CYLINDRICAL CONSTRAINT EVOLUTIONARY ALGORITHM
FOR MULTIOBJECTIVE OPTIMIZATION
Tohid Erfani and Sergei V. Utyuzhnikov
The University of Manchester, School of Mechanical, Aerospace and Civil Engineering
George Begg Building, M13 9PL, Manchester, U.K.
Keywords:
Cylindrical constraint method, Multiobjective optimization, Evolutionary optimization, Evolutionary algo-
rithms, Genetic algorithm.
Abstract:
This paper introduces a new iterative evolutionary algorithm, which is able to provide an evenly distributed
set of solutions in multiobjective context. The method is different from the other evolutionary algorithms in
two perspectives. First, instead of density information incorporated to find a diverse set of solutions, a hyper-
cylinder is introduced as a new constraint to the problem. Searching for the solution within this hypercylinder
guarantees the evenly generated solutions at the end of the optimization process. Second, a fitness function
is constructed to handle the problem constraints and meanwhile minimize the distance of the solution to the
true optimum frontier. In addition, the method is developed in such a way that it can be easily implemented in
searching the preferable region of the search space. The algorithm behaviour is tested on different test cases
and the results are compared in both convergence and diversity to those of other well known approaches to
demonstrate the efficacy of the proposed method.
1 INTRODUCTION
Most of the optimization problems in the real world
are inherently multiobjective. That is, for a single
problem, there exists possibly conflicting objectives
which need to be optimized simultaneously. This
leads to generating a so called trade-off surface. The
solution on this frontier is called a Pareto optimal so-
lution in a sense that no improvement can be made
with respect to one objective without deterioration
of at least one of the other. In the literature, there
are two principle approaches to solving such prob-
lems; classical directional (gradient) based methods
and evolutionary algorithms. In the first category,
all the objective functions are aggregated to form a
single objective optimization sub-problem. Differ-
ent sub-problems are then solved for a different po-
sition in the search space to obtain the Pareto fron-
tier. Normalized Normal Constraint method (NNC)
(Messac et al., 2003) and Directed Search Domain
method (DSD) (Erfani and Utyuzhnikov, 2010) are
amongst the directional methods capable in generat-
ing the whole Pareto surface. In the latter category,
the focus is to get the whole trade-off surface in a
single optimization run using evolutionary techniques
such as NSGA-II (Deb et al., 2002) and SPEA2 (Zit-
zler et al., 2001) as two well-studied algorithms. Gen-
erally, there are two issues which need to be accom-
plished in both categories. Firstly, the generated solu-
tions should have minimum distance to the true trade-
off surface and secondly, a diverse set of solutions
should be obtained, which guarantees a good approxi-
mation of the whole Pareto frontier. The latter issue is
specially important in engineering design, where the
designer is only capable in analysing a very limited
set of solutions.
With respect to the first task, the directional meth-
ods utilize the gradient based search for any single ob-
jective optimization sub-problem to move towards the
optimal solution quickly and efficiently (Jahn, 2004).
However, since these methods need the starting point
to initialize the sub-problem search, different start-
ing values may result in different solutions (Boyd and
Vandenberghe, 2004). That is, the possibility of get-
ting stuck to a local optimum brings an added diffi-
culty to the methods. On the other hand, in the evolu-
tionary techniques, with different detailed procedure
for different algorithms, a mating selection is used to
mainly rank the individuals based on the Pareto def-
inition (Zitzler et al., 2001). For example, in SPEA2
the the strength of a solution is defined by the num-
ber of solutions it dominates (Zitzler et al., 2001). In
184
Erfani T. and V. Utyuzhnikov S..
CYLINDRICAL CONSTRAINT EVOLUTIONARY ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION.
DOI: 10.5220/0003670101840189
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 184-189
ISBN: 978-989-8425-83-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

NSGA-II, the solutions are ranked in different fron-
tiers based on the Pareto definitions (Deb et al., 2002).
In both cases, optimizing the rank and strength of the
solutions, so far found by the algorithms, assures the
proximity of the current solution to the Pareto frontier.
The second task is also tackled differently by these
two classes of methods. In the directional methods,
a so called utopia plane is formed by an evenly dis-
tributed set of points (Das and Dennis, 1998). The
aim is to achieve an evenly distributed set of Pareto
solutions by solving numerous sub-problems corre-
sponding to each point. However, as shown by (Das
and Dennis, 1998; Messac et al., 2003; Erfani and
Utyuzhnikov, 2010), the points on utopia plane may
not cover the whole Pareto surface and hence loosing
some portion of solutions. To alleviate the inadequate
extent of the utopia plane, specifically NNC and DSD
methods utilize the extended utopia plane and rota-
tion technique, respectively. However, in the case of
degeneration of the utopia plane to lower dimension,
NNC technique may not sound promising. In the evo-
lutionary algorithms, on the other hand, the density
information is exploited to define an extra order be-
tween the individuals (Zitzler and Thiele, 2002). In
SPEA2, the k-th nearest neighbour method is used to
consider the density information of the individuals to
finally discard the solutions which are so close to each
other. In NSGA-II, a crowding distance is defined,
which is the average distance of the two points in ei-
ther side of an individual for each objective function.
This measure is then used to avoid a dense of solu-
tions in one part of the frontier in the case that some
other parts are isolated (Deb et al., 2002).
In this study we introduce a method which incor-
porates the desired strength of the evolutionary and
directional methodology in order to obtain a new al-
gorithm. In particular the main characteristics of the
method are:
First. To generate an evenly distributed set of solu-
tion, a hyperplane is constructed with evenly dis-
tributed points on it. Afterwards, a hypercylinder
constraint is defined for each point which its fea-
sibility in the optimization process guarantees the
evenly generated solution. This removes the bur-
den of sorting and clustering techniques for rank-
ing the solutions based on their density informa-
tion.
Second. With the mutation and recombination re-
main identical to those introduced in literature
(Haupt et al., 1998), a fitness function and sim-
ple sampling strategy is introduced to fill up the
mating pool for selection.
2 CYLINDRICAL CONSTRAINT
EVOLUTIONARY ALGORITHM
Our approach is based on generating an evenly dis-
tributed set of points m on a plane P defined by the
vectors of the Cartesian coordinate system ˆe
i
(e.g.
ˆe
1
= (1,0) and ˆe
2
= (0,1) for 2-objective case):
P =
n
∑
i=1
α
i
ˆe
i
,
n
∑
j=1
α
j
= 1, (1)
0 ≤ α
j
≤ 1.
In contrast to the other directional techniques in-
troduced in literature (Messac et al., 2003; Erfani and
Utyuzhnikov, 2010), we do not use the utopia hyper-
plane as it can not cover the whole Pareto surface. In-
stead, we define a direction from the utopia point U
∗
(the point which its components are the minimum of
each objective in each generation) to each m by
ν = m−U
∗
. (2)
This represents a set of rays emitting from U
∗
to-
wards the m on polygon P. Associated with each m,
a scalar optimization problem is solved. The point m
for each scalar problem serves as the problem con-
straint defined by a cylinder. The cylinder is con-
structed by a circle on a plane extended in direction
Feasible
point
Infeasible
point
Cylindrical
Constraint
m
Projected view
Normal view
Cylindrical base
Infeasible
point
Feasible
point
Projected
Cylinder
for m
ν vector
Figure 1: Explaining the CCEA method for a given point m
of ν (Look at Figure 1). Following the Figure 1, one
possible way to check whether the cylinder constraint
is satisfied is to project the current searching point on
the cylinder base and investigate whether or not the
projected point is inside the current circle centred on
m with predefined radius r. The cylinder base is a
plane with ν as its normal vector. Since the m ∈ P
are evenly generated, also the cylinders are and hence
(quasi) evenly distributed set of solutions are guaran-
teed by solving the optimization in each cylinder.
The Cylindrical Constraint Evolutionary Algo-
rithm (CCEA) is given below. In the first steps, the
CYLINDRICAL CONSTRAINT EVOLUTIONARY ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION
185

problem is initialized and individuals from pop is as-
signed to each m on P. This makes the size of the pop
to be the same as the number of the points m on P.
Therefore, each individual with dedicated point m
i
is
updated during the optimization of sub-problem i. In
addition, the utopia pointU
∗
is computed with respect
to the current population in objective space.
During the course of optimization of sub-problem
i corresponds to point m
i
, two points from pop are
randomly selected and binary tournament selection is
applied. The resultant p
b
from tournament is then re-
produced using genetic operators on p
b
and p
m
. The
offspring p
re
is checked for its fitness value. If it per-
forms better, it replaces the p
m
. In addition to this,
we have introduced one further checking. If the off-
spring p
re
does not perform better with respect to sub-
problem i, it may be a better solution for the other sub-
problems. That is, the genetic operators may produce
a solution near another cylinder which may perform
better with respect to that cylinder. Therefore, we
check the fitness of p
re
for the possible better perfor-
mance with respect to the other sub-problems. Since
the utopia point is changing from generation to gen-
eration, its value is updated using the new objective
values of the updated individuals.
To terminate the optimization procedure, it is rec-
ommended to follow two rules. If for point m
i
,
the average value of the best fitness function for a
given number of generation is unchanged with a pre-
scribed accuracy, the search in the cylinder i will be
stopped while its solution will still be used for the
other sub-problems’ reproduction and tournament se-
lection purposes. However, if the maximum gener-
ation is reached, the optimization procedure will be
terminated. Once the termination criteria is satisfied,
a Pareto filtering is implemented. The filtering aims to
eliminate the local Pareto solutions based on the non-
domination definition. One may find such a filtering
in (Erfani and Utyuzhnikov, 2010).
The tournament selection and fitness checking
steps described above are based on constructing the
following fitness function. The fitness function for
sub-problem i is defined with two separate terms in
order of their preference and importance:
Fitness
i
= α f
1
+ βf
2
. (3)
To define f
1
, we first describe the cylindrical con-
straint. Let F
c
= F(x
c
) be the current searching point
in objective space and F
p
c
be its orthogonal projection
on i-th cylinder base. Then,
d(F
p
c
,ν
i
) ≤ r (4)
is the cylindrical constraint for sub-problem i. d(,)
is the Euclidean distance between two points and r
is the predefined radius of the cylinder. To guarantee
one distinct solution in each cylinder, r is suggested
to be less than a half of the distance between any two
neighbour points m generated on P. To incorporate
this constraint into the fitness function, f
1
is defined
as
f
1
= max(0,d(F
p
c
,ν
i
) − r). (5)
f
1
, is the penalty function measuring the amount of
violation for all the constraints (if there is any) includ-
ing the cylinder one. One may calculate the penalty
to be larger outside the feasible space by multiplying
the f
1
by itself. Although some other constraint han-
dling techniques may be useful, the proposed penalty
function seems to work reasonably for our purpose.
The second term in Equation 3, f
2
, is introduced
by
f
2
= d(F
c
,U
∗
), (6)
which is the distance of the individuals from the
utopia pointU
∗
. SinceU
∗
is situated outside the feasi-
ble space, minimizing the f
2
guarantees the minimum
distance to the true Pareto frontier for minimization
problems. The order of the preference in Equation 3
can be incorporated by introducing a weight for each
term. We propose the higher weight for the first term,
α, to assure that the solution is obtained as closely
as possible inside the cylinder to guarantee the final
evenly generated set of Pareto solutions.
3 EXPERIMENTAL TEST SETUPS
In this section we study the behaviour of the CCEA
method on the ZDT and DTLZ selected problems in-
troduced in (Deb et al., 2005). The individuals are
coded as real value vectors. The population size is set
to n =52 for both two- and three-dimensions prob-
lems with MaxGen =50 to deal with lower computa-
tional cost. This leads to 2600 fitness evaluations for
each test problem.
For the matter of comparison with CCEA, we have
adopted NSGA-II and SPEA2 algorithms. In these
three algorithms with the same genetic operators, we
set crossover rate as 1 and mutation rate as 1/k where
k is the number of decision variables.
3.1 Performance Assessment
To evaluate the performance of the CCEA algorithm
on the test cases, 10 independent algorithm runs are
carried out. To account for both diversity and qual-
ity (divergence) of the solutions, the hypervolume in-
dicator(Deb, 2001) is computed. Hypervolume met-
ric estimates the volume (in the objective space) cov-
ered by the individuals in a given solution set. For
ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications
186

our case, we have used hypervolume metric to cal-
culate the ratio between the volume of obtain Pareto
solutions (PA
o
) and the volume of true Pareto frontier
(PA
f
) bounded by a reference point as
∆ =
H(PA
o
,ref )
H(PA
f
,ref )
. (7)
The reference points are computed based on the
worst value of each objective function for a given test
case using all three algorithms. Given the PA
f
as the
true set of Pareto solutions, ∆ for a perfect situation
equals 1. Therefore the closer the ∆ to 1, the better
the spread and divergence of solutions. For our pur-
pose |PA
f
|=300 and 500 evenly distributed points are
sampled from the Pareto frontier for two- and three
objective problems, respectively.
In the Figures, the best value recorded in 10 runs
are shown as the result. Table 1 records the ∆ metric
with the variance of the 10 algorithm runs for different
test cases.
3.2 Test Problems and Discussion of
Results
For the formulation of the test cases, the reader may
refer to (Deb et al., 2005). All the test cases have
k =10 decision variables. In all of them, the true
Pareto frontier is visualized in order to make the as-
sessment easier.
Table 1: Comparison between CCEA, NSGA-II and SPEA2
on ZDT test cases using hypervolume (∆) metric. (bold
shows better performance on average) The figures in paren-
theses are the variance.
∆ metric on average of
10 independent run for ZDTs
ZDT1 ZDT3 ZDT6
NSGAII 0.962 0.971 0.817
(0.013) (0.011) (0.092)
SPEA2 0.933 0.763 0.751
(0.033) (0.044) (0.052)
CCEA 0.996 0.991 0.995
(0.003) (0.008) (0.004)
Ref. [2,2] [2,4] [2,10]
ZDT1. Figure 2 demonstrates the generated points
using CCEA algorithm for this convex test case. In
addition, in Table 1 , the ∆ metric shows that esti-
mated solutions using the CCEA has better approxi-
mation than those produced by NSGA-II and SPEA2.
As demonstrated visually in Figure 2, it is evident that
the CCEA outperforms the other two methods with
small number of generation and population.
ZDT3 This test case is highly oscillated and has
a disconnected Pareto frontier. Table 1 shows that
the SPEA2 performs worst while NSGA-II obtained
a reasonably better approximation. However, as can
be seen in Figure 2, with the given number of gen-
eration, the algorithm fails to diverge perfectly to the
Pareto frontier. Meanwhile, looking at Table 1, the
results of CCEA indicate a diverse approximation of
Pareto frontier as well as good quality of solutions
with respect to divergence.
ZDT6. To account for the deceptive Pareto frontier,
this test case has been adopted. Such a deceptive
problems may encounter in real world optimization
problems such as aerodynamic design task, where
the fitness evaluation is also a costly job. There-
fore, an efficient algorithm should be able to gener-
ate the whole Pareto surface with less computation
cost. Having a non-convex Pareto set, the solutions
across the Pareto boundary of this test case are non-
uniformly distributed. As can be seen in Figure 2,
NSGAII and SPEA2 can not maintain a good diverse
set of solutions as well as reasonable divergence in
50 problem generations. CCEA, however, generates
evenly distributed set of solution on the Pareto fron-
tier demonstrated in Figure 2. Table 1 shows that the
larger volume is captured using the approximation by
CCEA compared to the other two methods.
DTLZ2. The Pareto surface in this test case is part
of the unit sphere corresponding to non negative coor-
dinates and is non-convex. By 52 different rays from
utopia point (updated in each generation), we were
able to cover the whole Pareto frontier without miss-
ing the peripheral areas; the difficulty faced in using
directional classical methods. In addition, Table 2 and
Figure 3 reveal that the results of CCEA is better in
terms of quality and spread. Although NSGA-II were
Table 2: Comparison between CCEA, NSGA-II and SPEA2
on DTLZ test cases using hypervolume (∆) metric. (bold
shows better performance on average) The figures in paren-
theses are the variance.
∆ metric on average of 10
independent run for DTLZs
DTLZ2 DTLZ5
NSGAII 0.923 0.985
(0.032) (0.002)
SPEA2 0.934 0.952
(0.031) (0.010)
CCEA 0.993 0.984
(0.002) (0.005)
Ref. [2,2,2] [2,2,2]
CYLINDRICAL CONSTRAINT EVOLUTIONARY ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION
187

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Pareto Front
NSGA−II
(a) ZDT1 using NSGA-II
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
Pareto Front
SPEA2
(b) ZDT1 using SPEA2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
f
1
f
2
Pareto Front
CCEA
(c) ZDT1 using CCEA
0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
0.5
1
1.5
f
1
f
2
Pareto Front
NSGA−II
(d) ZDT3 using NSGA-II
0 0.2 0.4 0.6 0.8 1
−1
0
1
2
3
4
f
1
f
2
Pareto Front
SPEA2
(e) ZDT3 using SPEA2
0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
0.5
1
1.5
f
1
f
2
Pareto Front
CCEA
(f) ZDT3 using CCEA
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
f
1
f
2
Pareto Front
NSGA−II
(g) ZDT6 using NSGA-II
0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
f
1
f
2
Pareto Front
SPEA2
(h) ZDT6 using SPEA2
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
f
1
f
2
Pareto Front
CCEA
(i) ZDT6 using CCEA
Figure 2: Results of NSGA-II, SPEA2 and CCEA on 2-dimentional test problems .
0
0.5
1
0
0.5
1
0
f
2
f
1
f
3
Pareto Front
NSGA−II
(a) DTLZ2 using NSGA-II
0
0.5
1
1.5
0
0.5
1
1.5
0
Pareto Front
SPEA2
(b) DTLZ2 using SPEA2
0
0.5
1
0
0.5
1
0
f
2
f
1
f
3
Pareto Front
CCEA
(c) DTLZ2 using CCEA
Figure 3: Results of NSGA-II, SPEA2 and CCEA on DTLZ2 test problem .
able to approximate a reasonable diverse set of so-
lutions, CCEA could manage to find a more evenly
distributed set of solutions due to the cylindrical con-
straints application.
ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications
188

DTLZ5. The Pareto solutions for this problem lies
on a curve as explained in (Deb et al., 2005). In ad-
dition, the anchor points which are used in classical
algorithms to generate the utopia plane coincide each
others and hence producing a problem for generating
algorithms. As is evident in Table 2, The CCEA and
NSGA-II perform pretty much the same in obtaining
the Pareto curve.
4 CONCLUSIONS AND FUTURE
WORK
In this paper, we introduced an evolutionary algo-
rithm, CCEA, which is able to generate the whole
Pareto frontier evenly distributed closely to the true
Pareto surface. The method is based on introducing
a uniformly distributed set of points on a plane and
constructing a cylinder for each point. The axis of
each cylinder is defined to be parallel to a vector con-
necting the utopia point to the corresponding point on
the plane. The cylinder serves as the problem con-
straint, which assures the search to be biased towards
its axis. By introducing a fitness function, the algo-
rithm seems to reach near the Pareto optimal frontier
while generating a diverse set of solutions. The ex-
perimental study shows that the solutions are reason-
ably distributed on the Pareto surface for a given test
cases in literature. With regards to the computational
cost while maintaining a good quality of diverse so-
lution, the CCEA seems to outperform the NSGAII
and SPEA2 on the given test cases. In addition and
for further improvement, one may note that chang-
ing and tailoring the GA parameters may also speed
up the performance of the CCEA algorithm. For fu-
ture investigation, CCEA should be evaluated based
on some other performance metrics and test cases.
The performance of the method should also be in-
vestigated on higher dimension problems. In addi-
tion, a new development for computing the direction
of the cylinders can increase the efficiency of the al-
gorithm for some adversed scaled problems. Never-
theless, CCEA method can be used easily as a ref-
erence based technique to incorporate the preference
into multiobjective optimization and design.
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