Content uploaded by Mihai Oltean
Author content
All content in this area was uploaded by Mihai Oltean on Sep 05, 2015
Content may be subject to copyright.
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.
References
[1] C. A. C. Coello, A comprehensive survey of evolutionary-based multiobjective optimization
techniques, Knowledge and Information Systems, 1(3), pp. 269–308, 1999.
[2] K. Deb, Multiob jective evolutionary algorithms: problem difficulties and construction of
test problems, Evolutionary Computation, 7, pp. 205–230, 1999.
[3] D. Dumitrescu, B., Lazzerini, L.C. Jain, A. Dumitrescu, Evolutionary Computation, CRC
Press, Boca Raton, FL, 2000.
[4] D. Dumitrescu, Genetic chromodynamics, Studia, Babes-Bolyai University, Ser. Informatica,
45, 2000, pp 39–50, 2000
[5] D. Dumitrescu, C. Grosan, M. Oltean, An evolutionary algorithm for detecting continuous
Pareto regions, Studia Babes-Bolyai University, Ser. Informatica, 45, pp. 51–68, 2000.
[6] D. E. Goldberg, Evolutionary Algorithms in Search, Optimization and Machine Learning,
Addison Wesley, Reading, 1989.
[7] J. Horn, N. Nafpliotis, Multiobjective optimization using niched pareto evolutionary algo-
rithms, IlliGAL Report 93005, Illionois Evolutionary Algorithms Laboratory, University of
Illinois, Urbana Champaingn.
[8] J. Horn, N. Nafpliotis, D. E. Goldberg, A niche Pareto evolutionary algorithm for multiob-
jective optimization, Proc. 1st IEEE Conf. Evolutionary Computation, Piscataway, vol 1,
pp. 82–87, 1994.
[9] J. D. Schaffer, Multiple objective optimization with vector evaluated evolutionary algo-
rithms, Evolutionary Algorithms and Their Applications, J.J. Grefenstette (Ed.), Erlbaum,
Hillsdale, NJ, pp. 93–100, 1985.
[10] N. Srinivas, K., Deb, Multiobjective function optimization using nondominated sorting evo-
lutionary algorithms, Evolutionary Computing, 2, pp. 221–248, 1994.
[11] D.A.V. Veldhuizen, Multiobjective Evolutionary Algorithms: Classification, Analyses and
New Innovations, Ph.D Thesis, Graduated School of Engineering of the Air Force Institute
of Technology, Air University, 1999.
[12] D.A.V. Veldhuizen, G. B. Lamont, Multiobjective evolutionary algorithms: analyzing the
state- of-the-art, Evolutionary Computation, 8, pp. 125–147, 2000.
[13] E. Zitzler, L. Thiele, Multiobjective evolutionary algorithms: A comparative study and the
strength Pareto approach, IEEE Trans on Evolutionary Computation, 3, pp. 257–271, 1999.
30 D. DUMITRESCU, CRINA GROS¸AN, AND MIHAI OLTEAN
[14] E. Zitzler, Evolutionary Algorithms for Multiobjective Optimization: Methods and Applica-
tions, Doctoral Dissertation, Swiss Federal Institut of Technology Zurich,Tik-Schriftenreihe
nr. 30, 1999.
“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