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1
An Efficient Population-Merging Execution
Model, for Evolutionary Algorithms
Javier Arellano-Verdejo, Salvador Godoy-Calderón, Adolfo Guzmán-Arenas, Senior Member, IEEE

Abstract—In this paper a coarse-grain execution model for
evolutionary algorithms is proposed and used for solving
numerical optimization problems. This model does not use
migration as the solution dispersion mechanism, in its place a
more efficient population-merging mechanism is used that
dynamically reduces the population size as well as the total
number of parallel evolving populations (i.e. islands). Even more
relevant is the fact that the proposed model eliminates the need
to set up an interconnection topology among islands. Extensive
numerical experimentation, using genetic algorithms over a
well-known set of typo problems, shows the proposed model to
be faster and more accurate than the traditional one.
Index Terms—Evolutionary algorithms, Genetic algorithms,
Optimization, Parallel algorithms, Parallel island model,
Spatially structured evolutionay algorithms.
I. INTRODUCTION
During the last few decades several evolutionary
algorithms (EAs) have been successfully applied in the
solution of many optimization problems [6]-[9]. Nevertheless,
as problems become more and more complex, the execution
time of these algorithms increases, making them prohibitively
slow for single-core execution. Of course, the populationbased nature of evolutionary algorithms, along with the
current accelerated development on multi-core hardware
technologies, has turned EAs into strong candidates for
parallel implementation. Besides the differences in the way
each algorithm manages its own heuristic search for optimum
values, all EAs share the need to evaluate the fitness of a
possibly large population, as well as the need to balance their
own mechanisms for global and local search (i.e exploration
and exploitation) over the search-space. Thus, several parallel
execution models have been proposed for virtually all types of
EAs [10]-[12], [14].
J. Arellano-Verdejo, S. Godoy-Calderón are with the Laboratorio de
Inteligencia Artificial, Centro de Investigación en Computación (CIC) del
Instituto Politécnico Nacional (IPN). Av. Juan de Dios Bátiz, Esq. Miguel Othón
de Mendizábal, Col. Nueva Industrial Vallejo Delegación Gustavo A. Madero
C.P
07738,
México
(email:
[email protected],
[email protected]).
Adolfo Guzmán Arenas is with the Laboratorio de Bases de Datos y
Tecnología de Software, Centro de Investigación en Computación (CIC) del
Instituto Politécnico Nacional (IPN), (email [email protected]).
Parallel evolutionary algorithms (PEAs) have been
classified into three groups according to their execution
model [5], [13], [15] (See figure 1): master-slave PEAs,
coarse-grain PEAs (also known as island-model PEAs) and
fine-grain PEAs (also known as cellular-diffusion PEAs).
Although the master-slave model offers a practical
organization and structure for the tasks of the algorithm, it
turns to be a terrible waste of the parallel capabilities of the
running system. In contrast, the fine-grain model requires
too much system capacity and provides no extra mechanisms
for controlling the way in which the search space is to be
explored and exploited. The island-model appears to be a
natural balance between requirements and performance, one
that allows leveraging the capabilities of the underlying
parallel system and at the same time retain the ability to
adjust the exploration/exploitation behavior of the EA.
One peculiar characteristic of traditional island-based
PEAs is that the population of each island is randomly
selected from an initial uniformly distributed random
population [16]. Such a selection method implies that an
island’s population might not be representative of any
particular region of the search space, and that it can
theoretically span over almost the same search space than the
original population but with significantly less individuals.
Since in the island-model the evolution process of each island
is executed by a different core, the performance of the
underlying parallel system is notoriously wasted, and the
algorithm is left dangerously prone to premature
convergence. Seemingly, this model uses parallelism only to
perform the search in less time, without this implying any
improvement in the way in which the problem search space is
explored or exploited.
As a way of compensating, a migration mechanism is used
to disseminate the best fitted solutions among all islands.
While this mechanism can be extremely effective in some
specific situations when properly configured [1]-[3] it
imposes the need to setup an interconnection topology among
the islands (i.e star, ring, toroid, etc.). This implies the
forcible inclusion of an extra parameter into the execution
model, one whose choice has a strong impact on the way the
search for the optimal value develops, as well as on the
overall performance of the evolutionary algorithm itself.
Unfortunately there are no easy to follow or universally
accepted criteria to make that choice.
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2
To reduce the number of parameters that need to be set, T.
Hiroyasu et al [19] propose an algorithm called DuDGA
(Dual Distributed Genetic Algorithm) which defines islands
with only two individuals in each of them. After the only two
individuals on each island are used for a one-point crossover
operation and their descendants are mutated, only the two
best fitted individuals (ancestors or descendants) survive for
the next generation. Unfortunately, the final result’s quality,
as well as the overall performance of this EA, turns out to be
very poor as it still depend on a proper choice for the rest the
parameters.
Fig. 1. a) Master-Slave model, b) Fine-Grain model
c) Coarse-Grain model
In this paper a new coarse-grain parallel execution model
is introduced, which was originally inspired by the traditional
island-based model but does not use migration as a solution
dispersion mechanism and consequently eliminates the need
to setup an interconnection topology among islands. In its
place a population merging mechanism is used with the
option to use several different criteria for selecting island
candidates to merge their populations. This option of the
model provides by itself an extra control layer that adds to the
exploration/exploitation capabilities of the running EA.
The rest of this paper is organized as follows: in Section II
related works are presented, in Section III, the proposed new
model is described, detailing each of the characteristics that
make it different and more versatile compared to a traditional
migration-based island-model. Section IV shows a set of
comparative experiments run with genetic algorithms over
the same populations in a migration-based parallel model and
in the proposed population-merging model. Finally, in
Section V, conclusions are offered.
II. RELATED WORKS
Every kind of EA requires multiple parameters to properly
function. The correct choice for the value of each parameter
provides the algorithm with the right balance of exploration
and exploitation capabilities. When implemented in parallel,
the resulting PEA works as a high level operational layer,
thus needing its own set of parameters. It is quite clear that
the larger the number of parameters that must be set for the
execution of the algorithm, the greater the difficulty to find a
combination of the right values for them. In fact, the selection
of parameters has been posed as an optimization problem
itself, with multiple solution proposals as in [4], [17], [18].
In [20], H. Sawai et al describe an algorithm called PfGA
(Parameter free Genetic Algorithm) in which the population
size changes as a function of the fitness value of individuals.
From every generation only two parents are selected for
multi-point crossover and there is no need to setup any other
parameters. Even traditional genetic algorithm’s parameters
like crossover and mutation rates are not needed since the
selection strategy always keeps a small population and a high
performance. Although the PfGA itself requires no
parameters, the parallel execution layer does and in fact, in
[21] it is shown that migration plays an essential role as a tool
to improve global search, yet the authors clearly showed that
the optimal migration model is different for each problem.
The same type of reduced population (two individuals) is
used in [22], where authors introduce an algorithm called
VIGA (Variant Island Genetic Algorithm). This algorithm
works with a dynamic number of islands and explicit
mechanisms called resize, copy and create operations to
incrementally take advantage of the convergence status of the
search process. Although experimental results show VIGA to
be computationally efficient, it relies on global fitness
information, which implies high communication overhead
when implemented as a distributed parallel algorithm. Also,
the fact that a new individual or a mutant is created with low
probability means that the parallel execution model
dangerously decreases the variance of the population and
increases the risk of premature convergence.
Parallel execution models have been proposed for many
kinds of EAs other than genetic algorithms. In [23], for
example, de la Ossa et al, describe a family of island-based
EDAs (Estimation of Distribution Algorithms) and use them
to solve numerical optimization problems. The authors select
star and ring topologies to experiment with and add a new
level of information exchange between islands when using
both individual and model migration. Also in [23], an
empirical comparison of several parallel Particle Swarm
Optimization (pPSO) algorithms is shown.
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The alternative execution model proposed in this paper
does not use migration nor a pre-defined connection topology
among islands, so the dissemination of the solutions is not
dependent on the interconnection topology selected. The
migration process has been replaced by a population-merging
mechanism between islands, which allows a dynamic way of
handling the number of islands as well as the size of the
global population. All those tunings and new elements have
resulted in a dramatic reduction in the number of times the
fitness function is evaluated along the development of the
execution and endows the proposed model with a better
control of the way in which the search space is explored and
exploited.
III. MERGING POPULATIONS
The Population-Merging Island Model (PMIM) proposed
herein slightly modifies the above parameter list and operates
on the parameters shown in Table 1.
It can be seen that in the PMIM the global
population size monotonically decreases over the course of
the heuristic search. This is achieved by eliminating all the
f population
g (ni )k
times, where
total number of generations the algorithm evolves,
population size of island
Ii
and
k
Parameter
Semantics
Initial population’s size.
Initial number of islands.
Number of generations to evolve before
searching for merging candidates.
Maximum number of evolution/fusion
rounds.
Island selection criterion for the merger.
n
k
m
r
fisland
Population’s replacement criteria for
merged islands.
f population
g
ni
fisland
The use of criterion
1) Create
a
uniformly distributed
P
population
2) Create
k
with
3) Execute evolutionary algorithm
of each island
with populations
Ii , i  1..k
A
during
on the population
m generations.
4) Pause evolution on all islands and using islandselection criterion,
fisland
select two or more
islands for merging their populations.
is the
5)
Merge the population of all selected islands into a
If
In contrast, the PMIM starts with the same population size
for each island but keeps it only during m generations.
When the merging process takes place the population of all
selection criterion
is merged and subsequently
initial
P1,..., Pk  randomly selected from P .
new island
fisland
random
individuals.
I1,..., I k 
islands
islands.
islands selected by
n
is the
is the total number of
for selecting islands to be
merged eliminates the need to setup interconnection
topologies while allowing more freedom in the information
exchange mechanism. If the islands selected for merging
have a similar fitness variance of their populations then
exploitation is promoted, but if islands with notorious
population fitness variances are selected, then more
exploration is induced into the heuristic search. All of the
above happens independently of the EA’s own mechanisms
for exploration and exploitation. The PMIM model is
specified as follows:
criterion
after the merging process. Consequently, as the global
population size decreases, so does the number of times the
fitness function is evaluated and, since the total number of
islands also decreases, one execution core is freed each round
for the parallel system to use for other tasks. In a traditional
migration-based island-model both the global population size
and the number of islands are kept constant, so the fitness
function is evaluated exactly
TABLE I
PMIM PARAMETERS
reduced as specified by the f population criterion.
It is a well-known fact that the quality of the solution found
by any EA is directly dependent on the value of its
parameters. In addition, the traditional island-model requires
the following parameters: number of populations to evolve in
parallel (i.e. number of islands), size of the population on
each island, interconnection topology and migration
frequency. There is also the need to establish a migration
policy to determine the number and selection of individuals
who will migrate, as well as a replacement policy for the
newly arrived individuals on each island.
individuals that were not selected by the
3
6)
Update
and reduce its population using
f population .
k  k 1 .
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7)
k  1 go back to step 3 and resume evolution on
all islands, else execute algorithm A on the last
island’s population during m more generations or
TABLE 2
PORTFOLIO OF TEST FUNCTIONS
If
until the preset termination conditions are met.
IV. EXPERIMENTAL RESULTS
Labe
l
Each problem was solved 50 times by the same parallel
genetic algorithm running separately in the traditional islandmodel and in the PMIM model. Each run generated exactly
the same initial population for both models so that results
could be fairly compared.
All experiments, a total of 1000 (10 functions, 50 runs and
2 models), were run on an 8-core Intel Xeon system with 32
GB RAM over a 64 bits Linux operating system. In both
models the evolution process for each island is run by a
different core.
Function name
and search interval
f1
100  xi  100
Schwefel’s problem 1
10  xi  10
f3
f4
f5
100  xi  100
Schwefel’s problem 3
100  xi  100
Generalized
Rosenbrock’s
30  xi  30
f6
100  xi  100
f7
1.28  xi  1.28
f9
For the PMIM model the f island criterion was set to
f10
However, results yield by each model are surprisingly close
to each other so one could be drawn to conclude that both
models are more or less equivalent in their precision. When
comparing such small quantities it is often best to work in
terms of percentage. Thus, second column of table 5 shows
the percentage of improvement implied in table 4. The third
column of table 5 shows the improvement percentage in
number of evaluations of the fitness function (remember that
in the traditional island-model the number of evaluations of
the fitness function is constant). As a direct consequence of
30
i 1
i 1
min( f 2 )  f 2 (0,...,0)  0
30  i

f3 ( x)     x j 
i 1  j 1

min( f3 )  f3 (0,...,0)  0
f 4 ( x)  max i | xi |,1  i  30
min( f 4 )  f 4 (0,...,0)  0
29
f5 ( x)  |100( xi1  xi2 )2  ( xi  1)2 |
i 1
min( f5 )  f5 (1,...,1)  0
f 6 ( x)   (| xi  0.5 |) 2
i 1
min( f 6 )  f 6 (0,...,0)  0
30
Quartic with noise
Generalized
Rastringin’s
Table number 4 shows the average best solution found by
the same GA running under each model with all test
functions. As it can be seen the PMIM model was closer to
the known optimum in all cases.
30
f 2 ( x)   | xi |  | xi |
30
Step
f8
criterion always selected the best fitted two thirds of the
individuals from a merged island. The rest of the parameters
used for both models are shown in table 3.
i 1
min( f1 )  f1 (0,...,0)  0
2
Schwefel’s problem 2
In the case of the traditional island-model a ring topology,
as is the most common case, was used. Migration is
accomplished by copying the eight best fitted individuals to
other islands according to the selected interconnection
topology.
f population
f1 ( x)   xi2
Sphere model
Generalized
Schwefel’s problem
randomly select two islands each time and the
Function
30
f2
To test the performance of the PMIM the complete
benchmark set of ten numerical optimization problems
proposed by De Jong and extended by Coello et al [24] were
used.
The complete set of functions, along with a
specification of their search interval and their known
optimum value is shown in Table 2.
4
500  xi  500
5.12  xi  5.12
Generalized
Griewank’s
600  xi  600
f 7 ( x)   ixi4  random[0,1]
i 1
min( f 7 )  f 7 (0,...,0)  0
30

f8 ( x)   xi sin( | xi |)
i 1

min( f8 )  f8 (420.9687,...,420.9687)  12569.5
30
f9 ( x)    xi2  10cos(2 xi )  10 
i 1
min( f9 )  f9 (0,...,0)  0
1 30 2 30
x 
cos  i   1
 xi  
4000 i1
 i
i 1
min( f10 )  f10 (0,...,0)  0
f10 ( x) 
the fewest number of fitness function evaluations, runtime
also decreases significantly. The last column of table 5 shows
the improvement of the PMIM model over traditional
migration-based island model in execution time.
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TABLE 3
COMPARISON OF PARAMETERS USED BY THE TWO MODELS
TABLE 5
IMPROVEMENT OF THE PMIM MODEL OVER TRADITIONAL ISLAND-MODEL
Traditional
island-model
PMIM
model
Size of global population ( n )
Size of each island’s population
( ni )
1000
125
1000
dynamic
Number of generations to evolve
before migrating or merging
(m)
Maximum number of
evolution/migration or
evolution/fusion rounds ( r )
Initial number of islands
Island selection criterion for
migration or fusion ( f island )
100
100
Parameter
Replacement criterion for
emigrant individuals/merged
islands (
f population )
8
8
8
ring
topology
8
random
8 newly
arrived
individuals
replace the
8 less
fitted ones.
best
fitted
two
thirds of
merged
populati
on
º
TABLE 4
AVERAGE BEST SOLUTION FOUND BY EACH MODEL
Label
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
5
Known
optimum
Best individual in
PMIM
Best individual
in traditional
island-model
0
0.0000025
0.000019
0
0.0000075
0.000017
0
0.0000036
0.00001
0
0.0000072
0.00001
0
0.000005
0.000008
0
0.0000049
0.000008
0
0.0000000080
0.000000013
-12569.5
-12568.8972
-12554.2173
0
0.00000162
0.0000045
0
0.0000027
0.0000064
From table 5 it can be seen that the same algorithm, runs
faster and gets closer to the global optimum on the PMIM
model than on the traditional island-model. Those effects can
be attributed to the smaller number of evaluations of the
fitness function and the faster dispersion of solutions
generated by the population merging process in contrast to
the migration process. Table 6 sums up the PMIM’s average
percentage of improvement observed during the numerical
experiments.
Label
Improvement
in fitness of
best individual
(%)
Improvement in
number of
evaluations of
fitness function
(%)
Improvement in
execution time
(%)
76
41.2
10.91
22.66
41.33
8.86
27.22
41.27
11.67
13.88
41.12
11.18
16.6
41.17
9.95
15.30
41.29
11.55
16.25
41.47
11.88
9.99
41.16
9.83
27.77
41.3
14.12
23.7
41.11
10.34
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
TABLE 6
AVERAGE IMPROVEMENT OF THE PMIM MODEL
Average improvement
Average improvement
in number of
Average improvement
in fitness of best
evaluations of fitness
in execution time
individual
function
76%
41.24%
10.91%
V. CONCLUSIONS
A new coarse-grain (parallel) execution model for
evolutionary algorithms was presented. The new model,
called Population Merging Island Model (PMIM) replaces
migration with a population merging system as the solution
dispersion mechanism used by the traditional island-model.
In doing so, the PMIM model eliminates the need for
choosing an interconnection topology among islands, which
is one of the parameters with the greatest impact on the
accuracy and performance of the evolutionary algorithm.
When the population of two or more islands is merged and
then reduced, the global population size decreases, as well as
the number of evolving islands. The immediate effect of this
is a reduction in the number of evaluations of the fitness
function with the consecuent decrease in execution time.
Also the PMIM model offers an extra layer to help on the
exploration and exploitation control capabilities of the
heuristic search which results in a higher precision on the
quality of the found solution.
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Extensive numerical experimentation, using the benchmark
function suite proposed by De Joung and a GA as the inner
EA shows the PMIM model to be faster and more precise than
the traditional island-model (see Table 6).
Acknowledgements The authors express their gratitude to
the COFAA, SIP, and CIC units of the National Polytechnic
Institute IPN, as well as CONACyT and SNI, for their
economic support to carry out this work.
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[23] Luis de la Ossa, José A. Gámez and José M. Puerta: Initial approaches to
the application of island-based parallel EDAs in continuous domains. vol., no.,
pp.4698-4705, 25-28 Sept. 2007.
[24] Efrén Mezura-Montes, Jesús Velázquez-Reyes, and Carlos A. Coello
Coello. 2006. A comparative study of differential evolution variants for global
optimization. In Proceedings of the 8th annual conference on Genetic and
evolutionary computation (GECCO '06). ACM, New York, NY, USA, 485-492
Javier Arellano-Verdejo Recibed the Masters
degree in science from Tecnológico de Estudios
Superiores de Ecatepec Mexico in 2008. He is
currentrly pursing the Ph.D. degree in
Computation
Science
from
Centro
de
Investigación
en
Computación
Instituto
Politécnico Nacional Mexico City.
His current research interests include parallel
bioinspired heuristics , Estimation of Distribution
Algorithms and Pattern recognition.
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Salvador Godoy Calderón Ph.D. in Computer
Science at CIC-IPN and Master in C.S. by
CINVESTAV-IPN. Full time researcher and Head
of the Artificial Intelligence Laboratory at CICIPN with research interests including:
Pattern Recognition, Evolutionary Computation,
Strategic and Tactical Knowledge Systems and
Formal Logics.
Adolfo Guzmán-Arenas is a Computer Science
professor at Centro de Investigación en Computación, Instituto Politécnico Nacional, Mexico
City, of which he was Founding Director. He
holds a B. Sc. in Electronics from ESIME-IPN,
and a Ph. D. from MIT. He is an ACM Fellow,
IEEE Life Fellow Member, member of MIT
Educational Council, member of the Academia de
Ingeniería and the Academia Nacional de Ciencias
(Mexico).
From the President of Mexico, he has received
(1996) the National Prize in Science and
Technology and (2006) the Premio Nacional a la
Excelencia “Jaime Torres Bodet.” He works in
semantic information processing and AI
techniques, often mixed with distributed
information
systems.
More
at
http://alum.mit.edu/www/aguzman
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