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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

(CEC 99), Volume 1, Piscataway , NJ, (1999), pp. 98 – 105. IEEE Press.

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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.

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E-mail address: ddumitr|cgrosan|moltean@cs.ubbcluj.ro