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In this paper a new evolutionary paradigm is proposed. A technique called Adaptive Representation Evolutionary Algorithm (AREA) based on this paradigm is designed. AREA involves dynamic alphabets for encoding solutions. Proposed adaptive representation is more compact than binary representation. Mutation is the unique variation operator. Mutations are usually more aggressive when higher alphabets are used. Therefore the proposed encoding ensures an efficient exploration of the search space. Numerical experiments seem to indicate that APA process better than the best multiobjective evolutionary algorithms. An AREA technique is used for solving multiobjective optimization problems. The resulting algorithm is called Adaptive Pareto Algorithm (APA).
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STUDIA UNIV. BABES¸–BOLYAI, INFORMATICA, Volume XLVI, Number 2, 2001
A NEW EVOLUTIONARY ADAPTIVE REPRESENTATION
PARADIGM
D. DUMITRESCU, CRINA GROS¸AN, MIHAI OLTEAN
Abstract. In this paper a new evolutionary paradigm is proposed. A tech-
nique called Adaptive Representation Evolutionary Algorithm (AREA) based
on this paradigm is designed. AREA involves dynamic alphabets for encod-
ing solutions. Proposed adaptive representation is more compact than bi-
nary representation. Mutation is the unique variation operator. Mutations
are usually more aggressive when higher alphabets are used. Therefore the
proposed encoding ensures an efficient exploration of the search space.
Numerical experiments seem to indicate that APA process better than
the best multiobjective evolutionary algorithms.
An AREA technique is used for solving multiobjective optimization
problems. The resulting algorithm is called Adaptive Pareto Algorithm (APA).
Keywords: Evolutionary Computation, Evolutionary Multiobjective
Optimization, Pareto Set, Higher Alphabet Encoding, Adaptive Representa-
tion.
1. Introduction
In this paper we propose a new evolutionary paradigm. An algorithm based on
this paradigm and using a powerful adaptive representation is designed. The al-
gorithm called Adaptive Representation Evolutionary Algorithm (AREA). AREA
technique operators are mutation and selection for survival.
Many multiobjective optimization techniques using evolutionary algorithms have
been proposed in recent years. Strength Pareto Evolutionary Algorithm (SPEA,
[9]), Pareto Archived Evolution Strategy (PAES, [4]), Pareto Envelope based Se-
lection Algorithm (PESA, [1]), Nondominated Sorting Genetic Algorithm (NSGA
II, [3]) and SPEA II ([10]) are the best present-day Multiobjective Evolutionary
Algorithms (MOEAs).
Multi-alphabet representation proposed in this paper induces a powerful diver-
sity maintaining mechanism. For this reason AREA technique seems to be very
suitable for evolutionary multiobjective optimization purposes. Considered adap-
tive encoding allows solutions in the final population realizing a realistic picture
of Pareto frontier.
2000 Mathematics Subject Classification. 68T05.
1998 CR Categories and Descriptors. I.2.8 [Computing Methodologies]: Artificial In-
telligence Problem Solving, Control Methods, and Search.
19
20 D. DUMITRESCU, CRINA GROS¸AN, MIHAI OLTEAN
Numerical experiments with APA technique include several test functions rep-
utated as difficult ([2], [7]) and comparisons with the best MOEAs.
The paper is structured as follows: Section 2 is a short resume of the principal
recent evolutionary techniques for multiobjective optimization. Section 3 describes
the proposed algorithm. In Section 4 a comparison of the proposed approach with
some very efficient multiobjective evolutionary techniques is realized.
2. Recent MOEAs
In the last years a number of evolutionary algorithms for multiobjective opti-
mization have been proposed. Some of them will be shortly reviewed here.
2.1. Strength Pareto Evolutionary Algorithm. Zitzler and Thiele proposed
an elitist evolutionary algorithm called Strength Pareto Evolutionary Algorithm
(SPEA) ([9, 7]). The algorithm maintains an external population at every gen-
eration storing all nondominates solutions obtained so far. At each generation
external population is mixed with the current population. All nondominated solu-
tions in the mixed population are assigned fitness based on the number of solutions
they dominate. Dominated solutions are assigned fitness worse than the worst fit-
ness of any nondominates solutions. A deterministic clustering technique is used
to ensure diversity among nondominates solutions.
Pareto Archived Evolution Strategy
Knowles and Corne [4] proposed a simple evolutionary algorithm called Pareto
Archived Evolution Strategy (PAES). In PAES one parent generates by mutation
one offspring. The offspring is compared with the parent. If the offspring dom-
inates the parent, the offspring is accepted as the next parent and the iteration
continues. If the parent dominates the offspring, the offspring is discarded and
the new mutated solution (a new offspring) is generated. If the offspring and the
parent do not dominate each other, a comparison set of previously nondominated
individuals is used.
For maintaining population diversity along Pareto front, an archive of nondom-
inated solutions is considered. A new generated offspring is compared with the
archive to verify if it dominates any member of the archive. If yes, then the off-
spring enters the archive and is accepted as a new parent. The dominated solutions
are eliminated from the archive. If the offspring does not dominate any member
of the archive, both parent and offspring are checked for their nearness with the
solution of the archive. If the offspring resides in the least crowded region in the
parameter space among the members of the archive, it is accepted as a parent and
a copy is added to the archive.
2.2. Nondominated Sorting Genetic Algorithm. Deb and his students [3]
suggested a fast elitist Nondominated Sorting Genetic Algorithm (NSGA II). In
NSGA II, for each solution x the number of solutions that dominate solution x is
calculated. The set of solutions dominated by x is also calculated. The first front
(the current front) of the solutions that are nondominated is obtained.
A NEW EVOLUTIONARY ADAPTIVE REPRESENTATION PARADIGM 21
Let us denote by S
i
the set of solutions that are dominated by the solution
x
i
. For each solution x
i
from the current front consider each solution x
q
from
the set S
i
. The number of solutions that dominates x
q
is reduced by one. The
solutions that remain nondominates after this reduction will form a separate list.
This process continues using the newly identified front as the current front.
Let P (0) be the initial population of size N. An offspring population Q(t) of size
N is created from current population P (t). Consider the combined population:
R(t) = P (t) Q(t).
Population R(t) is ranked according nondomination. The fronts F
1
, F
2
, ... are
obtained. New population P (t+1) is formed by considering individuals from the
fronts F
1
, F
2
, ..., until the population size exceeds N. Solutions of the last allowed
front are ranked according to a crowded comparison relation.
NSGA II uses a parameter (called crowding distance) for density estimation for
each individual. Crowded distance of a solution x is the average side-length of the
cube enclosing the point without including any other point in the population. So-
lutions of the last accepted front are ranked according to the crowded comparison
distance.
NSGA II works as follows. Initially a random population, which is sorted based
on the nondomination, is created. Each solution is assigned a fitness equal to its
nondomination level (1 is the best level). Binary tournament selection, recom-
bination and mutation are used to create an offspring population. A combined
population is formed from the parent and offspring population. The population
is sorted according to the nondomination relation. The new parent population is
formed by adding the solutions from the first front and the followings until exceed
the population size. Crowding comparison procedure is used during the population
reduction phase and in the tournament selection for deciding the winner.
3. AREA technique
In this paper we propose a new evolutionary paradigm, The main idea is to al-
low each solution be encoded on a different alphabet. Moreover representation of
a particular solution is not fixed. Representation is adaptive and may be changed
during the search process as an effect of mutation operator, An adaptive represen-
tation evolutionary algorithm (AREA) based on the new paradigm is designed.
AREA technique proposed in this paper uses a fixed population. Each AREA
individual (chromosome) consists of a pair (x, B), where x is a string encoding
object variables and B specifies the alphabet used for encoding x.
B is an integer number, B > 2 and x is a string of symbols from the alphabet
{0, 1, . . . , B-1}. If B= 2, the standard binary encoding is obtained. The alphabet
over which x is encoded may change during the search process.
Mutation is the unique variation operator. For mutation, a random number
between 0 and 1 is uniformly generated for each position, including the last one, of
the chromosom. Each position (gene) value is modified with a mutation probability
(p
m
).
22 D. DUMITRESCU, CRINA GROS¸AN, MIHAI OLTEAN
Mutation can modify object variables as well as last position (fixing the repre-
sentation alphabet). If the position giving B is changed, then the object variables
will be represented using symbols over the new alphabet, corresponding to the mu-
tated value of B. When the changing gene belongs to the object variable sub-string
(x part of the chromosome), the mutated gene is a symbol randomly chosen from
the same alphabet.
4. APA method
In this section a new MOEA technique called Adaptive Pareto Algorithm (APA)
is proposed. APA relies on the AREA method previously described.
AREA uses a unique population. No external or intermediary population is
needed.
Initial p opulation is randomly generated. Each individual is selected for muta-
tion, which is the unique variation operator. The offspring and parent are com-
pared. Dominance relation guides the survival.
If the offspring dominates the parent then the offspring enters the new popu-
lation and the parent is removed. If the parent dominates the offspring obtained
in ksuccessive mutations then another alphabet is chosen and the parent is repre-
sented in symbols over this alphabet. In this case only representation is changed
and the encoded solution does not change. Adaptive representation mechanism
and the survival strategy is generates an effective and efficient diversity preserving
mechanism.
APA algorithm
Proposed APA algorithm may b e outlined as follows:
APA ALGORITHM:
begin
Set t = 0;
Random initializes chromosome population P (0 );
Set to zero the number of harmful mutations for each individual in P(0);
while (t ¡ number of generations) do
begin
for k = 1 to PopSize do
Mutate the k
th
chromosome from P (t ). An offspring is obtained.
If the offspring dominates the parent then the parent is removed and the offspring
is added to P (t+1);
else begin
Increase the number of harmful mutations for current individual;
If the number of harmful mutations = MAX HARMFUL MUTATIONS
then begin
Change the individual representation;
Set to zero the number of harmful mutations for the current individual;
endif
Add individual to P(t+1);
A NEW EVOLUTIONARY ADAPTIVE REPRESENTATION PARADIGM 23
endif
endfor;
Set t = t + 1;
endwhile;
end.
Despite its simplicity APA is able to generate a population converging towards
Pareto optimal set. Moreover, the diversity of the population is automatically
maintained without any specialized mechanism.
Proposed APA algorithm realizes a realistic picture of Pareto optimal set. Nu-
merical experiments emphasizes that for considered problems, APA technique is
more effective then best present-day MOEAs. Moreover, APA’s complexity is a
reduced one with respect to the MOEAs techniques considered for comparison.
5. Comparison of Several Evolutionary Multiobjective Algorithms
In this section complexity of the proposed APA technique is compared with the
complexity of several evolutionary multiobjective optimization algorithms (SPEA,
PAES and NSGA II).
Let us denote by m the numb er of objectives and by N the population size.
SPEA uses an internal and an external population. The fitness is assigned
differently to the individuals from these p opulations. A deterministic clustering
technique is used to reduce external population size the population diversity. The
complexity of this algorithm implementation is mN
2
.
PAES uses a single parent, which generate an offspring. An archive, which
maintains the nondominated solutions, is created. Let be a the archive size. The
worst case complexity for the PAES is amN. Since the archive size is usually
proportional to the population size N,the overall complexity of the algorithm is
mN
2
.
NSGA II computes for each individual x the number of solutions that dominates
it and the number of solution, which x dominates. NSGA II uses for this N
2
computations. Identifying the fronts requires (in the worst case) N
2
computations.
The overall complexity is (mN
2
+ N
2
) or N
2
. So, the complexity may increase
from N to N
2
(the worst case). Computation complexity for density estimation
is mN logN. For sorting the combined population 2N log(2N ) computations are
necessary. Overall NSGA II complexity is thus mN
2
.
APA uses a unique fixed size population. Each individual is considered for
mutation. This requires mN operations. The algorithm does not use a superposed
mechanism for diversity maintaining. Overall complexity of SMEA algorithm is
thus O(mN ).
6. Numerical experiments
In this section we compare the performance of APA algorithm with the perfor-
mances of SPEA, NSGA II, PAES.
24 D. DUMITRESCU, CRINA GROS¸AN, MIHAI OLTEAN
For this purpose by using six test functions introduced by Deb, Zitzler and
Thiele [7] are used.
6.1. Test functions. Each test function considered in this section is built by
using three functions f
1
, g, h. Biobjective function T considered here is
T (x) = (f
1
(x), f
2
(x)).
The optimization problem is:
½
Minimize T (x), w here f
2
(x) = g(x
2
, . . . , x
m
)h(f
1
(x
1
), g(x
2
, . . . x
m
),
x = (x
1
, . . . , x
m
)
The five test functions used in this paper for comparison are:
Test function T
1
is defined using the following functions:
f
1
(x
1
) = x
1
,
g(x
2
, . . . .x
m
) = 1 + 9 ·
P
m
i=2
x
i
/(m 1),
h(f
1
, g) = 1
p
f
1
/g,
where m= 30 and x
i
[0,1] i = 1,2,. . . ,m.
Pareto optimal front for the problem T
1
is convex and is characterized by the
equation
g(x) = 1.
Test function T
2
is defined by considering the following functions:
f
1
(x
1
) = x
1
g(x
2
, . . . .x
m
) = 1 + 9 ·
P
m
i=2
x
i
/(m 1)
h(f
1
, g) = 1 (f
1
/g)
2
where m = 30 and x
i
[0,1], i = 1,2,. . . ,m.
Pareto optimal front is characterized by the equation
g(x)=1.
T
2
is the nonconvex counterpart to T
1
.
Pareto optimal set corresponding to the Test function T
3
presents a discreten
feature. Pareto optimal front consists of several noncontiguous convex parts. The
involved functions are:
f
1
(x
1
) = x
1
g(x
2
, . . . .x
m
) = 1 + 9 ·
P
m
i=2
x
i
/(m 1)
h(f
1
, g) = 1
p
f
1
/g (f
1
/g) sin (10πf
1
)
where m= 30 and x
i
[0,1], i = 1,2,. . . m.
Pareto optimal front is characterized by the equation
g(x) = 1.
The introduction of the function sin in the expression of function h causes
discontinuity in the Pareto optimal front. However, there is no discontinuity in
the parameter space.
A NEW EVOLUTIONARY ADAPTIVE REPRESENTATION PARADIGM 25
The test function T
4
contains 21
9
local Pareto optimal fronts and, therefore, it
tests the EA ability to deal with multimodality. The involved functions are defined
by:
f
1
(x
1
) = x
1
g(x
2
, . . . .x
m
) = 1 + 10(m 1) +
P
m
i=2
(x
2
i
10 cos(4πx
i
))
h(f
1
, g) = 1
p
f
1
/g
where m = 10, x
1
[0,1] and x
2
,. . . ,x
m
[-5,5].
Global Pareto optimal front is characterized by the equation
g(x) = 1.
The best local Pareto optimal front is described by the equation
g(x) = 1.25.
Note that not all local Pareto optimal sets are distinguishable in the objective
space.
f
1
(x
1
) = 1 exp(4x
1
) sin
6
(6πx
1
)
g(x
2
, . . . .x
m
) = 1 + 9 · (
P
m
i=2
x
i
/(m 1))
0.25
h(f
1
, g) = 1 (f
1
/g)
2
The test function T
5
includes two
difficulties caused by the nonuniformity of the search space. First, the Pareto op-
timal solutions are nonuniformly distributed along the global Pareto optimal front
(the front is biased for solutions for which f
1
(x) is neat one). Second, the density
of the solutions is lowest near the Pareto optimal front and highest away from the
front.
This test function is defined by using:
where m= 10, x
i
[0,1], i = 1,2,. . . m.
The Pareto optimal front is characterized by the equation
g(x) = 1
and is nonconvex.
6.1.1. Numerical comparisons. Several numerical experiments were performed with
APA. According to these experiments APA gives a good approximation of the
Pareto front for all considered test functions.
For both test functions T
1
and T
2
, the differences between the four considered
algorithms are very small (see Figure 1 and Figure 2).
The difference between APA and the other algorithms is significant for test
function T 4. APA gives the best arrangement on the front. Goo d solution distri-
bution is obtained also by NSGA II and SPEA. Solutions distribution realized by
NSGA II and SPEA are close to Pareto front. Moreover solution only distribution
supplied by APA is covers the real front.
For test function T
5
, APA also gives the best solution arrangement on the Pareto
front. PAES also gives distribution.
In these comparisons 25.000.000 function evaluations have been considered for
each algorithm. This ensures a realistic comparison of the algorithm outputs.
26 D. DUMITRESCU, CRINA GROS¸AN, MIHAI OLTEAN
Figure 1. Results for test function T
1
. Pareto optimal front is convex
Figure 2. Results for test function T 2. Pareto optimal front is nonconvex
Conclusions
In this paper a new evolutionary paradigm is proposed. An evolutionary algo-
rithm (AREA) based on the new paradigm is designed.
A new evolutionary algorithm (called APA) for multiobjective optimization is
also proposed. AREA uses a new, dynamic solution representation.
APA technique is compared with four well-known evolutionary multiobjective
optimization algorithms. The results show that APA performs better than con-
sidered algorithms.
A NEW EVOLUTIONARY ADAPTIVE REPRESENTATION PARADIGM 27
Figure 3. Results for test function T 3. All considered algo-
rithms give a good approximation of the Pareto front
Figure 4. Results for test function T
4
. For test function T
4
,
APA gives the best arrangement on the Pareto front. NSGA II
and SPEA converge toward global Pareto front. PAES did not
converge to the global Pareto front
References
[1] Deb, K., Multi-objective genetic algorithms: Problem difficulties and construction of test
functions. Evolutionary Computation, 7(3), (1999), pp. 205-230.
[2] Deb, K., S. Agrawal, Amrit Pratap and T. Meyarivan, A fast elitist non dominated sorting
genetic algorithm for multi-objective optimization: NSGA II. In M. S. et al. (Ed), Parallel
Problem Solving From Nature PPSN VI, Berlin, (2000), pp. 849 858. Springer.
28 D. DUMITRESCU, CRINA GROS¸AN, MIHAI OLTEAN
Figure 5. Results for test function T
5
[3] Dumitrescu, D., Gro¸san, C., Oltean, M., Simple Multiobjective Evolutionary Algorithm,
Seminars on Computer Science, Faculty of Mathematics and Computer Science, Babe¸s-
Bolyai University of Cluj-Napoca, 2001, pp. 3-12.
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algorithm for Pareto multiobjective optimization. In Congress on Evolutionary Computation
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Dortmund, (1998).
[6] Zitzler, E., Evolutionary algorithms for multiobjective optimization: Methods and Applica-
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Schriftenreiche Nr. 30, Diss ETH No. 13398, (1999), Shaker Verlag, Aachen, Germany.
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“Babes¸-Bolyai” University, Faculty of Mathematics and Computer Science, 1 M.
Kog
˘
alniceanu Street, RO-3400 Cluj-Napoca, Romania
E-mail address: ddumitr|cgrosan|moltean@cs.ubbcluj.ro
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