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STUDIA UNIV. BABES¸–BOLYAI, INFORMATICA, Volume XLVI, Number 1, 2001

GENETIC CHROMODYNAMICS FOR OBTAINING

CONTINUOUS REPRESENTATION OF PARETO REGIONS

D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

Abstract. In [5] an evolutionary algorithm for detecting continuous Pareto

optimal sets has been proposed. In this paper we propose a new evolutionary

elitist approach combing a non-standard solution representation and an evo-

lutionary optimization technique. The proposed method permits detection

of continuous decision regions. In our approach an individual (a solution) is

either a closed interval or a point. The individuals in the ﬁnal population

give a realistic representation of Pareto optimal set. Each solution in this

population corresponds to a decision region of Pareto optimal set. Proposed

technique is an elitist one. It uses a unique population. Current population

contains non- dominated solutions already founded.

Keywords: evolutionary multiobjective optimization, Pareto set, Pa-

reto frontier, Pareto interval

1. Introduction

Several evolutionary algorithms for solving multiobjective optimization prob-

lems have been proposed ([2], [5]–[10], [12], [13]; see also the reviews [1], [11] and

[14]).

Usually Pareto evolutionary algorithms aim to give a discrete picture of the

Pareto optimal set (and of the corresponding Pareto frontier). But generally

Pareto optimal set is a continuous region in the search space. Therefore a con-

tinuous region is represented by a discrete set. When continuous decision regions

are represented by discrete solutions there is loss of information. Moreover re-

constructing continuous Pareto set from a discrete picture is not an easy task

[11].

In [5] an evolutionary algorithm for detecting continuous Pareto optimal sets

has been proposed. The method proposed in [5] uses Genetic chromodynamics

evolutionary technique [4] to maintain population diversity.

2000 Mathematics Subject Classiﬁcation. 68T05.

1998 CR Categories and Descriptors. I.2.8 [Computing Methodologies]: Artiﬁcial In-

telligence – Problem Solving, Control Methods, and Search.

15

16 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

In this paper a new evolutionary approach, combing a non-standard solution

representation and a Genetic Chromodynamics optimization technique, is consid-

ered. Within the proposed approach continuous decision regions may be detected.

A solution (individual) is either a closed interval or a point (considered as a de-

generated interval). Mutation is the unique search operator considered.

The mutation operator idea is to expand each individual toward left and toward

right. In this respect both interval extremities are mutated. The left extremity is

mutated towards left and the right extremity is mutated towards right.

To reduce population size and to obtain the correct number of solutions within

the ﬁnal population the merging operator introduced in context of Genetic Chro-

modynamics is used.

The solutions in the ﬁnal population supply a realistic representation of Pareto

optimal set. Each solution in this population corresponds to a decision region (a

subset of Pareto set). A decision region will also be called a Pareto region.

The solutions are detected in two stages. In the ﬁrst stage a Genetic Chromo-

dynamics technique is used to detect all (local and global) Pareto solutions. In the

second stage the solutions are reﬁned. During the ﬁne tunning the sub-optimal

regions are removed.

The evolutionary multiobjective technique proposed in this paper is called Con-

tinuous Pareto Set (CPS) algorithm.

2. Problem statement

Let Ω be the search space. Consider nobjective functions f1,f2, . . . , fn,

fi: Ω → R,

where Ω ⊂ R.

Consider the multiobjective optimization problem:

½optimize f(x) = (f1(x), . . . , fn(x))

subject to x∈Ω

The key concept in determining solutions of multiobjective optimization prob-

lems is that of Pareto optimality. In what follows we recall some basic deﬁnitions.

Deﬁnition1. (Pareto dominance) Consider a maximization problem. Let x,y

be two decision vectors (solutions) from Ω. Solution x dominate y (also written

as xÂy) if and only if the following conditions are fulﬁlled:

(i) fi(x)≥fi(y),∀i= 1,2, . . . , n,

(ii) ∃j∈ {1,2, . . . , n}:fj(x)> fj(y).

Deﬁnition 2. Let S⊆Ω. All solutions, which are not dominated by any

vector of S, are called nondominated with respect to S.

GENETIC CHROMODYNAMICS FOR PARETO REGIONS 17

Deﬁnition 3. Solutions that are nondominated with respect to S,S⊂Ω, are

called local Pareto solutions or local Pareto regions.

Deﬁnition 4. Solutions that are nondominated with respect to the entire

search space Ω are called Pareto optimal solutions.

Let us note that when the search space is a subset of R, then Pareto optimal

set may be represented as:

(i) a set of points;

(ii) a set of disjoint intervals;

(iii) a set of disjoint intervals and a set of points.

Remark. In each of the cases (i), (ii) and (iii) a point or an interval represents

a Pareto region.

Evolutionary multiobjective optimization algorithms are intended for supplying

a discrete picture of Pareto optimal set and of Pareto frontier. But Pareto set is

usually a union of continuous set. When continuous decision regions are repre-

sented by discrete solutions there is some information loss. The resulting sets are

but a discrete representation of their continuous counterparts.

Methods for ﬁnding Pareto optimal set and Pareto optimal front using discrete

solutions are computationally very diﬃcult. However the results may be accepted

as the ‘best possible? at a given computational resolution. Methods for obtaining

the continuous representation using discrete outputs of evolutionary techniques

are considered in Veldhuizen [11].

The evolutionary method proposed in this paper directly supply the true (i.e.

possibly continuous) Pareto optimal set.

3. Solution representation and domination

In this paper we consider solutions are represented as intervals in the search

space Ω.

Each interval-solution kis encoded by an interval [xk, yk]∈ R. Degenerated

intervals are allowed. Within degenerate case yk=xkthe solution is a point.

In order to deal with proposed representation a new domination concept is

needed. This domination concept is given by the next deﬁnition.

Deﬁnition 5. An interval-solution [x, y] is said to be interval-nondominated if

and only if all points of that interval [x, y] are nondominated point-wise solutions.

An interval-nondominated solution will be called a Pareto-interval.Remarks.

(i) If x=ythis concept reduced towards ordinary non-domination notion.

(ii) If no ambiguity arise we will use nondominated instead of interval-nondomi-

nated.

18 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

4. Mutation

Problem solutions are detected in two stages. In the ﬁrst stage a Genetic

Chromodynamics technique is used to detect all (global and local) solutions. This

represent the evolution stage.

In the second stage (ﬁne tuning or reﬁnement stage) solutions that have been

detected in the ﬁrst stage are reﬁned. By using the reﬁnement procedure sub-

optimal Pareto regions are removed from the ﬁnal population.

Most of the multiobjective optimization techniques based on Pareto ranking

use a secondary population (an archive) denoted Psecond for storing nondomi-

nated individuals. Archive members may be used to guide the search process. As

dimension of secondary population may dramatically increase several mechanisms

for reducing archive size have been proposed. In [13] and [14] a population de-

creasing technique based on a clustering procedure is considered. We may observe

that preserving only one individual from each cluster implies a loss of information.

In our approach the population size does not increase during the search process

even if the number of Pareto optimal points increase. The population size is kept

low due to the special representation we consider.

When a new nondominated point is found it replaces another point solution in

the population or it is used for building a new interval solution with another point

in the population. This does not cause any information loss concerning Pareto

optimal set during the search process.

The algorithm starts with a population of degenerated intervals (i.e. a pop-

ulation of points). The unique variation operator is mutation. It consists of

normal perturbation of interval extremities. Mutation can also be applied to

point-solutions (considered as degenerated intervals). Each interval extremity is

mutated. The left extremity of an interval is always mutated towards left and the

right interval extremity is mutated only towards right.

For mutation two cases are to be considered.

a) Degenerated interval: The individual is mutated towards left or right

with equal probability. The obtained point represents the oﬀspring. Par-

ent and oﬀspring compete for survival. The best, in the sense of domi-

nation, enter the new population.

If parent and oﬀspring are not comparable with respect to domina-

tion relation then the two points deﬁnes an interval solution. The new

interval solution is included in the new generation. The point solution

representing the parent is discarded.

GENETIC CHROMODYNAMICS FOR PARETO REGIONS 19

b) Nondegenerated interval: (i) Firstly the left extremity of the in-

terval [u, v] is mutated towards left. A point-oﬀspring u0is ob-

tained. Consider the case when the oﬀspring u0and the parent u

do not dominate each other. In this situation a new interval solu-

tion [u0, v] is generated. The new solution has the oﬀspring u0as

its left extremity and vas its right extremity. If the oﬀspring u0

dominates the parent u, then the interval solution [u, v] enters the

new population.

(ii) A similar mutation procedure is applied to the right interval ex-

tremity of the previously obtained solution ([u, v] or [u0, v]).

5. Population model

For each generation every individual in the current population is mutated. Par-

ents and oﬀspring directly compete for survival. The domination relation guides

this competition.

For detecting the correct number of Pareto optimal regions it is necessary to

have, in the ﬁnal population, only one solution per Pareto optimal region.

In this paper we consider the merging operator of Genetic Chromodynamics

for implementing the population decreasing mechanism. Very close solutions are

fused and population size decreases accordingly.

In the framework of this paper the merging operator is described as bellow:

(i) if two interval solutions overlap the shortest interval solution is discarded.

Degenerated interval- solution included into non-degenerated interval-solu-

tions are removed too;

(ii) if two degenerated solutions are closer than a ﬁxed threshold, then the worst

solution is discarded.

The merging operator is applied each time a new individual enters the popula-

tion.

The method allows a natural termination condition. The algorithm stops when

there is no improvement of the solutions for a ﬁxed number of generations. Each

solution in the last population supplies a Pareto optimal region contributing to

the picture of Pareto optimal set.

6. Fine tuning

During the ﬁne tuning stage sub-optimal solutions (regions) are removed. For

this aim each continuous Pareto region is transformed into a discrete set. Dis-

cretized version is obtained considering points within each interval solution at a

ﬁxed step size.

20 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

Let us denote by ss the step size. Consider an interval solution [x, y]. From

this solution consider the points xj fulﬁlling the conditions:

•xj=x+j·ss, j = 0,1, . . .;

•xj≤y.

These points represent the discretized version (denoted D) of the interval solu-

tion [x, y].

Each point xjwithin the interval solutions is checked. If a point from the

discretized set Ddominates the point xjthen xjis removed from the Pareto

interval [x, y] together with a small neighboring region. The size of the removed

region is equal with ss.

The intervals obtained after this stage are considered as the true Pareto sets.

7. Algorithm Complexity

The complexity of the proposed algorithm is low. Let mbe the number of

objectives and Nthe population size. The ﬁrst stage requires

K1=O(m·N·IterationN umber)

operations.

Denote by Imax is the longest interval solution in the population. Consider a

function

F:R × R → N.

Admit that Ffulﬁlls the following conditions:

(i) Fis a linear function,

(ii) F([a, a]) = 1, for each a∈(?∞,∞).

Second stage (ﬁne tuning) requires

K2=O(N2·F(Imax)2)

operations.

8. CPS Algorithm

Using the previous considerations we are ready to design a new multiobjective

optimization algorithm.

The evolution stage of the CPS algorithm is outlined as bellow:

Algorithm CPS:

generates an initial population P(0) {all intervals are degenerated i.e. xk=yk}

t= 0;

while not (Stop Condition) do

GENETIC CHROMODYNAMICS FOR PARETO REGIONS 21

begin

for each individual kin P(t)do

begin

p=random {generate a random number between 0 and 1}

if p < 0.5

then

Left oﬀspr = MutateTowardsLeft(xk);

if dominate(Left oﬀspr, xk)

then

if xk=yk

then xk=yk= Left oﬀspr

else

if nondominated(Left oﬀspr, xk)

then xk= Left oﬀspr;

else

Right oﬀspr = MutateTowardsRight(yk);

if dominate(Right oﬀspr, yk)

then

if xk=yk

then xk=yk= Right oﬀspr

else

if nondominated(Right oﬀspr, yk)

then yk= Right oﬀspr;

endif

endfor

t=t+ 1

endwhile

Fine tuning part of CPS algorithm is obvious.

9. Numerical experiments

Several numerical experiments using CPOS algorithm have been performed.

For all examples the detected solutions gave correct representations of Pareto set

with an acceptable accuracy degree. Some particular examples are given below.

Example 1. Consider the functions f1, f2: [−10,13] → R deﬁned as

f1(x) = sin(x),

f2(x) = sin(x+ 0.7).

22 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

Consider the multiobjective optimization problem:

½optimize f(x) = (f1(x), f2(x))

subject to x∈[−10,13]

The initial population is depicted in Figures 1(a) and 1(b).

(a) (b)

Figure 1. (a) Initial population represented within solution

space; (b) Initial population represented within objective space

Consider the value

σ= 0.1

for the standard deviation of mutation operator. Solutions obtained after 3 gen-

erations are depicted in Figures 2(a) and 2(b).

The ﬁnal population, obtained after 42 generations, is depicted in Figures 3(a)

and 3(b).

The ﬁnal population after the reﬁnement stage is depicted in Figures 4(a) and

4(b).

Solutions in the ﬁnal population are:

s1= [−8.47,−7.86],

s2= [−2.26,−1.56],

s3= [4.01,4.69],

s4= [10.29,10.99].

GENETIC CHROMODYNAMICS FOR PARETO REGIONS 23

(a) (b)

Figure 2. (a) Population obtained after 3 generations repre-

sented within solution space; (b) Population obtained after 3 gen-

erations represented within objective space

(a) (b)

Figure 3. (a) Population obtained at convergence (after 42 gen-

erations) represented within solution space; (b) Population ob-

tained at convergence (after 42 generations) represented within

objective space

Example 2. Consider the functions f1, f2: [−10,20] → R deﬁned as

f1(x) = sin(x),

24 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

(a) (b)

Figure 4. (a) Final population obtained after ﬁne tuning stage

represented within solution space; (b) Final population obtained

after ﬁne tuning stage represented within objective space

f2(x) = 2 ·sin(x)+1.

Consider the multiobjective optimization problem:

½optimize f(x) = (f1(x), f2(x))

subject to x∈[−10,20]

The initial population is depicted in Figures 5(a) and 5(b).

For the value

σ= 0.1

solutions obtained after 14 generations are depicted in Figures 6(a) and (b).

The ﬁnal population, obtained after 70 generations, is depicted in Figures 7(a)

and 7(b).

Example 3. Consider the functions f1, f2: [0,24] → R deﬁned as

f1(x) = sin(x),

f2(x) =

−4·x

π+ 8 ·k, 2·k·π≤x < 2·k·π+π

2,

4·x

π−4·(2 ·k+ 1),2·k·π+π

2≤x < (2 ·k+ 1) ·π,

−2·x

π+ 2 ·(2 ·k+ 1),(2 ·k+ 1) ·π≤x < (2 ·k+ 1) ·π+3·π

2,

2·x

π−4·(k+ 1),2·k·π+3·π

2≤x < 2·(k+ 1) ·π+π

2.

where k∈Z+.

GENETIC CHROMODYNAMICS FOR PARETO REGIONS 25

(a) (b)

Figure 5. (a) Initial population represented within solution

space; (b) Initial population represented within objective space

(a) (b)

Figure 6. (a) Population obtained after 14 generations repre-

sented within solution space; (b) Population obtained after 14

generations represented within objective space

Consider the multiobjective optimization problem:

½optimize f(x) = (f1(x), f2(x))

subject to x∈[0,24]

26 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

(a) (b)

Figure 7. (a) Final population obtained after 70 generations rep-

resented within solution space; (b) Final population obtained after

70 generations represented within objective space

The initial population is depicted in Figure 8(a) and 8(b).

(a) (b)

Figure 8. (a) Initial population represented within solution

space; (b) Initial population represented within objective space

For the value

σ= 0.1

GENETIC CHROMODYNAMICS FOR PARETO REGIONS 27

solutions obtained after 4 generations are depicted in Figures 9(a) and 9(b).

(a) (b)

Figure 9. (a) Population obtained after 4 generations repre-

sented within solution space; (b) Population obtained after 4 gen-

erations represented within objective space

The ﬁnal population, obtained after 60 generations, is depicted in Figures 10(a)

and 10(b).

The ﬁnal population after the reﬁnement stage is depicted in Figure 11(a) and

11(b).

10. Concluding remarks and further research

A new evolutionary technique for solving multiobjective optimization problems

involving one variable functions is proposed. A new solution representation is used.

Standard search (variation) operators are modiﬁed accordingly. The proposed

evolutionary multiobjective optimization technique uses only one population. This

population consists of nondominated solutions already founded.

All known standard or recent multiobjective optimization techniques supply

a discrete picture of Pareto optimal solutions and of Pareto frontier. But Pareto

optimal set is usually non-discrete. Finding Pareto optimal set and Pareto optimal

frontiers using a discrete representation is not a very easy computationally task

(see [11]).

Evolutionary technique proposed in this paper supplies directly a continuous

picture of Pareto optimal set and of Pareto frontier. This makes our approach

28 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN

(a) (b)

Figure 10. (a) Population obtained at convergence (after 60 gen-

erations) represented within solution space; (b) Population ob-

tained at convergence (after 60 generations) represented within

objective space

(a) (b)

Figure 11. (a) Final population obtained after ﬁne tuning stage

represented within solution space; (b) Final population obtained

after ﬁne tuning stage represented within objective space

very appealing for solving problems where very accurate solutions detection is

needed.

GENETIC CHROMODYNAMICS FOR PARETO REGIONS 29

Another advantage is that CPS technique has a natural termination condition

derived from the nature of evolutionary method used for preserving population

diversity.

Experimental results suggest that CPS algorithm supplies correct solutions after

few generations.

Further research will focus on the possibilities to extend the proposed technique

to deal with multidimensional domains.

Another research direction is to exploit the solution representation as intervals

for solving inequality systems and other problems for which this representation

seems to be natural.

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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