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50 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
Multi-Objective Self-Organizing
Migrating Algorithm Applied to the
Design of Electromagnetic Components
Petr Kadlec and Zbyněk Raida
Department of Radio Electronics
Brno University of Technology
Purkyňova 118, 612 00 Brno, Czech Republic
E-mail: xkadle19@stud.feec.vutbr.cz; raida@feec.vutbr.cz
Abstract
Real-life design problems of electromagnetic components are usually highly nonlinear. These problems can be effi ciently
solved with stochastic global optimization algorithms. This paper deals with an application of a novel Multi-Objective Self-
Organizing Migrating Algorithm (MOSOMA) to the design of two electromagnetic components: layered dielectric fi lters
and Yagi-Uda antennas. The optimization of dielectric fi lters considers two objectives: the minimization of refl ection in
the pass band and its maximization in the stop band. Bandpass, low-pass, and band-stop fi lters having seven dielectric
layers are optimized here. An option for treatment of both continuous parameters and discrete parameters, and for
dealing with optimization constraints without any change in the optimization algorithm, are briefl y discussed. Yagi-Uda
antenna optimization deals with two objectives: the maximization of gain, and the minimization of the relative sidelobe
level, while impedance matching is considered as a constraint for the proposed designs. Yagi-Uda antennas are
analyzed using 4NEC2 software, based on the Method of Moments. Cooperation of a MATLAB optimization script and
the 4NEC2 software in a non-interactive mode is explained. Results for four- and six-element antennas are presented.
The results of both the problems are compared to results from available references.
Keywords: Multi-objective; optimization; MOSOMA; 4NEC2; MATLAB; Yagi-Uda antenna; fi lter; moment methods
1. Introduction
Since almost every optimization problem can be viewed
from more than one side, the importance of ef cient multi-
objective optimizers has grown. These algorithms are able to
nd the so-called Pareto front of the solved problem. This set is
built by solutions that are “optimal” from the viewpoint of all
the objectives at the same time. The Pareto front expresses the
tradeoff among individual objectives.
Aggregation methods (the summation of weighted objec-
tives) suffer from various problems [1]. For the most part, the
aggregation of multiple objectives into one large tness func-
tion brings the problem of the proper settings of weights for
individual objectives. You must consider that you have to set
the weights a priori, which is almost impossible without good
experience with a solved problem. For example, assuming
these weights to be equal (we do not prefer any of the objec-
tives), the optimization process usually does not nd a solu tion
from the middle of the Pareto front, as expected, but rather the
proposed solution signi cantly prefers one of the objectives
[1]. Individual objectives next have to be normal ized by their
maximum value, to keep all the entries of the aggregated
function in the same scale. The nding of the maximum for
individual objectives is again an optimization task. Moreover,
most of the aggregation methods are not able to nd solutions
from the concave part of the Pareto front, as shown in [2].
To the contrary, searching for so-called Pareto front of a
problem with multiple objectives gives the user some addi tional
knowledge about the limits and dependencies of the considered
objectives, and enables carefully choosing the nal solution
of the problem. This approach gives a designer a chance to
take into account the true importance of individual objectives.
The nal solution can be then selected according to extra
information about the solved problem, either manually or by
ISSN 1045-9243/2012/$26 ©2013 IEEE
AP_Mag_Dec_2013_Final.indd 50 3/7/2014 2:48:20 PM
IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 51
a machine. With the knowledge of the Pareto front, the chosen
tradeoff solution then truly respects the designer’s preferences
for particular objectives (the weights of individual objectives
are selected according to the shape of the found Pareto front).
On the contrary, in the case of the methods aggregating all
the objectives, the designer doesn’t know if the weights for
particular objectives (which have to be selected a priori before
the start of optimization) respect the true tradeoff between the
objective-function values of the found solution (just a single
one).
The Multi-Objective Self-Organizing Migrating Algo-
rithm (MOSOMA) is one of the most-recent multi-objective
optimizers. A two-objective version of MOSOMA was intro-
duced in [3]. In [4], MOSOMA was extended so that it was
able to ef ciently solve problems with any number of objec-
tives. MOSOMA has shown very good performance on vari-
ous mathematical benchmark problems [3, 4], where results of
various convergence metrics were compared with commonly
used multi-objective algorithms NSGA-II [5] and SPEA2 [6]:
MOSOMA achieved at least comparable results in all watched
metrics.
The following two paragraphs try to explain the differ-
ences between previous papers concerning MOSOMA
and this one. Paper [2] compared the pure multi-objective
algorithm MOSOMA with conventional aggregating methods
on benchmark problems, to justify the derivation of the pure
multi-objective optimization method, based on self-organizing
migration. Almost the whole paper [4] was focused on the
extension of MOSOMA to solve problems with more than two
objectives. The functionality of our approach has been shown
for the design of dielectric lters. The lter design has been
considered as a three-objective problem (re ection minimiza-
tion in the pass-band, re ection maximization in the stop-band,
and total-length minimization). MOSOMA was employed for
the control of the time-domain adaptive beam forming of an
array of slot antennas in [7]. As that was a joint paper covering
a much larger area than electromagnetic multi-objective
optimization, the MOSOMA application was dis cussed in
appropriate detail.
This paper deals with the application of the generally
working (within the meaning of the objective-space size)
MOSOMA algorithm for the design of electromagnetic com-
ponents: dielectric lters and Yagi-Uda antennas. First, the
operation of the algorithm is described, especially from the
programming point of view (MATLAB scripts of the algorithm
can be downloaded for free from [8]). Practical issues arising
during the design process, such as the handling of constraints
and working with a decision space built from continuous and
discrete variables at the same time, are then discussed. The
cooperation of our optimization algorithm with an external
electromagnetic solver is also explained here.
Venkatarayalu et al. formulated the optimization of the
widths and relative permittivities of the individual layers of a
lter as a constrained two-objective problem in [9], where the
authors proposed a new evolutionary algorithm (MOEA) for its
solution. Goudos et al. used a multi-objective algorithm based
on swarm intelligence (MOPSO) for the solution of bandpass,
low-pass, and band-stop lter design [10].
The optimal design of a Yagi-Uda antenna has been a
challenging problem for various authors from the 20th cen tury.
Cheng [11] put a lot of effort into the application of gra dient-
based methods for the search for optimal lengths of ele ments
and their spacing. Cheng observed that the gain of the Yagi-
Uda antenna is described by a highly nonlinear function, and
the solution provided by gradient-based methods therefore
strongly depends on an initial guess. Many authors therefore
tried to apply global optimization methods. Jones and Joines
employed the single-objective binary-coded genetic algorithm
(GA) [12]. Baskar et al. [13] used comprehensive-learning
particle-swarm optimization (CLPSO). All previously men-
tioned authors solved the Yagi-Uda design problem as a sin gle-
objective task, using one aggregated tness function com posed
of individual objectives, considering gain, relative sidelobe
level, and impedance matching. Nevertheless, several variants
of Yagi-Uda antennas were optimized as a con strained two-
objective problem (maximization of the gain and minimization
of the sidelobe level as objectives, and imped ance matching as a
constraint) in [14] by Venkatarayalu et al. Kuwahara considered
impedance matching as a third objective instead of a constraint
in [15].
In this paper, the properties of the multi-objective optimi-
zation are brie y reviewed, to be introduced to readers. The basic
principles of MOSOMA are then described, and an illustrative
example of MOSOMA on a simple benchmark problem is
derived. Finally, the experimental results of the application of
MOSOMA to both of the design problems are discussed, and
compared with solutions obtained by other authors.
2. Multi-Objective Optimization
Optimization can be understood as a process of nding
and comparing feasible solutions of a solved problem until the
best solution is found. The quality of the solutions is expressed
by means of an objective function. The solved problem usu ally
has to be described by more than one objective (e.g., price,
robustness, reliability, size of some device, etc.). If the 2
a
objectives are con icting, optimization leads to a set,
P
, of so
called Pareto-optimal solutions, which represent a trade off
among all objectives.
The multi-objective optimization problem can be formu-
lated as follows:
Minimize
()
m
Fx,
1, 2,...,mM=
,
subject to
()
0
j
g≥x,
1, 2,...,jJ=
, (1)
,,
n min n n max
x xx≤≤ ,
1, 2,...,nN=
,
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52 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
where
M
denotes the number of objective functions, m
F;
N
stands for the number of decision variables;
x
is the vector of
decision variables for an individual solution; and n,min
x and
n,max
x are lower and upper bounds for the nth variable from the
N
-dimensional decision space. The symbol
J
stands for the
number of constraint functions,
j
g
. Every proposed solu tion
x
, de ned in the decision space (see Figure 1), can be
represented in the objective space by means of values of the
objective functions. The constraint functions,
j
g
, divide the
objective space into a feasible and an infeasible subspace.
The operation of almost every stochastic multi-objective
optimization algorithm is based on a principle of dominance.
The principle of dominance compares two solutions from the
viewpoint of all objectives at the same time [1]:
Solution
1
x is said to dominate the other solution, 2
x, if
both conditions 1 and 2 are met:
1. Solution 1
x is no worse than 2
x in
all objectives.
2. Solution 1
x is strictly better than 2
x
in at least one objective.
Such a comparison can end with two different results: either
one solution dominates the other (it is at least the same or better
in all objectives), or both the solutions are non-domi nated (any
solution is better in all objectives).
The principle of dominance is observed in the objective
space (see Figure 2). We have here a set,
Q
, consisting of ve
solutions. The dashed lines mark out regions in the objective
space that are dominated by a corresponding solution. The
solution 1
x dominates the solutions 2
x and 3
x, and the solu-
tion 4
x dominates the solution 5
x. The solutions 1
x and 4
x
are non-dominated. These two solutions build a non-domi nated
set (front),
P
, of the rst order. The solutions 2
x, 3
x, and 5
x
are also non-dominated, and build a front of the sec ond order.
Obviously, if the researched set,
Q
, contains all possible
solutions from the whole objective space, then the non-domi-
nated set contains all the Pareto-optimal solutions,
PF
, of the
problem. The concept of dominance can therefore be used for
an effective classi cation of proposed solutions from the
viewpoint of multiple objectives.
The process of nding a non-dominated set, P, is crucial
for fast convergence of every multi-objective optimizer. Intui-
Figure 1. Pareto optimal solutions depicted in the decision
and objective spaces.
Figure 2. The principle of dominance in two-dimensional
objective space.
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IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 53
tively, the comparison of the whole set of solutions,
Q
, for
evaluating the dominance is ineffective. A continuously updated
algorithm for assigning the set
P
was therefore pro posed in
[5]. Its pseudocode is depicted in Figure 3, and the working
MATLAB code can be found in [8].
Every multi-objective optimizer is aimed at revealing the
whole Pareto front. This task contains two contradictory minor
goals. First, members from the found set,
P
, should be as close
to the true Pareto front,
PF
, as possible. On the con trary,
members from
P
should be uniformly distributed along the
whole Pareto front. Following both these goals is very important
for a proper understanding of the trade-offs among the objectives
of an examined problem.
3. MOSOMA
A novel Multi-Objective Self-Organizing Migrating Algo-
rithm (MOSOMA) extends the single-objective optimi zation
method SOMA (Self-Organizing Migrating Algorithm) [16].
The multi-objective variant of the algorithm was derived in [3,
4]. Here, the main principles of MOSOMA will be described.
We then walk through an example running of MOSOMA on a
simple benchmark problem, to illustrate the algorithm from the
implementation point of view.
3.1 Algorithm Overview
MOSOMA works in both the optimization domains. So-
called agents (vectors of state variables) migrate through the
N
-dimensional decision space, and evaluate positions being
researched with values of
M
objective functions. These val ues
are used for a non-dominated sorting of the whole group of
agents. All agents then migrate towards the members of a so-
called external archive. The migration thus leads all agents
closer to the true Pareto front, as indicated in Figure 4. The
overall functionality of MOSOMA can be described by the
following steps:
Step 1: De ning controlling parameters of the
algorithm.
Step 2: Generating an initial population, evaluating
objective functions.
Step 3: Choosing an external archive from the current
population.
Step 4: Migrating agents to the members of the
external archive. Evaluating objective functions
for new positions. Updating the external
archive. Selecting migrating agents for the next
migration loop.
Step 5: Testing stopping conditions. If no stopping
condition is accomplished, go back to Step 4.
Step 6: Choosing a nal non-dominated set from the
current external archive.
The user can enhance the convergence of the algorithm by
felicitous setting of its controlling parameters. These parame-
ters can be summarized as follows:
1Q
denotes the initial
population size,
T
stands for the number of migrating agents,
ST
is the number of steps in one migration,
PL
denotes a
relative length of a path for one migration,
PR
means the
Figure 3. The pseudocode of the continuously updated
algorithm for assigning set P.
Figure 4. The migration of agents towards the external
archive [3].
AP_Mag_Dec_2013_Final.indd 53 3/7/2014 2:48:21 PM
54 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
probability of perturbation, and ,ex min
N stands for the mini mal
size of the external archive. The recommended intervals derived
on behalf of our experience with the algorithm are summarized
in Table 1. These intervals are de ned in multi ples of
N
(the
number of optimized parameters) or 1Q (the initial population
size).
The procedure of MOSOMA is as follows. First, posi tions
of
1Q
agents are randomly de ned by the equation
( )
,,q n nmin q n nmax nmin
x x rnd x x
=+−
, (2)
where ,qn
x denotes the nth variable of the qth agent,
,,
;
n min n max
xx denotes the feasible interval for the nth vari-
able, and ,qn
rnd is a random number from the interval 0;1
with uniform distribution of probability.
Objective functions are then computed for the whole
group
1Q
, so that the non-dominated sorting can be per formed.
The non-dominated set,
P
, is saved into the external archive,
EXT
. If the size of the external archive is lower than its
minimal size, ,ex min
N, de ned by the user, the archive is
completed with best solutions from the fronts of advancing
orders.
The iterative process of nding the Pareto-optimal solu-
tions is now performed within the maximal
MIGS
migration
loops. During the ith migration loop,
T
selected agents migrate
towards current members of the external archive. According to
our experience, we should choose the group of migrating agents
partly randomly (to suppress premature con vergence) and
partly from the members of the current
EXT
(to carefully
research the region of the so-far-found best solu tions to speed
up the convergence). The migration procedure is indicated in
Figure 4.
The positions visited during the ith migration of the agent
t
x from
T
towards the agent
p
x
from
EXT
are calculated by
() () () ()
,,
1 11
ts t tt ts
s
i i i i PL
ST
= −+ −− −
TMP x x x PRTV ,
(3)
where ,
ts
TMP is the vector specifying the sth position during
the migration, and
ST
de nes the number of steps for one
migration (
1, 2,...,s ST=
). Parameter
PL
denotes the multi ple
of the distance between agents t
x and
p
x
. Parameters
PL
and
ST
should be set in such a way so that the migrating agent does
not visit the position of the agent from
EXT
:
1
PL
sST ≠, for
1, 2,...,s ST∀=
. (4)
The so-called perturbation vector,
PRTV
, has the same size as
the vector de ning the position of an individual
x
, and consists
of zeros and ones. The vector
PRTV
is de ned for each
migration by
N
randomly generated numbers:
()
()
1 if rnd
() 0 if rnd
n PR
nn PR
>
=≤
PRTV
, (5)
where
PR
denotes a probability of perturbation de ned by the
user. The perturbation has a similar in uence on MOSOMA as
does the mutation for genetic algorithms. The perturbation
protects the algorithm against a bottleneck in the local optimum
(e.g., the front of the advancing order). The migration in the
decision space with the explanation of the in uence of the
perturbation can be seen in Figure 5.
MOSOMA uses the absorbing boundary condition as
de ned in classical particle-swarm optimization [17] when the
Table 1. The recommended values for the
controlling parameters of MOSOMA.
Parameter Recommended Interval
1
QN
5;12
TN
5;10
PL 1.1;1.7
ST 2;5
PR 0.1; 0.4
,1
ex min
NQ
1 3;2 3
Figure 5. The migration of an agent in the decision space,
with and without the infl uence of the perturbation (
3ST =
,
1.3PL =
) [4].
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IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 55
,,
agent visits places outside the feasible decision space during the
migration. If any of the variables n
x
is lower than ,n min
x
(or
higher than ,n max
x
), it is set to the value ,n min
x
(or ,n max
x
).
Typically, the size of the external archive grows with
consecutive migration loops. This behavior slows the proce dure
of MOSOMA (T migrating agents migrate toward an increasing
number of agents in
EXT
). Therefore, three stop ping conditions
are combined: the total number of migration loops,
MIGS
; the
maximum size of the external archive, ,ex max
N
; and the limit
for objective-function computations,
FFC
. The algorithm
stops whenever any of these conditions is ful lled. The detailed
pseudo code of MOSOMA can be found in Figure 6. The nal
external archive usually contains many more solutions than
requested by the user. Working with too large a Pareto-optimal
set slows down the nal choice of the trade-off solution, if this
is made manually. Therefore, we have to save into the nal non-
dominated set,
P
, those solu tions from the
EXT
that uniformly
cover the Pareto front. The detailed description for the two-
objective and
M
-objective variants of this procedure can be
found in [3] and [4], respec tively.
3.2 An Illustrative Running of MOSOMA
In this subsection, we will walk through a typical running
of MOSOMA when solving a relatively simple benchmark
problem. Interested readers can also follow up with the codes
available from [8], where three MATLAB scripts can be found:
mosoma.m, which controls the running of the optimization;
crit_f.m, which evaluates the proposed solutions with
objective functions; and NDsort1.m, which performs the non-
dominated sorting of the current set of proposed solutions, as
shown in Figure 3. The variables mentioned in this text that
were used with the same name in the MATLAB codes [8] are
marked with “...” (quotation marks).
We describe here the running of MOSOMA on a two-
objective problem having two input variables:
11
Fx
=,
(6)
2
21
1x
Fx
+
=,
where 1
x
can vary within the interval 0.1;1.0 , while 2
x
varies within the interval 0.0;5.0 . Both the objective func-
tions are to be minimized.
The settings of MOSOMA are as follows: the initial popu-
lation has
15Q=
agents, the migration proceeds in
3ST =
steps on the relative path length
1.3PL =
, the probabil ity of
perturbation is
0.1PR =
, and the minimum size of the external
archive is 3
ex,min
N
=. The number of migrat ing agents is set to
3T=
. Two of the migrating agents are taken from the initial
population ( “T1”
2=
), and the position of the third agent is
randomly chosen. The optimization proc ess stops immediately
if the total number of migration loops
10MIGS =
proceeds, the
objective functions are computed 1000 times (
1000FFC =
), or
the size of the external archive is 10 times higher than the total
number of expected non-dominated solutions, ,5
ex f
N
=.
We start with the random generation of ve agents,
according to Equation (2). Following the MATLAB script
mosoma.m, the optimized variables are stored in matrix
“AGENTS” (one row per variable, one column per agent). Two
objectives are evaluated for all of the agents by executing the
function crit_f.m and are stored in the matrix “F” (one row
per objective, one column per agent):
F = crit_f(AGENTS);
The positions of the agents and the corresponding objective
function values are given in Table 2.
We now have to determine the current external archive.
Most members of this archive are the non-dominated solutions
of the rst order (see Figure 2) from the set described by “F.”
The function NDsort1.m implements the approach described
in Figure 3 for nding the non-dominated set:
PSORTED = NDsort1(F(:,1:Q1));
Here, the “PSORTED” are the indices of the non-dominated
solutions from the current set of agents. Considering the val ues
of objective functions from Table 2, the non-dominated set is
built by two solutions, “PSORTED”
{}
34
,=xx . Both of the
agents are saved into the current external archive “EXTARCH.”
The number of “PSORTED” elements is lower than the
minimum size of the external archive, ,3
ex min
N
=. The non-
Figure 6. The pseudocode of MOSOMA [4].
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56 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
dominated sorting of the set “F” without agents 3
x and 4
x
therefore has to be performed to determine the non-dominated
set of the second order. It contains just a single solution, 2
x, in
the case of our example, which is saved into the “EXTARCH,”
as well. The external archive now has the expected minimum
size and contains three agents: “EXTARCH”
{}
342
,,=xxx . In
the case where the number of non-dominated solutions of the
rst order is larger than ,ex min
N, they would be all saved into
the “EXTARCH.”
Furthermore, the three migrating agents (
3T=
) have to
be selected. Two of them are randomly selected from the cur-
rent population. Let us now assume that agents 1
x and 2
x were
chosen. The position of the third agent is generated ran domly
using Equation (2). In the MATLAB code mosoma.m, the
positions of these three agents are saved into the variable
“MIGRATORS,” which has the same form as the variable
“AGENTS.”
All the selected agents now migrate towards all members
of the external archive. If the distance between two agents from
the migrating pair is too low, the migrating agent travels towards
a randomly chosen position. Every migration pro ceeds in three
steps (
3ST =
). The temporary positions “TMP” are computed
according to Equations (3) and (5). The migration of solution
1
x
towards the rst member of “EXTARCH” – agent 3
x
– is
summarized in Table 3. Here, 1
rnd
and 2
rnd
are random
numbers from the interval 0;1 , which control the perturbation
process (see Figure 5), and 1
TMP
, 2
TMP
, and 3
TMP
are
positions of the agents in the rst, second, and third steps of the
migration, respectively. If any n
rnd
value is lower than the
probability of perturbation (in our case,
0.10PR =
), the
corresponding variable n
x
remains the same. This happens in
the case of the second vari able for temporary position 2
TMP
in
Table 3 ( 20.08 0.10
rnd
= < ), and therefore this variable
remains
()
223.80x=
TMP .
Using the same procedure, 27 temporary positions are
obtained (three migrating agents × three members of the
external archive × three steps per every migration). All these
positions have to be checked to see if they remain in the feasi ble
part of the decision space. For example, in Table 3, the variable
1
x
of the temporary position 3
TMP
is
0.21−
, which is lower
than the minimum limit for this variable ( 1, 0.10
min
x
=). The
absorbing condition therefore has to be applied, and the value is
set to the value
1,min
x
. Intuitively, if any temporary position
value was higher than the maximum limit, it would be set to
,n max
x
.
After the boundary condition is applied, objective func-
tions are computed for all the “TMP” positions. The new
external archive is selected by applying non-dominated sorting
on a union of the objective values of “TMP” positions and the
previous external archive:
{}{ }
1 2 27 3 4 2
, ,..., ,,TMP TMP TMP x x x.
Selected agents then migrate towards all members of the new
external archive, objective values are evaluated in temporary
positions, and so on, until any of the stopping conditions is
ful lled.
After this procedure stops, the last task is to choose the
de ned number, ,ex f
N
, of nal solutions “REDUCT” from
current external archive. The objective-function values of the
nal external archive are stored in variable “FINAL.” Usually,
the nal external archive contains many more members than
,ex f
N
(if this is not true, we have to ll the rest of the
“REDUCT” set with agents from advancing fronts, or be con-
tent with a lower number of Pareto-optimal solutions than
,ex f
N
).
Let us assume that MOSOMA stopped with the external
archive having seven agents. They are depicted as blue crosses
in Figure 7. Their objective values are summarized in Table 4.
Here, the whole set is in ascending order according to values of
the rst objective, 1
F
(the index i is for the ordered set). The
Table 2. The positions of the randomly generated agents
and their objective-function values.
Agent
()
1
x−
()
2
x−
()
1
f−
()
2
f−
1
x0.60 3.80 0.60 8.00
2
x
0.48 1.88 0.48 6.00
3
x0.25 0.25 0.25 5.00
4
x
0.70 1.10 0.70 3.00
5
x0.78 4.46 0.78 7.00
Table 3. The migration of agent 1
x
towards agent 3
x
.
()
1
x−
()
2
x−
()
1
rnd −
()
2
rnd −
1
x 0.60 3.80 ––
3
x 0.25 1.88 ––
1
TMP
0.45 2.97 0.82 0.24
2
TMP
0.30 2.13 0.71 0.08
3
TMP
–0.21 1.30 0.96 0.34
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IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 57
Euclidian distance, i
d, between the neighbors in the ordered
set is computed. The total length of the external archive, tot
d,
so found can thus be computed as a sum of all i
d:
1
1
EXT
tot i
i
dd
−
=
=∑. (7)
The distance between two ideally placed agents can be then
computed:
()
,
1
tot
ideal
ex f
d
dN
=−
. (8)
In our case, the total length of the Pareto front found is
9.08
tot
d
=, and the distance between two ideal neighbors in
the ordered set is 2.27
ideal
d
=. First, the best and worst agent
according to 1
f
is saved into the nal set “REDUCT.” We now
have “REDUCT”
{}
35
,=xx
. We then have to choose three
other agents. We always pick the following agent from the
sorted list until the sum of the Euclidian distances between
neighbors from the rst agent to the actual agent (having index
i) is greater than the ideal length (
ideal
d
multiplied by
1i−
).
When this is satis ed, the ith agent is saved into the “REDUCT”
set. If the ith agent has already been saved there, the
()
1i+th
agent is saved into the nal set. Following this procedure for
our example, the nal set “REDUCT”
{ }
35 7 41
,,,,=xxxxx will
be formed. These solu tions are marked with red plus signs in
Figure 7. As can be seen here, the set “REDUCT” uniformly
covers the true Pareto front.
4. Experiments
This subsection describes the design of two electromag-
netic structures: a multilayer dielectric lter, and a Yagi-Uda
antenna. Both of these problems have been solved with other
optimization techniques. The results obtained by MOSOMA
can therefore be compared with independent references. All of
the tests were done such that MOSOMA computed the objec-
tive function (the most time-consuming part of the whole
optimization) in the maximum same times as was set in the
other references in order to keep the comparisons fair. The
signi cant differences among the various global optimization
tools presented here may have been caused by the fact that both
the solved problems have very large and complicated decision
spaces, and the times devoted to the optimizations were not
satisfactory to reach the global optimums. The main problems
faced during the optimization process (implementa tion of
constraints, discrete input variables, etc.) and their solutions
will be discussed here.
4.1 Multilayer Dielectric Filter
The design of a dielectric lter for the microwave fre-
quency bands involves an optimization of the relative permit-
tivity and the width of the individual layers of the lter. Con-
sidering the lter as having
N
layers,
2N
parameters are
changed during the optimization process. The layered medium
is depicted in Figure 8. Here,
0
k
stands for the wave vector of
the impinging wave, n
l denotes the width of the nth layer, ,
rn
ε
denotes the relative permittivity of the nth layer,
n
α
is the
incident angle for the nth interface, and
n
R
is the re ec tion
coef cient of the nth interface. The interface between the rst
and second dielectric layer is denoted by
2
R
.
Considering homogeneous lossless nonmagnetic materi-
als (
0
σ
=
, 1
r
µ
=), the generalized recursive re ection coef -
cient,
n
R
, for the nth interface can be derived [18]:
Table 4. The migration of agent 1
x towards agent 3
x :
experiments.
i
()
1
F−
()
2
F−Agent
()
i
d−
1 0.11 9.96 3
x
2.19
2 0.13 7.77 7
x1.50
3 0.16 6.27 6
x0.84
4 0.19 5.43 4
x
2.29
5 0.32 3.14 1
x1.89
6 0.78 1.31 2
x0.37
7 0.99 1.01 5
x–
Figure 7. The choice of the fi nal non-dominated set
(“REDUCT”) from the external archive (“FINAL”).
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58 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
()
()
1
1
exp 2
1 exp 2
nn nn
n
nn nn
rR jl
RrR j l
+
+
+
=+
k
k, (9)
where the wave vector,
n
k
, in the nth layer can be computed
using the equation [18]
,
2
nrn
f
c
πε
=k. (10)
After a few simpli cations, the re ection coef cient can be
derived for the TE mode,
( ) ()
( ) ()
22
, -1 1,
,TE
22
, -1 1,
1 sin 1 sin
1 sin 1 sin
rn n rn n
n
rn n rn n
r
ε αε α
ε αε α
−
−
− −−
=
− +−
, (11)
and for the TM mode,
( ) ()
( ) ()
22
, , -1 1 , -1 ,
,TM
22
, , -1 1 , -1 ,
1 sin 1 sin
1 sin 1 sin
rn rn n rn rn n
n
rn rn n rn rn n
r
εε α ε ε α
εε α ε ε α
−
−
−− −
=
−+ −
(12)
where the angle of incidence for the nth layer is de ned by [18]
,1
11
,
sin sin
rn
nn
rn
ε
αα
ε
−
−−
=
.
(13)
The coef cient between free space and the rst medium,
denoted 1
R, then expresses the re ection properties of the lter.
In [9, 10], two objective functions for the design of a lter
with seven layers were de ned:
() ()
22
1 1,TE 1,TM
1
P
pp
p
F Rf R f
=
= +
∑,
(14)
() ()
22
21,TE 1,TM
1
2
S
ss
s
F Rf R f
=
=−−
∑,
where
p
f
and s
f
respectively denote the pass and stop fre-
quencies of the lter, and
P
and
S
stand for the size of fre-
quency vectors used. The objective function, 1
F
, minimizes the
re ection of the layered media in the pass band, while the other
function, 2
F
, maximizes the re ection in the stop band. Under
this de nition, the Pareto fronts obtained by different authors
cannot be compared, because the values of the objec tive
functions are in uenced by the discretization of the fre quency
axis. We therefore propose a slight modi cation of the objective
functions:
() ()
22
11,TE 1,TM
1
1
P
pp
p
F Rf R f
P
=
= +
∑
,
(15)
() ()
22
21,TE 1,TM
1
12
S
ss
s
F Rf R f
S=
=−−
∑.
Both the functions are now normalized to the number of fre-
quency points examined, and are fully comparable.
The de nition of the optimization problem is fully com-
pleted by the formulation of the constraint functions for the
pass band and the stop band [9]:
()
1,TE
20log 10
pc
Rf<−
dB,
()
1,TM
20log 10
pc
Rf<− dB,
(16)
()
1,TE
20log 5
sc
Rf<−
dB,
()
1,TM
20log 5
sc
Rf<− dB,
where pc
f
and sc
f
denote the pass and stop frequencies con-
sidered for constraints, respectively. The constraint limits were
selected exactly as in [9], to keep the comparison between
MOSOMA and other references fair.
The so-called penalty-function approach [19] can be used
for handling the problem with constraints. The violation of any
constraint has an impact on worsening the objective func tions.
First, the violation, j
V
, from the normalized jth con straint
function, ,jn
g
, for the ith solution, i
x
, is calculated:
Figure 8. A description of the layered medium [4].
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IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 59
= +
=−−
= +
=−−
() () ()
,, if 0,
0, otherwise.
jn i j i
ji
gg
V
<
=
xx
x (17)
Objective functions ,cm
F
, considering the violations from all
constraint functions, are thereafter slightly modi ed to
() () ()
,
1
J
cmi mi m ji
j
F f RV
=
= + ∑
xx x
, (18)
where m
R stands for a penalty parameter, which is introduced
to equalize the magnitudes of both addends in Equation (18).
Since the values of both of the objective functions should vary
in the interval
0; 2
, the penalty operator was set for both the
objective functions to
5R=
. This procedure disquali es the
solutions violating any constraint from further searching by the
algorithm.
The design of the seven-layer lter evolved as the optimi-
zation of 14 parameters. The incidence angle was xed to
045
α
= ° [9]. The width of every layer,
17
x−, could vary in the
interval 1 mm;10 mm . The relative permittivity of all layers
could be chosen from commercially available dielectric
materials: {1.01, 2.20, 2.33, 2.50, 2.94, 3.00, 3.02, 3.27, 3.38,
4.48, 4.50, 6.00, 6.15, 9.20, 10.20} [9]. MOSOMA worked
initially only with continuous input variables. The problem
with discrete input variables could be solved using a relatively
simple approach. As we had 15 discrete values, the input vari-
ables
8 14
x
− were set from the interval 0;15 . This interval was
uniformly divided into 15 subintervals, each correspond ing to
one value of an available dielectric permittivity (e.g., value
8
6.35x=
corresponded to the seventh value from the list:
3.02). The variables within the algorithm were treated as
continuous. Only objective functions were evaluated with the
corresponding value of the relative permittivity.
This procedure obviously had some shortcomings. If both
the agents that participated on the migration had similar values
of the input variable, all steps of the migration could cause the
continuous variable not to leave the original subinterval, and
the same permittivity was examined again. Another problem
was caused by the fact that different values of the input vari able
meant one value of the relative permittivity. The result of the
migration then depended on the initial value of the migrating
agent in the subinterval. Let us consider Equa tion (3) with no
in uence of perturbation, parameters
1.3PL =
and
3ST =
,
and with two different migrating agents having only one
variable,
0.10
q
x=
and 0.99
q
x∗=. Now, let them migrate
towards the member of the external archive
3.5
p
x=
. The
resulting steps of the migration corre spond to different dielectric
materials,
{ }
1.62,3.13,4.65=TMP and
{ }
2.51, 4.02,5.54
∗=TMP . Beside all the shortcomings, the
algorithm was able to simul taneously solve problems with
continuous and discrete vari ables, without any change in the
MOSOMA program. The only change comprised the evaluation
of the objective func tions: discrete values were used according
to subintervals of the input variable.
The controlling parameters of MOSOMA were set in
such a way so that the results could be compared with results
published in [9, 10]. The settings are summarized in Table 5.
4.1.1 Bandpass Filter
The rst experiment was aimed at designing a bandpass
lter for frequencies from 28 GHz up to 32 GHz. The fre-
quency bands for the lter and for the constraint functions are
summarized in Table 6.
The Pareto front of the optimized problem is depicted
in Figure 9. Obviously, some of the Pareto-optimal solutions
violated the constraint functions, because their values of
the objective function were higher than two. Three solutions
are highlighted here: the best solution according to the rst
objec tive (red marker) and the second (green) objective, and
the tradeoff solution (blue). Figure 10 depicts the frequency
behaviors of the re ection coef cients for these solutions.
Here, colors correspond to markers in Figure 9. The red solu-
tion ideally satis ed the rst objective, but the last two con-
straints were violated. To the contrary, the green solution met
the second objective, but violated rst two constraint func tions.
Finally, the blue solution respected both the objectives and did
not violate any constraint function.
The tradeoff solution, composed of layers having widths
of {4.686 mm, 1.995 mm, 4.739 mm, 1.001 mm, 1.003 mm,
1.002 mm, 8.663 mm} and relative permittivities of {10.20,
1.01, 10.20, 1.01, 1.01, 2.94, 2.35} was chosen as the nal
Table 5. The settings of the MOSOMA parameters for the
dielectric fi lter design.
Parameter FFC PL ST Q1T,ex min
N
Value 15000 1.3 5 30 20 15
Table 6. The frequency bands for the
bandpass fi lter optimization.
Band Lower Bound
(GHz)
Upper Bound
(GHz)
p
f
28 32
s
f24; 32 28; 36
pc
f29 31
sc
f
24; 34 26; 36
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60 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
Figure 9. The Pareto front of the bandpass fi lter multi-
objective optimization using MOSOMA. The detailed plot
depicts the tradeoff solutions that do not violate the con-
straints.
Figure 10. The TE (solid line) and TM (dashed) refl ection
coeffi cients for three bandpass fi lters designed by
MOSOMA: red line (the best solution according to 1
F),
green (the best 2
F) and blue (tradeoff).
tradeoff solution. Figure 11 compares the re ection coef cients
of this tradeoff solution with those of solutions pub lished in [9,
10]. The total width of our design was 23.08 mm, compared to
33.44 mm [9] and 21.35 mm [10]. The re ection coef cient for
our solution remains below
16−
dB for the TE mode and
22−
dB for the TM mode in the operational band of the lter. The
coef cients TE
R
and TM
R
of our design decreased more
steeply at the boundaries of the desired fre quency band. The
results obtained by MOSOMA seemed to be better than the
results obtained by other global algorithms published in [9, 10],
although the same number of objective-function evaluations
was made. This showed that MOSOMA converged faster in this
case.
4.1.2 Low-Pass Filter
The frequency bands for the low-pass lter, operating up
to 30 GHz and corresponding to the constraint functions’ fre-
quency bands, are summarized in Table 7.
Figure 12 depicts the Pareto front obtained by MOSOMA.
Again, the best solutions according to the 1
F and 2
F objectives
are marked with the red and green colors, respec tively. The
solutions that did not violate the constraints are emphasized in
the detailed subplot. The solution chosen for the nal design is
here marked with the blue cross. This solution was composed of
layers having widths of {8.195 mm, 1.489 mm, 1.758 mm,
1.001 mm, 1.001 mm, 1.153 mm, 1.003 mm}, and relative
permittivities of {2.20, 1.01, 10.20, 3.02, 1.01, 10.20, 6.15}.
Figure 13 compares the distinguished solutions from the Pareto
front at Figure 12. We could see that the red and blue solutions
were similar for the pass band ( 1
F was just slightly better for
the red solution), while the blue solution was signi cantly
better in the stop band. Further, we could see that the solution
that was best according to 2
F re ected almost across the whole
frequency band.
The frequency response of the re ection coef cient for the
tradeoff solution is again depicted in Figure 14. The fre quency
response is compared here with solutions obtained by the
MOEA [9] and MOPSO [10] algorithms. The re ection in the
pass band was approximately under
20−
dB for the TE mode
and
18−
dB for the TM mode. Obviously, our solution was
better than the solution proposed in [9], because the re ection
was lower in the pass band and higher in the stop band. The
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IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 61
Figure 12. The Pareto front of the low-pass fi lter multi-
objective optimization using MOSOMA. The detailed plot
depicts the tradeoff solutions that do not violate the con-
straints, with the highlighted solution chosen for the design.
Figure 11. A comparison of the TE (solid line) and TM
(dashed) refl ection coeffi cients for the bandpass fi lter design
obtained by MOSOMA (blue), MOEA [9] (green), and
MOPSO [10] (red).
Table 7. The frequency bands for the
low-pass fi lter optimization.
Band Lower Bound
(GHz)
Upper Bound
(GHz)
p
f
24 30
s
f30 36
pc
f24 28
sc
f32 36
Figure 13. The TE (solid line) and TM (dashed) refl ection
coeffi cients for three low-pass fi lters designed by MOSOMA:
red line (the best solution according to 1
F
), green (the best
2
F
), and blue (tradeoff).
Figure 14. A comparison of the TE (solid line) and TM
(dashed) refl ection coeffi cients for low-pass fi lter designs
obtained by MOSOMA (blue), MOEA [9] (green), and
MOPSO [10] (red).
Table 8. The frequency bands for the
band-stop fi lter optimization.
Band Lower Bound
(GHz)
Upper Bound
(GHz)
p
f
24; 32 28; 36
s
f28 32
pc
f24; 34 26; 36
sc
f29 31
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62 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
solution proposed in [10] exhibited similar re ec tion properties,
but the total width of the lter was 16.24 mm, while the total
width of our lter was only 15.58 mm.
4.1.3 Band-Stop Filter
The next example considered the design of a band-stop
lter for the frequency band from 28 GHz to 32 GHz. The
frequency bounds for the ltering properties and constraint
functions are summarized in Table 8.
We selected the solution composed from dielectric layers
having widths of {3.090 mm, 5.358 mm, 1.001 mm, 1.000 mm,
5.585 mm, 1.000 mm, 6.020 mm}, and relative permittivities
of {2.50, 3.38, 10.20, 2.33, 3.38, 10.20, 2.50}. This solution is
marked in Figure 15 of the Pareto front obtained by MOSOMA.
Obviously, the constraints for the optimization were too strict,
which caused an increase in the objective-function values of the
Pareto-optimal solutions.
Figure 16 compares the re ection properties of three dis-
tinguished solutions: the best solution according to 1
F (red),
the best solution according to 2
F (green), and the tradeoff
solution (blue). It was obvious that both the extreme solutions
considered just one of the objectives: the red solution passed in
the whole frequency band (just 1
F was considered), while the
green solution stopped. Finally, the blue solution consid ered
both the objectives at the same time.
The re ection coef cient of this tradeoff solution was then
compared with solutions published in [9, 10] in Fig ure 17. The
re ection coef cient was very high in the whole stop band, and
fell to under
15−
dB in both parts of the pass band for both the
modes. The re ection properties were com parable for all three
algorithms. The total length of our lter was 23.05 mm,
compared to 29.25 mm in [9], and 20.53 mm in [10].
4.2 Yagi-Uda Antenna
The design of a Yagi-Uda antenna requires the optimiza-
tion of the lengths of the individual elements and the spacing
between them. The Yagi-Uda antenna is depicted in Figure 18.
Here, n
d stands for the total length of the nth element, and n
s
denotes the spacing between the nth and
()
1n+th element.
Considering
N
elements,
21N−
parameters were optimized.
Figure 15. The Pareto front of the band-stop fi lter multi-
objective optimization using MOSOMA. The detailed plot
depicts the tradeoff, with the highlighted solution chosen
for the fi nal design.
Figure 16. The TE (solid line) and TM (dashed) refl ection
coeffi cients for three band-stop fi lters designed by
MOSOMA: red line (the best solution according to 1
F),
green (the best 2
F), and blue (tradeoff).
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IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 63
Figure 17. A comparison of the TE (solid line) and TM
(dashed) refl ection coeffi cients for the band-stop fi lter
design obtained by MOSOMA (blue), MOEA [9] (green),
and MOPSO [10] (red).
Figure 18. A description of the N-element Yagi-Uda antenna
with a description of the optimized parameters.
Figure 19. The connection between the optimization script
in MATLAB and the 4NEC2 analysis tool.
The operating frequency was set to 30 GHz, as indicated in
Figure 21; however, all antenna dimensions are presented with
respect to the corresponding free-space wavelength,
λ
. The
width of every element could vary in the interval
0.30;0.70
,
while the spacing between neighboring elements could vary
within the interval 0.1; 0.35 .
The settings of MOSOMA were chosen so that it com-
puted objective functions a maximum of 36000 times, and our
results could be compared with results published in [12-15].
The controlling parameters are summarized in Table 9.
Two objectives were considered for the optimization of
the Yagi-Uda antenna: the maximum gain and the minimum
relative sidelobe level [14]:
1
FG
= − ,
(17)
2
F SLL=,
where
G
denotes the gain and
SLL
stands for the relative
sidelobe level of the antenna. Two constraint functions were
de ned to ensure a proper impedance matching of the designed
antenna [11]:
()
50 Re 5
in
Z
−<
,
(18)
()
Im 10
in
Z<,
where in
Z is the input impedance of the antenna.
The analysis necessary for the computation of the objec-
tives was performed by the 4NEC2 software, based on the
Method of Moments [20]. This software is freely available at
the Web site http://home.ict.nl/~arivoors/. The interconnection
between the MATLAB script that provided the optimization
of the input parameters and the 4NEC2 software is described
in Figure 19. Information about the analyzed structure had to
be saved in an ASCII text le input.nec in an appropriate
for mat de ned in [21]. An example for a four-element Yagi-
Uda antenna with the description of individual entries can be
viewed in Figure 20. The 4NEC2 software could be executed
from MATLAB in a non-interactive mode using the expression
! …\4nec2.exe …\input.nec –I
The results could be read from the ouptut.out text le gen-
erated by the 4NEC2 software in the folder …4nec2\out.
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64 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
Table 9. The settings of the MOSOMA parameters for
the design of the Yagi-Uda antenna.
Parameter FFC PL ST Q1T,ex min
N
Value 36000 1.3 5 30 20 15
Figure 20. An example of the 4NEC2 input fi le for the four-element Yagi-Uda antenna analysis.
4.2.1 Four-Element Design
The radius of all the wires was set to
0.00225
λ
[12]. The
best Pareto front from ten runs of MOSOMA is depicted in
Figure 21. Here, the Pareto-optimal solutions are compared
with solutions obtained by MOEA published in [12]. The
MOSOMA algorithm achieved signi cantly better results,
because most of the solutions proposed in [12] were domi nated
by the solutions obtained by MOSOMA, except for a few
solutions in a region of the minimum of the second objec tive.
The distinguished solutions from the obtained Pareto front
are highlighted in Figure 22. Their radiation patterns are
depicted with corresponding colors in Figure 22. The solution
depicted in red represented the maximal gain
10.35G=
dBi.
The green solution corresponded to the minimum level of
sidelobes (
54.29SLL = −
dB), but exhibited a very poor gain,
5.64G=
dBi. The chosen tradeoff solution showed a very good
gain,
10.08G=
dBi, and a satisfactory sidelobe level,
11.81SLL = −
dB.
The parameters of the designs depicted in Figure 22 are
summarized in Table 10. The results obtained by MOSOMA
are compared here with the designs proposed in [12-14]. The
design of the Yagi-Uda antenna was treated as a pure multi-
objective problem only in [14, 15]; however in [15], the
impedance matching was considered as the third objective
function, which prevents us from comparing those results with
ours. In [12, 13], two objectives were aggregated to build one
tness function, which was then solved by single-objective
modi cations of genetic algorithms and particle-swarm opti-
mization, respectively. None of the solutions from all the ref-
erences was not better in all considered objectives than our nal
tradeoff solution.
4.2.2 Six-Element Design
The six-element design had the same settings as the previ-
ous four-element problem; only the radius of the wire was
increased to
0.003369
λ
[12]. Pareto-optimal solutions obtained
by MOSOMA are depicted in Figure 23. Here, the extreme
solutions according to both the objectives (the maxi mum gain,
12.67G=
dBi, and the maximum sidelobe-level suppression,
38.84SLL = −
dB) are highlighted with red and green markers,
respectively. One tradeoff solution was also chosen (
12.65G=
dBi,
12.66SLL = −
dB). The radiation pat terns of those three
designs are depicted in Figure 24. Obvi ously, the radiation
patterns of the tradeoff solution and that of the best solution
according to 1
F were almost overlapping, but the tradeoff
solution had a slightly better sidelobe level and signi cantly
better impedance matching.
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IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 65
Figure 21. The Pareto front of the four-element Yagi-Uda
antenna design obtained by MOSOMA (black dots) and by
MOEA [12] (blue dots).
Figure 22. The H-plane radiation patterns of chosen solu-
tions from the Pareto front obtained by MOSOMA for the
four-element Yagi-Uda antenna.
Figure 23. The Pareto front of the six-element Yagi-Uda
antenna design obtained by MOSOMA (black dots), with
highlighted solutions from Table 10.
Figure 24. The H-plane radiation patterns of chosen solu-
tions from the Pareto front obtained by MOSOMA for the
six-element Yagi-Uda antenna optimization.
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66 IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013
Table 10. The four-element Yagi-Uda antenna design parameters obtained from the
MOSOMA, MOEA [14], GA [12], and CLPSO [13] algorithms.
MOSOMA MOEA [14] GA [12] CLPSO [13
Best 1
FBest 2
FTradeoff Best 1
FBest 2
FAggregated Aggregated
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
1 0.473 0.313 0.494 0.245 0.474 0.295 0.480 0.270 0.628 0.204 0.490 0.283 0.476 0.311
2 0.445 0.350 0.469 0.196 0.463 0.260 0.474 0.186 0.488 0.195 0.472 0.179 0.466 0.205
3 0.439 0.288 0.402 0.272 0.440 0.249 0.436 0.274 0.436 0.114 0.442 0.279 0.434 0.279
4 0.434 –0.651 –0.433 –0.434 –0.582 –0.424 –0.412 –
G (dBi) 10.35 5.64 10.08 9.6 5.5 9.84 9.44
SLL (dB) –7.6 –54.29 –11.81 –12.14 –62.6 –14.5 –15.02
Zin (Ω) 45.18 + 6.96i 45.32 – 1.70i 45.36 + 3.65i 47.59 – 5.67i 45.51 + 7.83i 38.50 – 2.30i 49.56 + 0.11i
VSWR (–) 1.19 1.11 1.13 1.13 1.21 1.3 1.01
Table 11. The six-element Yagi-Uda antenna design parameters obtained from the MOSOMA,
Pareto GA [15], GA [12], and CLPSO [13] algorithms.
MOSOMA MOEA [14] GA [12] CLPSO [13
Best 1
FBest 2
FTradeoff Best 1
FBest 2
FAggregated Aggregated
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
()
d
λ
()
s
λ
1 0.473 0.257 0.352 0.300 0.476 0.259 0.474 0.355 0.490 0.184 0.478 0.182 0.476 0.245
2 0.462 0.192 0.449 0.289 0.462 0.190 0.462 0.243 0.476 0.100 0.450 0.152 0.462 0.196
3 0.444 0.304 0.473 0.150 0.440 0.302 0.438 0.386 0.447 0.198 0.448 0.229 0.442 0.272
4 0.430 0.334 0.459 0.342 0.430 0.337 0.427 0.348 0.433 0.317 0.434 0.435 0.430 0.334
5 0.422 0.345 0.459 0.298 0.421 0.343 0.427 0.402 0.405 0.392 0,434 0.272 0.422 0.345
6 0.428 –0.382 –0.419 –0.422 –0.347 –0.440 –0.428 –
G (dBi) 12.67 6.58 12.56 13.70 11.30 12.58 12.65
SLL (dB) –10.02 –38.84 –12.66 –11.60 –25.40 –10.01 –9.12
Zin (Ω) 45.99 + 2.32i 54.45 – 6.56i 45.56 + 0.34i 21.61 + 17.63i 33.41 – 8.76i 49.64 – 5.08i 50.01 – 0.01i
VSWR (–) 1.10 1.18 1.09 2.66 1.58 1.11 1.01
AP_Mag_Dec_2013_Final.indd 66 3/7/2014 2:48:25 PM
IEEE Antennas and Propagation Magazine, Vol. 55, No. 6, December 2013 67
All the proposed solutions did not violate any of the con-
straints used, and exhibited very good impedance matching
(maximum
1.18VSWR =
). The parameters of all three pro-
posed antennas are summarized in Table 11, where we com-
pared the MOSOMA results with solutions published in [12, 13,
15]. In [15], the impedance matching was not treated as a
constraint, but was considered as a third objective function.
Very good values of the gain presented in [15] are therefore
paid for with worsening of the impedance matching. No refer-
enced solution was better than our tradeoff solution in all con-
sidered objectives.
5. Conclusions
MOSOMA is a very ef cient multi-objective optimizer.
MOSOMA can cope both with continuous and discrete
parameters at the same time. The constraint functions can also
be considered for the solved optimization problem.
The implementation of MOSOMA is easier than for
genetic algorithms (no binary coding of optimized parameters
is needed), and is very similar to particle-swarm optimization.
For the case of the investigated electromagnetic structures,
MOSOMA showed performance better (or comparable) to
other commonly used multi-objective optimizers.
MOSOMA was successfully applied to the design of
dielectric lters and Yagi-Uda antennas. MOSOMA signi -
cantly outperformed the algorithm published in [14] in the
four-element Yagi-Uda antenna multi-objective design. In
comparison with aggregation methods that converted a multi-
objective problem into a single-objective problem, MOSOMA
brought in additional information about the limits of the con-
sidered objectives. The tradeoff solution chosen from the
revealed Pareto front was better or comparable in all consid ered
objectives.
6. Acknowledgements
The research described in this paper was nancially
sup ported by Czech Science Foundation under grant No.
P102/12/1274. The support of projects CZ.1.07/2.3.00/20.0007
and CZ.1.05/2.1.00/03.0072 is also gratefully acknowledged.
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Introducing the Feature Article Authors
Petr Kadlec was born in Brno, Czech Republic, in 1985.
He received the BS and MS from the Brno University of
Technology, Brno, Czech Republic, in Electronics and Com-
munications, in 2007 and 2009, respectively.
Zbyněk Raida received Ing (MSc) and Dr (PhD) degrees
from the Brno University of Technology in 1991 and 1994,
respectively. Since 1993, he has been with the Department of
Radio Electronics, FEEC BUT, as Assistant Professor (1993
to 1998), Associate Professor (1999 to 2003), and Professor
(since 2004). In 1997, he spent six months at the Laboratoire
de Hyperfrequences, Universite Catholique de Louvain, Bel-
gium working on variational methods of numerical analysis
of electromagnetic structures. Since 2006, he has been the
head of the Department of Radio Electronics. Zbyněk Raida
has been working together with his students and colleagues
on numerical modeling and optimization of electromagnetic
structures, exploitation of arti cial neural networks for solving
electromagnetic compatibility issues, and the design of special
antennas. Prof. Raida is a member of the IEEE Microwave
Theory and Techniques Society.
AP_Mag_Dec_2013_Final.indd 68 3/7/2014 2:48:26 PM
