponding to the value in that is

centered the PDF on average only in 10% of the

choices made. This means that ants do not

effectively use the information memorized in

pheromone trails. So the next two variants of elitist

behavior were inserted in DASA Algorithm.

2.2 Pure Elitist DASA

In this variant of DASA algorithm, one of the m ants

will use the same path/differences from the previous

iteration, if that iteration has generated an

improvement. In DASA, the differences

corresponding to the path constructed by an ant, are

weighted with a random value. The variant of

algorithm, that use in (2) for elitist ant a weight

random generated is named DASA-elitist-A

algorithm. In the case that the elitist ant use the same

weight that generate the improvement, at the

previous iteration, the variants of algorithm is named

DASA-elitist-B.

2.2 Probabilistic Elitist DASA

The probabilistic elitist DASA approach directly

controls the percent in which an ant chose the node

corresponding to the maximum value of pheromone.

This type of elitism was successfully applied in Ant

Colony System for Travelling Salesman Problem. In

this variant of DASA algorithm, every ant chose

with probability α∈(0, 1) the node in which the

Cauchy probability density function is centered.

TWO ELITIST VARIANTS OF DIFFERENTIAL ANT-STIGMERGY ALGORITHM

137

With complementary probability (1 - α), an ant

chose a node by sampling the Cauchy PDF. This

variant of algorithm is named DASA-elitist-C.

The parameter α controls the importance of the

information given by ant pheromones. If α is small,

the ants are able to achieve more choices different

from the optimal value on which is centered Cauchy

distribution. The choices made by ants, however, are

not purely random but are also based on the Cauchy

distribution, thus achieving a oriented search.

This type of elitism increases the local search. In

the case of DASA-elitist-C algorithm, the local

search action on the difference vector. So, this local

search tries to keep the same speed in improving the

current solution and not to search around the current

solution.

3 EXPERIMENTAL RESULTS

3.1 The Experimental Environment

and the Benchmark Suite

The computer platform used to perform the

experiments was based on Intel dual core 2.13-GHz

processor, 2 GB of RAM. The DASA was

implemented in VisualC.

The proposed variants of DASA algorithm was

tested on a set of six benchmark functions defined

for CEC 2008 Special Session on Large Scale

Global Optimization. The six functions are sphere,

Schwefel, Rosenbrock, Rastrigin, Griewank and

Ackley. To prevent exploitation of the symmetry of

the search space and of the typical zero value

associated with the global optimum, local optima of

these functions are shifted to values different from

zero, and the function values in the global optima

are non-zero. A definition of them can be found in

(Tang et al, 2007). The six functions are defined on

a search space with D=100 dimensions (number of

parameters) and the minima is searched.

3.2 Algorithm Parameter Settings

The DASA has six parameters: the number of ants,

m; the discrete base, b; the pheromone dispersion

factor, ρ; the global scale-increasing factor, s+; the

global scale-decreasing factor, s-; and the maximum

parameter precision, ε. For the basic form of DASA

and for its elitist variants it was used the default

parameter settings: m = 10, b = 10, ρ= 0.2, s+= 0.02,

s-= 0.01, and ε = 1.0E-15. This values was selected

based on recommended values (Korosec, 2006).

3.3 Testing Procedure

The experimental results are recorded over 25 trials

on each pair, benchmark function and algorithm.

Every trial used different seed for random number

generator.

The function error, Error=f(x)-f(x*), where x* is

the optimum, is recorded after 50xD, 500xD, and

5000xD function evaluations (FEs). The Error is

collected for n= 25 runs and then the trials are

ordered from best to worst. The results of the 1st

(Best) and 25th (Worst) trial, as well as the trial

mean (Mean), standard deviation (StDev) and root

relative squared error (RRSE) are presented in tables

3 and 5. Here, the RRSE is defined as:

()

Error

RRSE

rror Mean

−

∑

3.4 Results

To evaluate the quality of elitist ants in pure elitist

versions of DASA, elitist ant's performance was

compared with those of a normal ant. In tests

performed, it was counted the number of iteration in

which the elitist ant is the best, and number of

improves of temporary solution generated by elitist

ant. In table 2 is presented as a percentage the

efficiency of the two types of pure elitist ants.

If we consider the basic form of DASA

algorithm, all ants are equals and they have the same

chance to be the best of the iteration. Therefore the

chance of one from the 10 ants, used in tests, to be

the best of the iteration should be around 10%.

Table 2 shows that elitist-A ant is the best of the

iteration in 35.01% of iterations. This result is

repeated in the case of the percentage of temporary

solution improvements generated by elitist-A ant

reported to the total number of improvements. The

average percentage for the 6 functions is 35.49%.

The elitist-B ant got better results. Thus, in

41.83% of iterations the elitist-B ant is the best of

iteration. If a normal ant is the best of the iteration

on average in a percentage (100% -41.83%) / (m-1)

= 6.46% iterations, that means the elitist-B ant is

better by 41.83/6.46 = 6.47 times than a normal ant.

The error evolution presented in table 3, prove

that the elitist-A and elitist-B maintain the

convergence of DASA for the six functions

considered in test. The recommended number

function evaluations, 50xD, 500xD and 500xD, to be

used in paper for CEC 2008 Special Session on

Large Scale Global Optimization, do not permit to

rank the DASA variants. For all 3 variants of DASA

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

138

Table 2: The quality of elitist ant.

Alg.

Measure

Function

f1 f2 f3 f4 f5 f6 Mean

DASA-

elitis

-A

(1) number of iteration in which the elitist ant is the best of iteration [%] 33.93 43.84 37.88 30.86 33.29 30.25 35.01

(2) number of iteration in which the elitist ant improves the temporary best

solution [%]

21.39 31.80 24.60 19.71 20.89 18.97 22.89

(3) number of iteration in which the temporary best solution is improved [%] 62.47 66.97 66.91 63.08 62.02 64.28 64.29

(4) percentage of improvements generated by elitist ant 2/3 [%] 34.24 47.48 36.77 31.25 33.68 29.51 35.49

DASA-elitis

(1) number of iteration in which the elitist ant is the best of iteration [%] 39.90 51.60 44.54 37.56 40.19 37.19 41.83

(2) number of iteration in which the elitist ant improves the temporary best

solution [%]

26.42 39.10 30.50 24.76 26.58 24.38 28.62

(3) number of iteration in which the temporary best solution is improved [%] 62.67 66.99 66.92 63.19 62.62 64.39 64.46

(4) percentage of improvements generated by elitist ant (2)/(3) [%] 42.16 58.37 45.58 39.18 42.45 37.86 44.27

Alg.

Measure

Function

f1 f2 f3 f4 f5 f6 Mean

DASA-

elitis

-A

(1) number of iteration in which the elitist ant is the best of iteration [%] 33.93 43.84 37.88 30.86 33.29 30.25 35.01

(2) number of iteration in which the elitist ant improves the temporary best

solution [%]

21.39 31.80 24.60 19.71 20.89 18.97 22.89

(3) number of iteration in which the temporary best solution is improved [%] 62.47 66.97 66.91 63.08 62.02 64.28 64.29

(4) percentage of improvements generated by elitist ant 2/3 [%] 34.24 47.48 36.77 31.25 33.68 29.51 35.49

DASA-elitist-B

(1) number of iteration in which the elitist ant is the best of iteration [%] 39.90 51.60 44.54 37.56 40.19 37.19 41.83

(2) number of iteration in which the elitist ant improves the temporary best

solution [%]

26.42 39.10 30.50 24.76 26.58 24.38 28.62

(3) number of iteration in which the temporary best solution is improved [%] 62.67 66.99 66.92 63.19 62.62 64.39 64.46

(4) percentage of improvements generated by elitist ant (2)/(3) [%] 42.16 58.37 45.58 39.18 42.45 37.86 44.27

analyzed in table 3, the mean of error over 25 trials

are under 1E-10 for function f1, f4, f5 and f6, after

500000 FEs. Usually, it is considered that for an

error under 1.E-9 the searched optima is founded, so

the 3 DASA variants from table 3 are equivalent for

function f1, f4, f5 and f6. For this 4 function, a

supplementary test, record the number of iteration

needed to obtain an error under 1.E-9. The minimum

number of iterations, the maximum number of

iterations and the mean number of iterations over 25

trials are presented in table 4. If we analyze the

mean number of iteration over the 25 trails, than we

can observe that the standard variant of DASA

perform better like elitist variants for function f1 and

f6. For function f4 and f5 the DASA elitist-A variant

performs better.

For function f2 and f3, that have non-separable

parameters, the performance of standard DASA and

elitist-A DASA are equivalent.

The least performing, between the three variant of

DASA, is the elitist-B. However, the performance

difference is not significant. The errors obtained by

DASA elitist-B have the same order of magnitude

with those of the other two variants of DASA.

The experimental results for DASA-elitist-C are

presented in table 5 and 6. Evolution of the average

error obtained after 500,000 FEs shows for all six

functions that a higher value of alpha increases the

algorithm convergence. Analysis of the results table

5 recommends for α a value of 0.8. The performance

of DASA elitist-C is equivalent to those of DASA

elitist-B, the error after 500,000 Fes having the same

order of magnitude.

The maximum, minimum and average numbers

of iterations required by DASA-elitist-C algorithm

to obtain an error less than 1E-9 are given in table 6.

The associated execution time is also recorded. The

time needed by DASA elitist-C, to obtain an error

less than 1E-9, decrease when parameter α

increases. This is happening because of the time

needed to sample Cauchy PDF, that is non negligible

if it is compared with time needed to evaluate the

optimized function. The results from table 6

recommend also a value of α=0.8

4 CONCLUSIONS

The tests results prove that using elitism in DASA

algorithm can improve for same function the

convergence of algorithm. The expected results for

the elitist strategies are confirmed through the

experimental results, presented in the tables above.

The number of iterations necessary to reach the

optima is smaller for the DASA-elitist-A, and

DASA-elitist-B. The future work is to analyze the

use of booth type of elitism in parallel.

TWO ELITIST VARIANTS OF DIFFERENTIAL ANT-STIGMERGY ALGORITHM

139

Table 3: Error values produced with DASA standard, DASA elitist-A and DASA elitist-B for function f1-f6.

FEs Error Algorithm

Functions

f1 f2 f3 f4 f5 f6

5,000

Best

standard

2,21E+003

6,03E+001

7,70E+007 1,35E+002 7,40E+000 9,25E+000

elitist-A 3,00E+003 6,62E+001 1,17E+008 1,47E+002 9,41E+000 9,38E+000

elitist-B 2,57E+003

5,99E+001

1,20E+008 1,43E+002 1,09E+001 1,02E+001

Worst

standard

5,40E+003 7,80E+001 3,25E+008 2,06E+002 2,00E+001

1,75E+001

elitist-A 6,87E+003 8,91E+001 5,08E+008 2,47E+002 2,73E+001 1,75E+001

elitist-B 6,00E+003 8,91E+001 5,99E+008 2,54E+002 2,85E+001

1,63E+001

Mean

standard

3,51E+003 6,96E+001 1,70E+008 1,80E+002 1,37E+001

1,34E+001

elitist-A 4,38E+003 7,47E+001 2,56E+008 1,91E+002 1,60E+001

1,31E+001

elitist-B 4,40E+003 7,62E+001 3,59E+008 1,99E+002 1,73E+001 1,35E+001

StDev

standard

8,24E+002 4,14E+000 6,65E+007 1,73E+001 3,12E+000

1,97E+000

elitist-A 1,02E+003 5,99E+000 9,00E+007 2,44E+001 4,51E+000 2,19E+000

elitist-B 9,04E+002 7,37E+000 1,08E+008 2,75E+001 4,78E+000

1,71E+000

50,000

Best

standard

7,49E-011 1,18E+001

1,55E+002

4,24E-009 1,27E-011 4,05E-006

elitist-A 3,81E-009 1,84E+001

1,49E+002

4,90E-009 4,53E-010 1,56E-005

elitist-B 5,62E-008 2,03E+001 1,79E+002 1,93E-007 8,91E-009 4,36E-005

Worst

standard

2,95E-009 1,69E+001

1,65E+004 1,99E+000 7,07E-002

3,07E-005

elitist-A 1,10E-007 2,55E+001 1,61E+004 3,20E+000

6,58E-002

9,83E-005

elitist-B 2,57E-006 2,71E+001 1,61E+004 1,99E+000 7,86E-002 3,08E-004

Mean

standard

7,33E-010 1,43E+001 3,81E+003 4,78E-001

1,14E-002

1,15E-005

elitist-A 2,53E-008 2,16E+001 5,44E+003 6,06E-001

8,14E-003

3,86E-005

elitist-B 5,09E-007 2,41E+001 3,90E+003 7,96E-001 1,11E-002 1,36E-004

StDev

standard

5,95E-010 1,17E+000 4,99E+003 5,72E-001

1,79E-002

6,53E-006

elitist-A 2,43E-008 1,57E+000 6,78E+003 8,70E-001

1,45E-002

2,03E-005

elitist-B 5,16E-007 1,74E+000 5,94E+003 7,96E-001 1,95E-002 6,80E-005

RRSE

standard 1,59E+000 1,23E+001 1,26E+000 1,30E+000 1,19E+000 2,03E+000

elitist-A 1,44E+000 1,38E+001 1,28E+000 1,22E+000 1,15E+000 2,15E+000

elitist-B 1,40E+000 1,39E+001 1,20E+000 1,41E+000 1,15E+000 2,23E+000

500,000

Best

standard

3,52E-012 1,67E-002

2,21E-001

5,80E-012

3,75E-012 6,11E-012

elitist-A 5,29E-012 7,19E-001

6,30E-002

6,76E-012

3,64E-012

7,30E-012

elitist-B 5,17E-012 1,55E+000 2,49E-001 6,42E-012 3,67E-012

6,05E-012

Worst

standard 1,42E-011

3,54E-002

1,41E+003 2,73E-011

1,17E-011

1,15E-011

elitist-A 1,46E-011 1,10E+000 1,30E+003 1,99E-011 1,43E-011 1,19E-011

elitist-B

1,24E-011

2,11E+000

5,43E+002 1,92E-011

2,15E-011

1,13E-011

Mean

standard 9,51E-012

2,44E-002 1,66E+002

1,19E-011 6,55E-012

8,17E-012

elitist-A 9,47E-012 8,47E-001 2,52E+002 1,22E-011 6,74E-012 9,07E-012

elitist-B

8,54E-012

1,86E+000 1,77E+002

1,10E-011 6,06E-012

8,19E-012

StDev

standard 2,49E-012

4,39E-003

2,77E+002 4,96E-012

1,84E-012 1,07E-012

elitist-A 2,34E-012 8,59E-002 3,24E+002 3,76E-012 2,54E-012 1,31E-012

elitist-B

2,26E-012

1,35E-001

1,37E+002 3,66E-012

3,51E-012 1,37E-012

RRSE

standard 3,95E+000 5,66E+000 1,17E+000 2,60E+000 3,70E+000 7,71E+000

elitist-A 4,17E+000 9,91E+000 1,27E+000 3,39E+000 2,84E+000 6,98E+000

elitist-B 3,92E+000 1,39E+001 1,63E+000 3,16E+000 1,99E+000 6,05E+000

Table 4: Number of iteration needed by DASA elitist-A and DASA elitist-A to obtain an error under 1E-9.

Number of

iteration

Function

Algorithm f1

relativ[%]

f4 relativ[%] f5 relativ[%] f6 relativ[%]

Minimum

standard

46.951

0,00

52.301

0,00

44.081

0,00

75.371

0,00

elitist-A 52.382

11,57

53.072

1,47

48.732

10,55

81.072

7,56

elitist-B 57.211

21,85

57.111

9,20

55.381

25,63

88.701

17,69

Maximum

standard 51.781

0,00

124.222

0,00

263.975

14,01

82.981

0,00

elitist-A 57.812

11,65

144.583

16,39

231.545

0,00

93.962

13,23

elitist-B

64.291

24,16

232.023

86,78

>500.000

>115,94

100.351

20,93

Mean

standard

49.195,80

0,00

71.870,76

0,79

103.731,64

7,29

79.688,60

0,00

elitist-A 55.054,40

11,91

71.306,48

0,00

96.685,56

0,00

86.699,20

8,80

elitist-B 61.141,00

24,28

78.804,72

10,52

136.245,36

40,92

94.807,80

18,97

ACKNOWLEDGEMENTS

This work was supported by the CNCSIS –

UEFISCDI, project number IDEI-506/2008.

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

140

Table 5: Error values produced with DASA standard and DASA elitist-C for function f1-f6.

FEs Error

Algorithm Function

f1 f2 f3 f4 f5 f6

500,000

Best

standard

3,52E-012 1,67E-002

2,21E-001 5,80E-012 3,75E-012 6,11E-012

elitist-C

0,5 6,54E-012 4,38E-002 1,85E-001 8,07E-012

2,76E-012

6,34E-012

0,6 4,32E-012 5,29E-002 2,18E-001 7,05E-012 3,21E-012 6,51E-012

0,7 5,46E-012 1,13E-001

5,88E-003

7,16E-012 3,38E-012

6,00E-012

0,8 3,69E-012 3,20E-001 3,26E-001 7,05E-012 3,10E-012 6,48E-012

0,9 4,49E-012 2,20E+000 1,40E+001

5,06E-012

3,67E-012 6,99E-012

Worst

standard 1,42E-011

3,54E-002

1,41E+003 2,73E-011 1,17E-011 1,15E-011

elitist-C

0,5 1,72E-011 9,09E-002 1,33E+003 2,25E-011

8,92E-012 1,10E-011

0,6

1,27E-011

1,17E-001 1,29E+003 2,43E-011 1,34E-011 1,14E-011

0,7 1,49E-011 2,27E-001 7,44E+002 2,16E-011 9,86E-003

1,10E-011

0,8 1,35E-011 6,40E-001 1,22E+003

2,02E-011

1,21E-011 1,39E-011

0,9 1,57E-011 3,83E+000

5,66E+002

2,12E-011 1,83E-011 1,42E-011

Mean

standard 9,51E-012

2,44E-002 1,66E+002

1,19E-011 6,55E-012

8,17E-012

elitist-C

0,5 1,05E-011 6,46E-002 3,00E+002 1,40E-011

5,71E-012

8,74E-012

0,6 8,90E-012 9,30E-002 1,75E+002 1,25E-011 5,93E-012 8,46E-012

0,7 9,61E-012 1,70E-001 1,85E+002 1,26E-011 6,90E-004 8,70E-012

0,8

8,72E-012

4,47E-001 2,59E+002 1,18E-011 6,15E-012 9,28E-012

0,9 9,41E-012 2,97E+000 2,01E+002

1,15E-011

6,49E-012 9,78E-012

Table 6: Number of iteration needed by DASA elitist-C to obtain an error under 1E-9.

Function f1 f4 f5 f6

FEs time[s] FEs time[s] FEs time[s] FEs time[s]

Minimum

0,5 48.211 1,9960 49.111 2,4960 44.251 2,6520 75.951 3,9630

0,6 47.611 1,8560 52.931 2,5740 44.651 2,5430 78.321 3,8220

0,7 48.261 1,7470 54.771 2,5270 44.641 2,4490 77.551 3,5880

0,8 48.661

1,6530

55.451 2,4180 46.111

2,4180

80.711

3,5410

0,9 53.331 1,6690 55.391

2,2780

51.341 2,5740 86.161 3,5720

Maximum

0,5 52.701 2,2000 119.482 6,2710 328.526 20,7480 84.241 4,5710

0,6 52.921 2,0440 123.682 6,1460 392.797 23,5870 84.391 4,1180

0,7 54.011 1,9500 141.902 6,5990 >500.000 >19,0630 86.801 4,0090

0,8 54.981 2,4650 146.092 6,5050 350.256 18,9700 89.361 4,4150

0,9 59.631 2,4500 214.523 9,1100 376.516 20,0770 96.571 4,1810

Mean

0,5 50.041,40 2,0802 63.460,24 3,2392 84.289,28

5,1910

80.400,20 4,2001

0,6 50.349,00 1,9524 72.306,76 3,5294 103.546,80 6,0865 81.451,00 3,9786

0,7 50.755,80 1,8377 67.517,44 3,1000 >139.199,08 7,5730 82.566,60 3,8201

0,8 52.055,40

1,7890

68.731,88 3,0052 108.038,00 5,8419 83.873,40

3,7091

0,9 56.746,60 1,8433 70.185,88

2,9072

132.655,44 6,8709 92.127,00 3,8401

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