Domain Decomposition Evolutionary Algorithm for MultiModal Function Optimization
ABSTRACT We here proposed some selfadaptive methods to choose the results of Gaussian and Cauchy mutation, and the dimension of subspace. We used the better of Gaussian and Cauchy mutation to do local search in subspace, and used multiparents crossover to exchange their information to do global search, and used the worst individual eliminated selection strategy to keep population more diversity. Judging by the results obtained from the above numerical experiments, we conclude that our new algorithm is both universal and robust. It can be used to solve function optimization problems with complex constraints, such as NLP problems with inequality and (or) equality constraints, or without constraints. It can solve 01 NLP problems, integer NLP problems and mixed integer NLP problems. When confronted with different types of problems, we don't need to change our algorithm. All that is needed is to input the fitness function, the constraint expressions, and the upper and lower limits of the variables of the problem. Our algorithm usually finds the global optimal value. In the paper we analyze the character of the multiparent genetic algorithm, when applied to solve the optimization of multimodal function, MPGA works in different forms during different phases and then forms twophase genetic algorithm. The experiments indicate that DDEA is effective to solve the optimization of multimodal function whose dimension is no
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Conference Paper: Selfadaptive penalties for GAbased optimization
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ABSTRACT: This paper introduces the notion of using coevolution to adapt the penalty factors of a fitness function incorporated in a genetic algorithm for numerical optimization. The proposed approach produces solutions even better than those previously reported in the literature for other (GAbased and mathematical programming) techniques that have been particularly finetuned using a normally lengthy trial and error process to solve a certain problem or set of problems. The present technique is also easy to implement and suitable for parallelization, which is a necessary further step to improve its current performanceEvolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on; 02/1999  [Show abstract] [Hide abstract]
ABSTRACT: Thesis (Ph. D.)University of Michigan, 1975. Includes bibliographical references (leaves 253256). Photocopy.01/1975;  The University of Michigan Press, Ann Arbor. 01/1975;
Page 1
9
Domain Decomposition Evolutionary Algorithm
for MultiModal Function Optimization
Guangming Lin1, Lishan Kang2, Yongsheng Liang1 and Yuping Chen2
1Shenzhen Institute of Information Technology, Shenzhen 518029,
2School of Computer, China University of Geosciences, Wuhan,
PRC.
1. Introduction
The Simple Genetic Algorithm (SGA) is applied more and more extensively since it was
proposed by J. H. Holland [1] in 1970’s. SGA is an optimization method based on
population by emulating the evolvement disciplinarian of the nature. It has showed the
great advantage of quick search for optimal solutions while applied in the optimization of
singlemodal functions. But as we all know many problems in reality belong to the
optimization of multimodal function, and if SGA is applied to solve this kind of problems,
it has the confliction between the search space and convergence speed: the expansion of
search space will slow down the convergence speed and the acceleration of convergence
speed will reduce the search space, lead to early convergence and as a result stop research at
some local optimal solutions.
Evolutionary algorithms have been used regularly to solve multimodal function
optimization problems, due to their populationbased approach and their inherent
parallelism, e.g. a crowding factor model proposed by De Jong[2], a sharedfunction model
proposed by Goldberg and Richardson[3], an artificial immune system method, a split ring
parallel evolutionary algorithm, etc., all of which have attempted to maintain the diversity
of the population during the process of evolution. In this chapter, we introduce a new
‘Domain Decomposition Evolutionary algorithm (called DDEA) which can solve not only
simple nonlinear programming problems effectively and efficiently, but can also find the
multiple solutions of multimodal problems in a single run. The DDEA employs dual
strategy approach that searches at two levels of detail (namely global then local). In the first
(global) step, a Selfadaptive Mutations with Multiparent Crossover Evolutionary
Algorithm (SMMCEA)[4] is employed to perform a global search to divide the
(chromosome) population into several subpopulations or niches in subdomains, which is
domain decomposition. In the second (local) step, an evolutionary strategylike algorithm is
employed to perform a local search on each isolated niche independently. Then the best
solutions of the multimodal problem are exploited.
The remainder of the chapter is organized as follows. Section 2 introduces a Selfadaptive
Mutations with Multiparent Crossover Evolutionary Algorithm (SMMCEA); Section 3
introduces Domain Decomposition evolutionary algorithm (DDEA); Section 4 presents the
successful results of applying DDEA to several challenging numerical multimodal
optimization problems; Section 5 concludes.
Open Access Database www.itechonline.com
Source: Advances in Evolutionary Algorithms, Book edited by: Witold Kosiński, ISBN 9789537619114, pp. 468, November 2008,
ITech Education and Publishing, Vienna, Austria
Page 2
Advances in Evolutionary Algorithms
168
2. Introduction of SMMCEA
2.1 The Problem to Solve
The general nonlinear programming (NLP) problem can be expressed in the following
form:
Minimize f(X,Y)
s.t. hi(X,Y)= 0 i = 1,2,...,k1 ， gj(X,Y) ≤0 j=k1+1, k1+2,...,k
Xlower ≤ X ≤ Xupper ， Ylower ≤ Y ≤ Yupper
(1)
where X∈Rp, Y∈Nq, and the objective function f (X,Y), the equality constraints hi(X,Y) and
the inequality constraints gj(X ,Y) are usually nonlinear functions which include both real
variable vector X and integer variable vector Y.
Denoting the domain D = {(X,Y)  Xlower ≤ X ≤ Xupper，Ylower ≤ Y ≤ Yupper }, we introduce the
concept of a subspace V of the domain D. m points (Xj,Yj), j＝1,2,…,m in D are used to
construct the subspace V, defined as :
V ＝{(Xv,Yv)∈D(Xv,Yv)= ∑=
m
i
iii
Y,Xa
1
)(
}
where ai is subject to ∑=
Because we deal mainly with optimization problems which have real variables and
INequality constraints, we assume k1 = 0 and q = 0 in the expression (1).
⎪
⎨
otherwise ),( X
gi
Then problem (1) can be expressed as follows:
m
i
i a
1
= 1, 0.5≤ai ≤1.5.
Denoting wi (X)＝
⎪⎩
⎧
≤
0)( 0,X
gi
and W(X)＝
)(
1
XW
k
i
i
∑
=
Minimize f(X) X∈D (2)
Subject to
W(X)=0 X∈D
We define a Boolean function “better” as:
better (X1, X2) ＝
⎪
⎩
⎪
⎪
⎨
⎪
⎧
>∧=
≤∧=
>
<
FALSE
2121
TRUE
121
FALSE
21
TRUE
2
1
))()(()) W( )(W(
)) ()(()) W( )(W(
) W( ) W(
)( )(
2
X
f
X
f
XX
X
f
X
f
XX
XX
X
WXW
If better (X1, X2) is TRUE，this means that the individual X1 is “better” than the individual
X2.
2.2 Related Work
Page 3
Domain Decomposition Evolutionary Algorithm for MultiModal Function Optimization
169
In 1999, Guo Tao proposed a multiparent combinatorial search algorithm (GTA) for solving
nonlinear optimization problems in his PhD thesis [5]. Later it was developed as a kind of
subspace stochastic search algorithm [6], that can be described as follows:
Guo Tao’s Algorithm (GTA)
Begin
initialize popln P ＝ {X1, X2,…, XN }; Xi ∈D since (q = 0 implies no integer variables)
generation count t := 0;
X best ＝arg
)(
1
Ni≤≤
X worst ＝ arg
) (
1Ni
≤≤
while abs(f (X best)f (X worst)) >ε do
select randomly m points X 1′, X 2′,…, X m′ from P to form the subspace V;
select randomly one point X′ from V;
If better (X′, X worst) then Xworst: = X′;
t := t + 1;
Xbest = arg
)(
1
Ni≤≤
Xworst ＝ arg
)(
1
Ni≤≤
end do
output t , P ;
End
where N is the size of population P, (m –1) is the dimension of the subspace V (if the m
points (vectors) that construct the subspace V are linearly independent)，t is the number of
generations, ε is the accuracy of solution. Xbest = arg
X
f
Min
i
;
X
Max
i
f
;
X
f
Min
i
;
X
f
Max
i
)(
1
X
f
Min
i≤≤
i
N
means that Xbest is the
variable (individual) in Xi (i=1, 2,…, N) that makes the function f (X) have the smallest value.
The subpopulation in GTA is families which reproduce sexually through the number of m
individuals randomly selected from P. The best individual in the subpopulation takes part
in competition to replace the worst individual in P, therefore the pressure of elimination
through selection is minimum. There is no mutation operator, only using multiparents
crossover in GTA.
2.3 A selfadaptive evolutionary algorithm
Since Guo’s algorithm deals mainly with continuous NLP problems with Inequality
constraints, to make it a truly universal and robust algorithm for solving general NLP
problems, we extend Guo’s algorithm by adding to it the following improvements:
(1) Guo selected randomly only one candidate solution from the current subspace V.
Although he used the concept of a subspace to describe his algorithm, he did not really use a
subspace search, but rather a multiparent crossover. Because he selected randomly only one
individual in the subspace, this action would tend to ignore better solutions in the subspace,
and hence influence negatively the quality of the result and the efficiency of the search. If
however, we select randomly several individuals from the subspace, and substitute the best
one for the worst one in the current population, the search should be better. So we replace
the instruction line in Guo’s algorithm:
Page 4
Advances in Evolutionary Algorithms
170
“select randomly one point X′from V; ”
with the two instruction lines:
“ select randomly s points
*
1
X ，
*
2
X ，…，
*
s
X from V;
X′= arg
()
1
i
f X
Min
i
≤ ≤
s
∗
;”
(2)The dimension m of the subspace in Guo’s algorithm is fixed (i.e. m parents reproduce).
The algorithm always selects a substitute solution in subspaces which have the same
dimension, regardless of the characteristics of the solutions in the current population. Thus,
when the population is close to the optimal value, the searching range is still large. This
would apparently result in unnecessary computation, and affect the efficiency of the search.
We can in fact reduce the search range, that is to say, the dimension of the subspaces. We
therefore use subspaces with variable dimensions in the new algorithm, by adding the
following instruction line to Guo’s algorithm:
if abs ( f (Xbest) – f (Xworst)) ≤ η .and. m ≥3 then m := m – 1;
where η depends on the computation accuracy ε, and η > ε. For example, if the computation
accuracy ε = 1014, then we can set η = 102 or 103.
(3) We know in principle that Guo’s algorithm can deal with problems containing EQuality
constraints. For example, we can use the device of setting two INequality constraints
0≤hi(X ,Y) and hi(X ,Y)≤0 to replace the equality constraint hi(X ,Y) = 0, but the experimental
results when employing this device are not ideal. However, equality constraints are likely to
exist in realworld problems, so we should find methods to deal with them. One such
method is to define a new function W(X, Y)
k
YX
i
1
Where W(X, Y) = ∑
=
i
W
)
,(
⎪⎩
⎪⎨
⎧
++=
=
=
.,, 2
1
, 1
1
)},,(,max{
,, 2 , 1 , ),(
),(
kkkiYX
i
go
ik
iYX
ih
YX
Wi
?
?
(4) The penalty factor r is usually fixed. However, some people use it as a variable, such as
Cello[7], who employed a selfadaptive penalty function, but his procedure was rather
complex (using two populations). We also make r a variable namely r = r (t), where t is the
iteration count. It can selfadjust according to the reflection information, so we label it a
“selfadaptive penalty operator”. Since the constraints have been normalized, r is relative
only to the range of the objective function, which ensures a balance between the errors of the
fitness function and the objective function, in order of magnitude.
(5) Guo’s algorithm can deal only with continuous optimization problems. It cannot deal
directly with integer or mixed integer NLP problems. In our algorithm, when we are
confronted with such problems, we need only replace the integer variables derived from the
Page 5
Domain Decomposition Evolutionary Algorithm for MultiModal Function Optimization
171
range of the float of the fitness function with “integer function” int(Y*), where int(Y*) is
defined as the integer part of Y*. No other changes to the algorithm are needed.
(6) The only genetic operator used in Guo’s algorithm was crossover. However, we can add
self –adaptive mutations in it, we introduce a better of Gaussian and Cauchy mutation
operator into the subspace search. For Gaussian density function fG with expectation 0; and
variance σ2 is
Gf =
2
2
2
2
1
σ
πσ
x
e
−
, －∞ < x < +∞
For Cauchy density function fC with scale parameter t>0 is,
Cf =
22
11
π
xt +
, －∞ < x < +∞
2.4 A Selfadaptive mutations with multiparent crossover evolutionary algorithm
Considering the above points, we introduce a new algorithm as follows:
Denoting Z = (X, Y*), where Z∈D*, and
D* = {(X, Y*)Xlower≤X≤Xupper, Ylower≤Y*≤Yu, X ∈Rp, Y*∈Rq}, we define integer vector
Y=int(Y*), where Yu = Yupper+0.999…9I
Denoting W(Z)＝W(X, int(Y*)),
we define the Boolean function “better” as follows:
better(Z1 ,Z2) ＝
⎪
⎩
⎪
⎪
⎨
⎪
⎧
>∧=
≤∧=
>
<
FALSE
2121
TRUE
121
FALSE
21
TRUE
2
1
))()(( )) W( )(W(
))()(()) W( )(W(
) W( ) W(
)( )(
2
X
f
X
f
XX
X
f
X
f
XX
XX
X
WXW
The general NLP problem (1) can be expressed as follows:
Minimize f(X,int(Y*)) in D* S.t. (3)
W(Z)=0 , Z∈D*
The new algorithm can now be described as follows:
SMMCEA :
Begin
initialize P ＝ {Z1,Z2,…,ZN }; Zi∈
t := 0;
Zbest ＝
arg
1Ni≤≤
*
D ;
)(
i
Zf
Min
;
Page 6
Advances in Evolutionary Algorithms
172
Zworst ＝
)( arg
1
i
Zf
Max
i≤≤
N
;
while not abs ( F (Zbest) – F (Zworst)) ≤ε do
select randomly M points Z1′, Z2′,…, ZM′from P to form the subspace V;
*
1
Z ,
Z …
for i=1,…s do
for j=1,…p+q do
*
Gi
Z
(j) :=
Z
(j) :=
) 1 , 0 (
N
τ
endfor
*
Gi
Z
,
Ci
Z
) then
:
i
ZZ
=
endfor
Z′=
)(arg
1
i
Ni≤≤
if better (Z′, Z worst) then Zworst := Z′;
t := t + 1;
Zbest ＝
)(arg
1
i
Ni≤≤
Zworst ＝
)( arg
1
i
Ni≤≤
if abs (f (Zbest) f (Zworst)) ≤η .and. M ≥3 then
M := M 1;
endwhile
output t , Zbest , f(Zbest) ;
end
*
Gi
Z
(j),
Ci
Z
(j) and
respectively. N(0,1) denotes a normally distributed onedimensional random number with
mean zero and standard deviation one. Nj(0, 1) indicates that the Gaussian random
select s points randomly
*
2
*
s
Z from V;
*
i Z (j)+
i σ (j)Nj(0, 1)
i σ (j)Cj(1)
*
Ci
*
i Z (j)+
i σ (j)exp(
i σ (j) :=
)) 1 , 0 (
j
'
N
τ+
if better(
*
: else
*
Ci
'*
i
*
Gi
'*
ZZ
=
;
Zf
Min
;
Zf
Min
;
Zf
Max
;
Where
*
i σ (j) denote the jth component of the vectors
*
Gi
Z
,
*
Ci
Z
and
i σ ,
number is generated anew for each value of j. Cj(1) denotes a Cauchy distributed one
dimensional random number with t=1.
The factors τ and ' τ have commonly set to
⎜⎝
The new algorithm has the two important features:
1. This algorithm is an ergodicity search. During the random search of the subspace, we
employ a “nonconvex combination” approach, that is, the coefficients ai of Z’=∑
1
)(2
−
⎟⎠
⎞⎛
+ qp
and ()
1
)( 2
−
+ qp
.
=
m
i
iiZa
1
'are
random numbers in the interval [0.5，1.5] This ensures a nonzero probability that any
point in the solution space is searched. This ergodicity of the algorithm ensures that the
optimum is not ignored.
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Domain Decomposition Evolutionary Algorithm for MultiModal Function Optimization
173
2. The monotonic fitness decrease of the population (when the minimum is required). Each
iteration (t→t+1) of the algorithm discards only the individual having the worst fitness in
the population. This ensures a monotonically decreasing trend of the values of objective
function of the population, which ensures that each individual of the population will reach
the optimum.
When we consider the population P(0), P(1), P(2),…, P(t),… as a Markov chain, we can prove
the convergence of our new algorithm. See [12].
3. Introduction of DDEA
Experiments indicate that if SMMCEA is directly applied to the optimization of multi
modal function, it is easy to encounter the following two conditions:
1. If keep searching with relatively large population size and crossover size, the
individuals of the population will spread around near different modals, but it’s difficult
for population to get any more improvement and to reach all the modals exactly.
2. If keep searching with relatively small population size and crossover size, the
individuals of the population will converge rapidly and reach a few modals, but lose
many other modals.
To adopt it to the optimization of multimodal functions, we combine the above two
conditions together and forms twophase evolutionary algorithm. we divide the
optimization procedure into two phases: the first phase is called global optimization, which
keeps searching with relatively large population size and crossover size in order to
determine the neighborhood of all modals; the second phase is called local optimization,
which begins search from each of the neighborhoods which is determined by the global
optimization and then keep searching with relatively small subpopulation size and
crossover size in order to converge rapidly and reach the modals respectively.
In addition, we introduce the following strategies to make the algorithm suitable to the
different tasks of the two phases:
1. During the phase of global optimization, in order to avoid the loss of some obtained
modals we introduce the strategy of good individuals isolation: before each evolvement
all the individuals in the current population are sorted by their fitness value and then
some of the good individuals are limited not to be parents in the next multiparent
crossover.
2. During the phase of local optimization, in order to make all the subpopulations
converge to their modals respectively more quickly, we introduce the strategy of best
individual exemplar: the best individual of the current population will be compelled to
be one of the parents in the next multiparent crossover.
3. During the phase of local optimization, in order to begin search based on the result of
the global optimization and to keep the search around the neighborhood of all the
modals, to each modal we will construct a local feasible area η, which is to be modified
during the evolvement.
The detailed procedures of the optimization DDEA are as the following:
Phase 1: Global optimization (using SMMCEA)
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Advances in Evolutionary Algorithms
174
Phase 2: Local optimization
for k= 1 to N1 do
The new algorithm employs a zoomed (global to local) dual strategy (two steps) approach.
The first (global) step employs a global search, i.e. it divides the (chromosome) population
into L (L ≤ k) niches, each of which includes at least one of the k optimal solutions (if the
objective function is continuous in D*). This step uses a SMMCEA [4]. If the number of
parents M in the multiparent recombination operator is large enough, for example, M ≥ 8,
Randomly initialize population P(0)= {P1,P2,…, PN1},Evaluate P(0),t1=0
while t1< MAXT1 do
randomly select m1 parents from P(t1) with the strategy of good individuals isolation
produce a child by multiparent crossover and selfadaptive Gaussian and Cauchy
mutation
if the child is better than the worst individual of P(t1) then
replace the worst individual of P(t1) with the child
end if
t1= t1+1
end while
initialize local feasible area η, which is the rectangle area around Pk with the
radium r
Randomly initialize subpopulation SUBP(0) within the area of η
SUBP(0)={ SUBP 1, SUBP 2,…, SUBP N2 }
t2=0
while (t2< MAXT2 and individuals of SUBP(t2) are different )do
randomly select m2 parents from SUBP(t2) with the strategy of the best
individual exemplar
produce a child by multiparent crossover
if the child∈η and it is better than the worst individual of SUBP(t2)
then replace the worst individual of SUBP(t2) with the child
end if
evaluate the best individual of SUBP(t2), which is named as
SUBPbest
modify local feasible area η, make it as the rectangle area around
SUBPbest
with the radium r
t2= t2+1
end while
output the best individual of SUBP(t2)
end for
Page 9
Domain Decomposition Evolutionary Algorithm for MultiModal Function Optimization
175
then after sufficient large generations the population is decomposed into subpopulations
(each of which approaches to an optimal solution), else it will converge to only one solution
[11].
The second (local) step employs an evolution strategy [13] to search for the local optima in
the chosen L subspaces determined by the subpopulations. Since the L optimal solutions are
located in separate subspaces, the local strategy consists of two substeps:
a). Rank the individuals of the population obtained from the first (global) step according to
their fitness values. Then choose the best L individuals from the population, ensuring that
they are not close to each other like hedgehogs.
b). Generate L subspaces with the chosen individual at the center of each. Search these
niches locally until each subspace converges to an optimal solution. If one does not know
how many optimal solutions a given problem has, one can predict the number k, for
example, by using the number of individuals whose fitness values are larger than the
average fitness value.
The algorithm has different limiting behaviors for different problems, namely:
a). When the problem has only k = 1 solution, i.e. the only globally optimal solution.
Following the nature of population descent, all of the individuals will descend together to
the bottom of the valley.
b). When the problem has k > 1 solutions, i.e. if k ≤ N, where N is the size of the population, k
solutions may be generated in the population. The algorithm will then find multisolutions
in a single run.
4. Numerical experiments and analysis
Example 1 Humpback function (the function has six local optimal solutions, two of which
are global optimal solutions)
2
2
2
221
2
1
4
1
2
12
∈
1
)44( ) 3/ 1 . 2
−
4 (
−
),( min
∈
xxxxxxxxxf
+−+++=
where
Example 2 Typical function with many global optimal solutions(the function has increasing
number of global optimal solutions while j is increased)
] 2 , 2[ ], 3 , 3
−
[
21
xx
2
2
2
121
))(sin())(sin(3),( min
jxjxxxf
=
−−=
,
where
Example 3 Absolute value function(the function has a plenty of local optimal solutions, 16 of
which are global optimal solutions)
?
, 2 , 1],6 , 0 [
∈
,
21
jxx
∏
=
j
∏
=
i
−+−=
4
1
2
4
1
121
43),(min
jxixxxf
,
where
Example 4 Ndimension Shubert function[8](when n=2，the function has 720 local optimal
solutions, 18 of which are global optimal solutions)
]17 , 0 [
∈
],13, 0 [
∈
21
xx
Page 10
Advances in Evolutionary Algorithms
176
∏∑
=
i
1
=
++=
n
j
in
jxjjxxxf
5
1
21
)) 1 cos((),,,(min
?
where
[ 10,10],
∈ −
1,2,,
ixin
=
?
Fig. 1. Shubert function
All the examples mentioned above are representatives of different kinds of functions.
Example 1, Example 2 and Example 4 are cited from [9]. Example 1 is the representative of
glossy function with only a few modals, Example 2 is the representative of glossy function
with many modals, Example 3 is the representative of nonglossy function, and example 4 is
the representative of highdimension function. Generally we can get satisfying optimal
solutions when we set the parameters according to the following principle:
The phase of global optimization: N1≈10*the number of actual optimal solutions
2000 < MAXT1< 100*N1
6 ≤ m1 ≤ 10
The phase of local optimization: 10 < N2 < 20, r = 2.0
2000 < MAXT2 < 5000
3 ≤ m2 ≤ 5
The following figures show population distribution in different phases for each example,
which indicate the optimization procedures of different examples. Each figure has three
parts: (a) is the distribution of population after randomly initialization; (b) is the distribution
of population after global optimization; (c) is the distribution of the found modals after local
optimization. The horizontal coordinate is the value of x1 and the vertical coordinate is the
value of x2.
Page 11
Domain Decomposition Evolutionary Algorithm for MultiModal Function Optimization
177
3210123
2
1.5
1
0.5
0
0.5
1
1.5
0.20.15 0.10.0500.05 0.10.15
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
0.10.080.060.040.0200.020.040.060.080.1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
(a) after randomly initialization (b) after global optimization (c) after local optimization
Fig. 2. Population distribution for example 1
0123456
0
1
2
3
4
5
6
0123456
0
1
2
3
4
5
6
0123456
0
1
2
3
4
5
6
(a) after randomly initialization (b) after global optimization (c) after local optimization
Fig. 3. Population distribution for example 2 when j=5
02468101214
0
2
4
6
8
10
12
14
16
18
24681012 14
2
4
6
8
10
12
14
16
18
246810 1214
2
4
6
8
10
12
14
16
18
(a) after randomly initialization (b) after global optimization (c) after local optimization
Fig. 4. Population distribution for example 3
1086 42 0246810
10
8
6
4
2
0
2
4
6
8
10
8 642 02468
8
6
4
2
0
2
4
6
8
8642 0246
8
6
4
2
0
2
4
6
(a) after randomly initialization (b) after global optimization (c) after local optimization
Fig. 5. Population distribution for example 4 when n=2
Additionally, the following tables list parameters and results for different experiments:
Parameters
Example No.
N1 MAXT1
Actual modals
results
Found modals fitness of
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Advances in Evolutionary Algorithms
178
all modals
1.031628
1.000000
0.000000
186.730909
Example 1
Example 2（j=5）
Example 3
Example 4（n=2）
20
600
200
100
2000
60000
20000
10000
2[9]
100[9]
16
18[9]
2
100
16
18
Table 1. Experiment parameters and results for each example (other parameters are:
m1=7,m2=5,N2=10,MAXT2=2000)
The value of j
Actual modals
Found modals
2
16
16
3
36
36
4
64
64
5 6 7 8 9 10
361
361
100
100
121
121
169
169
225
225
289
289
Table 2. Experiment results for example 2 with different value of j (The fitness of all modals
is 1.000000)
Parameters Results
Example No.
N1 MAXT1
Found
modals
81
fitness of
all modals
2709.09350
Example 4（n=3）
800 500000~1000000
Table 3. Parameters and experiment results for example 4 when n=3 (other parameters are:
m1=7,m2=5,N2=10,MAXT2=10000).
From population distribution of the optimization procedures showed in Fig2, Fig3, Fig4 and
Fig5, as well as the experiment results showed in Tables 1 and Table 2, we can see that
DDEA is very efficient for the optimization of low dimension multimodal function, usually
we can reach all the modals exactly. But Table 3 indicates that when the dimension of the
function is increased to higher than two, the efficiency is decreased because of the search
space is expanded sharply.
5. Conclusion
We here proposed some selfadaptive methods to choose the results of Gaussian and
Cauchy mutation, and the dimension of subspace. We used the better of Gaussian and
Cauchy mutation to do local search in subspace, and used multiparents crossover to
exchange their information to do global search, and used the worst individual eliminated
selection strategy to keep population more diversity.
Judging by the results obtained from the above numerical experiments, we conclude that
our new algorithm is both universal and robust. It can be used to solve function
optimization problems with complex constraints, such as NLP problems with inequality and
(or) equality constraints, or without constraints. It can solve 01 NLP problems, integer NLP
problems and mixed integer NLP problems. When confronted with different types of
problems, we don’t need to change our algorithm. All that is needed is to input the fitness
function, the constraint expressions, and the upper and lower limits of the variables of the
problem. Our algorithm usually finds the global optimal value.
In the paper we analyze the character of the multiparent genetic algorithm, when applied to
solve the optimization of multimodal function, MPGA works in different forms during
different phases and then forms twophase genetic algorithm. The experiments indicate that
DDEA is effective to solve the optimization of multimodal function whose dimension is no
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179
higher than two, but to highdimension function, the efficiency is not eminent and it needs
to be improved much more.
6. Acknowledgements
This work was supported by the National Natural Science Foundation of China
(No.60772163) and the Natural Science Foundation of Hubei Province (No. 2005ABA234).
Thanks especially give to the anonymous reviewers for their valuable comments.
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