A Grid-based Genetic Algorithm for Multimodal Real Function
Optimization
Jose M. Chaquet and Enrique J. Carmona
Dpto. de Inteligencia Artificial, Escuela Técnica Superior de Ingeniería Informática,
Universidad Nacional de Educación a Distancia, Madrid, Spain
Keywords:
Genetic Algorithm, Multimodal Real Functions, Grid-based Optimization, Integer-real Representation.
Abstract:
A novel genetic algorithm called GGA (Grid-based Genetic Algorithm) is presented to improve the optimiza-
tion of multimodal real functions. The search space is discretized using a grid, making the search process
more efficient and faster. An integer-real vector codes the genotype and a GA is used for evolving the pop-
ulation. The integer part allows us to explore the search space and the real part to exploit the best solutions.
A comparison with a standard GA is performed using typical benchmarking multimodal functions from the
literature. In all the tested problems, the proposed algorithm equals or outperforms the standard GA.
1 INTRODUCTION
Optimization of real problems are normally hard to
solve because deals with multimodal functions and
complex fitness landscapes. Issues as multiple local
optimum, premature convergence, ruggedness or de-
ceptiveness are some of the difficulties (Weise et al.,
2009). To face this kind of problems, the use of
Evolutionary Computing (EC) paradigms is very at-
tractive. One of the most used EC paradigms for
the tuning of multimodal real functions are Evolution
Strategies (ES). Real coding representation and self-
adaptation of the optimal mutation strengths make
ES suitable to these type of domains. However, in
this work, we want to investigate how to improve the
performance of a standard Genetic Algorithm (GA)
based on real-valued or floating-point representation.
In fact, since this type of representation was pro-
posed (Davis, 1991; Janikow and Michalewicz, 1991;
Wright, 1991), there have been many works in the lit-
erature that have been devoted to this purpose. Each
of them was focused in different aspects as, for ex-
ample, new mutation operators (Deep and Thakur,
2007b; Korejo et al., 2010), new crossover opera-
tors (Deep and Thakur, 2007a; Garcia-Martinez et al.,
2008; Tutkun, 2009), or new self-adaptive selection
schemes (Affenzeller and Wagner, 2005).
The main two ideas of this paper are to use an
integer-real vector for individual representation and
to discretize the search space by using a grid. Such
mixed representation allows breaking down the stand-
ard search process in two types of search made simul-
taneously. One of them is constrained to the grid and
allows making a global search in the domain tuning
the integer part (exploration). The other one tunes the
real part making a local search of the best individuals
(exploitation). The new algorithm implemented using
that methodology is called Grid-based Genetic Algo-
rithm (GGA).
The idea of using a grid to facilitate the search
process has been also reported in the so-called cell-
to-cell mapping method (Hsu, 1988). In a similar
way, other approaches based on the subdivision of the
search space into boxes were presented in (Dellnitz
et al., 2001). In both works, stochastic search is in-
troduced for the evaluation of the boxes, but each box
is considered once during the search process which is
not the spirit of GAs. On the other hand, a mixture of
different type of numbers for representation was also
used in (Li, 2009), where the coding involves using
real, integer and nominal values. Nevertheless, in that
work, each vector component represents a different
dimension in the search space, that is, two or more
components are not treated as forming a unique en-
tity. Conversely, in GGA, each couple of integer-real
components represents implicitly one dimension.
The rest of the paper is organized as follows. Sec-
tion 2 describes the new proposed algorithm. Next,
section 3 presents the problems used as benchmark-
ing and the final configuration (parameters and opera-
tors) used for our GGA and a real-coded standard GA
employed for comparison. Section 4 presents a com-
158
M. Chaquet J. and J. Carmona E..
A Grid-based Genetic Algorithm for Multimodal Real Function Optimization.
DOI: 10.5220/0004114401580163
In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 158-163
ISBN: 978-989-8565-33-4
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

parative of the results obtained using both algorithms.
Finally, in section 5, the main conclusions and future
works are given.
2 ALGORITHM DESCRIPTION
The objective of multimodal real function optimiza-
tion, R
N
→ R : F (x) with x = (x
1
,x
1
,··· ,x
N
), is to
find the global optimum in a N-dimensional space
R
N
. In formal notation, the problem to be solved is
the following:
x
min
|F (x
min
) ≤ F (x),∀x ∈ R
n
. (1)
The proposed algorithm works with a grid in the
N-dimensional space defined by (∆
1
,∆
2
,...,∆
N
) ∈
R
N
, where each ∆
i
represents the uniform grid
step size in each dimension. In GGA, the geno-
type of each individual is an integer-real vector,
(s
1
,s
2
,...,s
N
,
α
1
,
α
2
,...,
α
N
), where s
i
∈ Z, and
α
i
∈
R must fulfil the constraint 0 ≤
α
i
< ∆
i
. Then the
phenotype of each individual is computed in the fol-
lowing way:
x
i
= s
i
·∆
i
+
α
i
, i = 1,... ,N. (2)
The main idea of this special encoding is to discretize
the search space in order to facilitate the global op-
timum seeking using two types of search simultane-
ously: global and local search. The former uses the
integer part of each individual and is constrained to
the grid. It enables the exploration in the search space.
On the other hand, the later uses the real part of each
individual to exploit those solutions whose integer
part is close to an optimum. Fig. 1 shows an exam-
ple of the mentioned encoding in a two-dimensional
problem (N = 2). The grid step size, ∆
i
, is defined by
the user implicitly. Thus, the user chooses the number
of grid intervals n
∆
. This value will be the same for
all dimensions. If the definition domain range of each
variable is x
i
∈ [x
i,min
,x
i,max
], each ∆
i
is computed by
∆
i
= (x
i,max
−x
i,min
)/n
∆
, (3)
being ∆
i
fixed for all the individuals and for all gener-
ations, i. e., it is not evolved by the algorithm.
The following describes the parameters and oper-
ators used by the GGA. The initial population is ini-
tialized randomly covering all the domain space, i. e.,
s
i
∈ [⌊x
i,min
/∆
i
⌋,⌊x
i,max
/∆
i
⌋] and
α
i
∈ [0,∆
i
], where
⌊x⌋ denotes the function which returns the largest in-
teger not greater than x. As parent selection, it is used
the tournament method with size t
size
. Once parents
are selected, the crossover operator is performed with
a defined probability p
cross
. If the cross operator is not
invoked, the children are just an exact copy of their
∆
∆
α
α
1
1
2
2
(s ,s )
1 2
x
2
x
1
Figure 1: Graphical example of encoding in a two-
dimensional problem. The point represents an individ-
ual, being its relative coordinates (
α
1
,
α
2
) respect to the
grid node (∆
1
s
1
,∆
2
s
2
). The absolute coordinates are
(s
1
∆
1
+
α
1
,s
2
∆
2
+
α
2
) where ∆
1
and ∆
2
are the grid step
sizes in each dimension.
parents. Standard one-point crossover is used for the
integers s
i
and real numbers
α
i
and the same crossover
point is used for both.
After crossover, the mutation operator is applied
over each gene of each child with a specified prob-
ability p
mut
. Unlike the crossover operator, two dif-
ferent mutation operators have been implemented for
the real and integer part, called
α -mutation and s-
mutation, respectively. The first one is chosen with
a predefined probability p
α /s
, whereas the second op-
erator is chosen with a probability 1− p
α /s
. So, for
each gene, one of them is chosen if p < p
mut
. In other
case, the gene is not mutated. Being U (a,b) a sample
of a random variable with a uniform distribution in
the interval [a, b], the
α -mutation operator is defined
as follows:
α
′
i
← α
i
+U (−∆
i
σ
α
,∆
i
σ
α
)
if
α
′
i
< 0 ⇒

α
′′
i
← α
′
i
+ ∆
i
s
′
i
← s
i
−1
if
α
′
i
> ∆
i
⇒

α
′′
i
←
α
′
i
−∆
i
s
′
i
← s
i
+ 1
, (4)
where the parameter
σ
α
≪ 1 controls the mutation
strength. As we can see, first a low random value is
added to
α
i
. Then, it is checked if the new value is in-
side the feasible range. If not, both values s
i
and
α
i
are
updated accordingly. With this strategy, a fine tuning
can be done during the exploitation phase because the
mutation operator is not disruptive and allows chang-
ing the grid node if necessary.
Because the s
i
components of each individual are
thought to explore the search space, the s-mutation
operator only modifies s
i
and maintains invariable
the
α
i
values. Here the idea is to use the typical
nonuniform mutation with normal distribution but,
because we are working with integers, that distribu-
AGrid-basedGeneticAlgorithmforMultimodalRealFunctionOptimization
159

tion is substituted by the difference of two geometri-
cal distributed variables Z ( σ
s
), which it is more suit-
able for integer numbers (Rudolph, 1994):
s
i
´ ← s
i
+ Z
1
(
σ
s
) −Z
2
(
σ
s
). (5)
According to (Rudolph, 1994), a geometrical dis-
tributed variable Z with dispersion
σ can be generated
from a uniform distribution with the following proce-
dure:
u ←U (0,1)
ψ ← 1−
σ
1+
√
1+ σ
2
Z ←
j
ln(1−u)
ln(1− ψ )
k
(6)
The dispersion of the two geometrical distributions,
σ
s
, is a control parameter of the mutation strength.
The objective of this operator is to create enough di-
versity in the population to facilitate the exploration
of the search space.
After the variation operators are applied, it is em-
ployed a generational model, i.e., the whole popula-
tion is replaced by its offspring, which forms the next
generation. All these steps are repeated for several
generations until the stop condition is fulfilled. Two
stop conditions are checked: a maximum number of
generations or a fitness value close to the global opti-
mum below a predefined threshold,
θ
solution
.
Once the GGA is described, here are some notes
about the implemented standard GA used for com-
parison with our algorithm. Real-valued representa-
tion has been employed. The population initializa-
tion is performed randomly over the predefined range
for each dimension. As in GGA, tournament se-
lection is employed for choosing the parents. Once
all the parents are selected, the cross operator is ap-
plied with certain probability p
cross
over parent cou-
ples. If crossover is not selected, the children are
just a copy of their parents. As crossover opera-
tor, intermediate recombination is employed: x
child
i
=
ϕ x
parent1
i
+ (1− ϕ )x
parent2
i
for some ϕ ∈ [0,1]. The
mutation operator is used with a certain probability
p
mut
for each gene and each individual. Nonuniform
mutation with Gaussian distribution N (0,1) is used:
x
′
i
← x
i
+
σ N(0,1). In order to facilitate the explo-
ration in the first part of the algorithm and a correct
exploitation at the final generations, a linear variation
of the standard deviation between the first generation
(with a value
σ = σ
initial
) and a prescribed genera-
tion G
linear
(decreasing until the value
σ = σ
final
) is
employed. From G
linear
generation till the end, the
standard deviation is maintained constant and equal
to
σ
final
. As in GGA, a generational model is used
and the same stop criterion is adopted.
-10 -5 0 5 10
x
0
50
100
F(x)
Sphere
Ackley
Rastrigin
-400 -200 0 200 400
x
0
200
400
600
800
F(x)
Schwefel
Figure 2: Sphere, Ackley and Rastrigin (top) and Schwefel
(down) functions in one-dimensional problem N = 1.
3 BENCHMARKING AND
ALGORITHM SETUP
In this section the multimodal functions used for
benchmarking are described. Also the specific pa-
rameters employed in the GA and GGA are given.
Four functions extracted from the literature are used:
Sphere (unimodal), Ackley, Rastrigin and Schefwel:
F
Sphere
(x) =
N
∑
i=1
x
2
i
, (7)
F
Ackley
(x) = 20+ e −20exp

−0.2
q
∑
N
i=1
x
2
i
N

−
−exp

∑
N
i=1
cos(2
π x
i
)
N

,
(8)
F
Rastrigin
(x) = 10N +
N
∑
i=1
x
2
i
−10
N
∑
i=1
cos(2
π x
i
), (9)
F
Schwef el
(x) = 418.982988N−
N
∑
i=1
x
i
sin
p
|x
i
|. (10)
All the functions, except the Schwefel one, have a
global minimum in the origin with zero value, be-
ing the definition range in all dimensions [−10, 10].
Fig. 2 (top) shows the first three functions in one-
dimensional problem. Schwefel function is defined
in −500 ≤ x
i
≤ 500 and its global minimum (as well
with zero value) is located in x
i
= 420.968746. See
Fig. 2 (down) for one-dimensional representation.
The search of the global optimum for the three func-
tions that have the global minimum in the origin
should produce an individual with all its α
i
equal to
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
160

zero. This fact is extremely unusual in real applica-
tions. So in order to study the behaviour of the GGA
in more realistic scenarios, three new functions, called
π -functions, are built doing a shift in all dimensions
in the following way:
F
π
(x) = F (x−Π), (11)
where F is the original Sphere, Ackley or Rastrigin
function and Π is a vector with all its components
equal to
π .
On the other hand, all the above functions are
isotropic (same behaviour in all dimensions). Some
real problems are non-isotropic. Therefore it has been
implemented a new function family called modified
functions doing a change of variable
x
i
= 2
−i+1
y
i
. (12)
In this way, for example, the modified Sphere or m-
Sphere function is defined as:
F
m−Sphere
(y) =
N
∑
i=1
2
−2i+2
y
2
i
. (13)
Likewise, obtaining the other modified functions
is immediate. For all the modified functions, the
global minimum is in y
i
= 0, except for the m-
Schwefel, where the minimum is in y
i
= 420.968746·
2
i−1
. The definition ranges for the four modified func-
tion are now x
i,min
2
i−1
≤ y
i
≤ x
i,min
2
i−1
, being x
i,min
and x
i,max
the range limits of the original functions.
Note that for the first dimension x
1
, the original range
is preserved.
Finally a new family of functions called modified
π or m- π functions are defined shifting and scaling the
variables simultaneously applying Eq. (11) and (12) .
Once the test functions are introduced, Tables 1
and 2 give the parameter settings used in GA and
GGA. Note that the two algorithms have the most
similar parameter values as possible in order to do
a fair comparison. The values have been experimen-
tally chosen. The parameter tuning is straightforward,
and the same values used for all the runs demonstrate
the robustness of the method.
4 RESULTS
A total of 14 test functions have been employed, in-
cluding the original Sphere, Ackley, Rastrigin and
Schwefel, and the variants
π , modified and π -
modified functions. The number of dimensions con-
sidered in all the cases are N = 10. Each run has been
repeated 20 times using different seeds for the ran-
dom number generator and averages were taken. Ta-
bles 3 and 4 give the percentage of successful runs
Table 1: GA parameters.
Parameter Value
Population size 200
Tournament t
size
3
p
cross
0.8
p
mut
0.05
Max. generations 2000
G
linear
1000
σ
initial
/(x
i,max
−x
i,min
) 0.3
σ
final
/(x
i,max
−x
i,min
) 10
−6
Threshold
θ
solution
≤ 10
−4
Table 2: GGA parameters.
Parameter Value
Intervals n
∆
20
Population size 200
Tournament t
size
3
p
cross
0.8
p
mut
0.05
p
α /s
0.9
σ
α
10
−2
σ
s
6
Max. generations 2000
Threshold
θ
solution
≤ 10
−4
and the average number of generation for achieving
the stop condition both for GA and GGA respectively.
As well it is given the standard deviation of the gen-
eration number. A run is considered successful when
the fitness of the best individual is less than
θ
solution
in
a generation number less or equal than the maximum
generation number. Only successful runs are consid-
ered for the generation number averages.
According to the results, the proposed algorithm
equals or outperforms the GA, both in percentage
of successful runs and in number of generations re-
quired. Because of the linear variation of the stan-
dard deviation in the GA during the first 1000 gen-
erations, the required generations for achieving the
stop condition is larger than 1000. This limitation is
not observed in GGA, because no variation in the pa-
rameters is imposed with the generation number. A
shorter linear variation step has been tested in GA us-
ing G
linear
= 500, but although the average number of
generations is reduced in some cases, the successful
rates are reduced as well.
Fig. 3, 4 and 5 shows a typical run of GGA for
Schwefel function. For this example ∆
i
= 1000/20 =
50 and the optimum is placed in x
i
= s
i
∆
i
+
α
i
=
8 ·50 + 20.968746. In Fig. 3, the fitness of the best
individual and the population average fitness are plot-
ted versus the generation. We can differentiate two
phases. The first one where the majority of the inte-
AGrid-basedGeneticAlgorithmforMultimodalRealFunctionOptimization
161

Table 3: Experimental results obtained for the 14 test func-
tions using GA. Percentage of successful runs, average
number of generations, and standard deviations of the gen-
erations are provided.
Case % Success Generations σ
generations
Sphere 100 1013 43
Ackley 100 1212 73
Rastrigin 25 1134 375
Schwefel 0 - 0
π -Sphere 100 1003 1
π -Ackley 100 1233 108
π -Rastrigin 10 1310 208
m-Sphere 100 1003 1
m-Ackley 100 1231 82
m-Rastrigin 35 1205 383
m-Schwefel 0 - 0
m- π -Sphere 100 1003 1
m- π -Ackley 100 1256 99
m- π -Rastrigin 20 1140 345
Table 4: Experimental results obtained for the 14 test func-
tions using GGA. Percentage of successful runs, average
number of generations, and standard deviations of the gen-
erations are provided.
Case % Success Generations σ
generations
Sphere 100 410 58
Ackley 100 601 87
Rastrigin 100 322 61
Schwefel 100 678 293
π -Sphere 100 402 45
π -Ackley 100 592 77
π -Rastrigin 100 355 64
m-Sphere 100 399 49
m-Ackley 100 623 74
m-Rastrigin 100 325 72
m-Schwefel 100 640 125
m- π -Sphere 100 444 48
m- π -Ackley 100 621 77
m- π -Rastrigin 100 372 75
ger numbers s
i
are adjusted (see Fig. 4, first 100 gen-
erations), and a second phase where the fine tuning of
the reals
α
i
are sought (Fig. 5, from generations 100
to 700). Note that in the second phase, when a
α
i
exits
from the feasible range [0,∆
i
], the integer counterpart
s
i
is updated accordingly and the
α
i
can evolve in the
new interval (Figs. 4 and 5).
5 CONCLUSIONS
A new algorithm for multimodal real optimization,
called GGA, is presented. Here the definition domain
0 200 400 600 800
Generations
0.0001
0.01
1
100
10000
Best and Average Fitness
Figure 3: Convergence history of a typical run of Schwe-
fel function: fitness of the best individual and population
average fitness.
0 200 400 600 800
Generations
-20
-10
0
10
20
s
i
Figure 4: Convergence history of a typical run of Schwefel
function: steps s
i
of the best individual.
of the optimization problem is discretized using a grid
and each individual is represented by integer and real
number couples. This frame facilities the search pro-
cess allowing two types of search simultaneously: a
global search for exploration and a local search for ex-
ploitation. The global optimum of 14 test multimodal
functions have been correctly found with a 100% suc-
cessful rate. A comparison with a standard real-coded
GA has been also performed. The proposed algorithm
equals or outperforms the standard GA in percentage
of successful runs and number of generations needed
to reach the global optimum.
These preliminary results are encouraging. Nev-
ertheless, the new method should be tested in more
functions and real problems, and compared with other
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
162

0 200 400 600 800
Generations
0
0.2
0.4
0.6
0.8
1
α
i
/△
i
Figure 5: Convergence history of a typical run of Schwefel
function:
α
i
/∆
i
ratios of the best individual.
paradigms of evolutionary algorithms. A sensitiv-
ity analysis of the algorithm to best tune the param-
eters should be necessary. As future work, auto-
adaption techniques for some of the algorithm pa-
rameters could be investigated, such as Evolution-
ary Strategies do for the mutation strengths. Auto-
adaptation is attractive because simplifies the setup
of the algorithm (low number of parameters are re-
quired) and better results can be obtained (the param-
eters are automatically evolved using the best one ac-
cording to the environment).
ACKNOWLEDGEMENTS
This work was suported by the Spanish Ministe-
rio de Economía y Competitividad under the Project
TIN2010-20845-C03-02.
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