Neighborhood Strategies for QPSO Algorithms to Solve Benchmark
Electromagnetic Problems
Anton Duca
1
, Laurentiu Duca
2
, Gabriela Ciuprina
1
and Daniel Ioan
1
1
Politehnica University of Bucharest, Faculty of Electrical Engineering, Bucharest, Romania
2
Politehnica University of Bucharest, Faculty of Computer Science, Bucharest, Romania
Keywords: PSO, QPSO, Neighborhood Strategies, Optimization, Electromagnetic Field.
Abstract: Several neighborhood strategies for QPSO algorithms are proposed and analyzed in order to improve the
performances of the original methods. The proposed strategies are applied to some of the most well known
QPSO algorithms such as the QPSO with random mean, the QPSO with Gaussian attractor and of course the
basic QPSO. To prevent the premature convergence and to avoid being trapped in local minima the
neighborhoods are dynamically changed during the optimization process. For testing the efficiency of the
neighborhood techniques two benchmark optimization problems from the electromagnetic field computation
have been chosen, Loney’s solenoid and TEAM22.
1 INTRODUCTION
The PSO (Particle Swarm Optimization) algorithms
are part of stochastic optimization methods which
use a population of candidate solutions that evolve
over time. Comparable in terms of performance with
the genetic algorithms, these algorithms are problem
independent and are suitable for solving difficult
optimization problems where the analytical
expression of the objective function is not known.
First proposed by Kennedy and
(Kennedy,
Eberhart, 1995)
, the original PSO (classic) has its
root in biology and is inspired by social behavior
within fish schools or bird flocks. Each particle in
the swarm (population) is characterized by its
current position and velocity. The position
encapsulates the potential solution of the
optimization problem, while the velocity influences
how that position will be changed at the next
iteration.
The main issues of the classical PSO are the high
probability to get stuck in a local minimum and the
large number of iterations required to find the global
solution. Over time, to improve the performance of
the PSO algorithm several solutions have been
proposed in the literature (Ciuprina et al., 2002), but
the most efficient options currently available are
based on SPSO (Standard PSO) (Bratton, Kennedy,
2007) (Clerc, 2012) and QPSO (Quantum-behaved
PSO) (Sun et al., 2004).
The SPSO and QPSO algorithms have been
successfully used to solve a variety of problems,
such as (Li et al., 2007) (Zhang, Zuo, 2013), but also
to solve electromagnetic optimization problems
(Mikki, Kishk, 2006) (Coelho, 2007) (Coelho,
Alotto, 2008). In (Mikki, Kishk, 2006) authors
solve the LAA problem (Linear Antenna Array)
proposing a method for the optimal control of the
QPSO algorithm parameters. In (Coelho, 2007) and
(Coelho, Alotto, 2008) the author makes a
comparison between the PSO and QPSO algorithms
for the TEAM22 problem and the Loney's solenoid
problem. Even if the solutions mentioned in
(Coelho, Alotto, 2008) do not mention/verify the
quench condition, the conclusions of both articles
highlight the superiority of the QPSO algorithms
over the PSO.
Although the latest QPSO versions proposed by
Sun&others (Xi et al., 2008) (Sun et al., 2011) (Sun
et al., 2012) are better than the SPSO algorithm
when optimizing CEC benchmark functions, in case
of statistical studies conducted on electromagnetic
optimization problems the SPSO algorithm is more
stable (Duca et al., 2014) (Duca et al., 2014, 2). In
some cases the performance offered by the QPSO
based algorithms provide better solutions (smaller
values of the objective function) but statistically the
QPSO based methods are outperformed by the SPSO
algorithm, which provides a smaller mean and a
148
Duca, A., Duca, L., Ciuprina, G. and Ioan, D.
Neighborhood Strategies for QPSO Algorithms to Solve Benchmark Electromagnetic Problems.
DOI: 10.5220/0006040901480155
In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 1: ECTA, pages 148-155
ISBN: 978-989-758-201-1
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
smaller standard deviation.
To improve the QPSO algorithms performances
when solving electromagnetic problems this paper
proposes and studies several neighborhood
strategies.
Starting from the idea used in SPSO, in the
current paper different neighborhood strategies are
applied to enhance the performances of the best
QPSO algorithms available at present time, QPSO-
WM (weighted mean) (Xi et al., 2008), QPSO-GA
(Gaussian attractor) (Sun et al., 2011), QPSO-RO
(ranking operator) (Sun et al., 2012), QPSO-RM
(random mean) (Sun et al., 2012), and basic (Sun et
al., 2004). To avoid the premature convergence and
local minima the neighborhoods are dynamically
changed during the optimization process. The
influence of the change frequency over the
performances is analyzed for each neighborhood
scheme and each QPSO algorithm. The tests are
carried for two benchmark electromagnetic
optimization problems, Loney’s solenoid and
TEAM22.
2 QPSO ALGORITHMS
Unlike PSO and SPSO algorithms, where the
particle trajectories are according to Newton's
mechanic laws, QPSO is a quantum system
proposed by Sun & others (Sun et al., 2004). In
QPSO the behavior of each particle is described by a
wave function Ψ (Schrödinger's equation) |Ψ|
2
being
the probability density function for the particle
position. On the other hand, while in the PSO
algorithm the particles converge to the solution
through the global best position, in QPSO the
particles exert a greater influence on each other
through an average of the personal best positions, so
the probability to get stuck in a local minimum is
smaller.
The coordinate j for a particle i at the step (t + 1)
is given by:
)1ln()()(
)()1(
,,
,,
jijij
jiji
utxtm
tptx
,
(1)
)()1(
)()(
,
,,,
tx
txtp
jGBji
jiPBjiji
,
(2)
where p is called attractor, u and α are random
generated numbers uniformly distributed in the
interval [0, 1) (for each component of a particle),
and β is a contraction–expansion factor linearly
decreased at each iteration with values between 1
and 0.5. The particles in a QPSO algorithm exert
great influence on each other through a mean best
(m) calculated using the formula:
M
i
niPB
M
i
iPB
n
tx
M
tx
M
tmtmtmtm
1
,
1
1,
21
)(
1
),(
1
)](),(),([)(
,
(3)
where M is the swarm size (the number of particles)
and n is the number of coordinates for the position
of a particle.
During time, to enhance the performances of the
QPSO several variants have been proposed. The
most effective QPSO based algorithms available
today are: QPSO with weighted mean (QPSO-WM),
QPSO with Gaussian attractor (QPSO-GA), QPSO
with ranking operator (QPSO-RO), and QPSO with
random mean (QPSO-RM).
In QPSO-WM (Xi et al., 2008), at each iteration,
the particles are sorted according to their fitness
values. Each particles is assigned a weight (α)
related to its ranking position (better solution means
larger weight), to enforce an elitism behavior. The
mean is calculated as a weighted sum as follows:
M
i
niPBni
M
i
iPBi
txtx
M
tm
1
,,
1
1,1,
)(),(
1
)(
.
(4)
For improving the convergence, diversify the
swarm, and escaping the local minima, in (Sun et al.,
2011) the authors propose a new QPSO-GA
algorithm which changes the attractor (p) from the
position formula to a Gaussian probability
distribution. The new attractor (np) is evaluated with
the formula:
))(),(()(
,,,
tpmtpNtnp
jijjiji
,
(5)
)1ln()()(
)()1(
,,
,,
jijij
jiji
utxtm
tnptx
,
(6)
where N is a function with Gaussian distribution. In
the beginning the particles are scattered in a wider
space, and the standard deviation is larger, while in
the late stage of the search the particles converge
toward the mean, and the standard deviation
decreases toward zero.
In (Sun et al., 2012), the authors demonstrate
using the probabilistic metric spaces theory that
QPSO can converge to a global optimum, and
propose two new improvements QPSO-RM and
QPSO-RO. The QPSO-RM algorithm replaces the
mean best position with a personal best position of a
random selected particle. This change diversifies the
Neighborhood Strategies for QPSO Algorithms to Solve Benchmark Electromagnetic Problems
149
swarm and enhances the ability of the global search.
Unlike the previous algorithms, where the
particles move toward global best, in the QPSO-RO
each particle is guided by its personal best and the
personal best of a random selected particle. The
selected particle is chosen from the particles with a
better fitness value, and its selection is based on a
ranking operator. The formula for the attractor
becomes:
)()1(
)()(
,,
,,,
tx
txtp
jqPBji
jiPBjiji
,
(7)
where q is the random selected particle with a better
fitness value.
3 QPSO NEIGHBORHOOD
STRATEGIES
Three different neighborhood strategies are proposed
and analyzed for combining with the QPSO
algorithms, one with unidirectional links and two
with bidirectional links between particles.
For the first strategy, which is inspired from
Clerc’s SPSO (Clerc, 2012), the particles of the
swarm are connected, each connection representing
a unidirectional link between two particles. A
unidirectional link has an informed and an informing
particle, the informed particle knowing the position
and the personal best of the second particle. Thus,
each informed particle has a set of informing
particles called neighborhood (Figure 1). This
strategy will be referred as INF (from
informed/informing).
Figure 1: INF neighborhood strategy for a swarm with 6
particles and 3 informants.
Figure 2: Subswarm neighborhood strategy for a swarm
with 6 particles and 2 subswarms.
When adapted to our QPSO algorithms, the INF
strategy will compute for each particle the mean
only using particle’s neighborhood, and the attractor
using the local best a from the same neighborhood.
In the case of QPSO-RO/RM the random chosen
particles for mean and attractor components will
only be from the neighborhood.
If for a given number of iterations the current
structure of the INF swarm does not improve the
global best the structure is regenerated randomly.
The pseudocode for this strategy is the following:
;generate links between particles
noImprovmentSteps=0
foreach iteration
foreach particle i
;calculate localbest for Ni
;calculate medium for Ni
;calculate attractor for Ni
;calculate new position
;evaluate
foreach particle i
;update position
;update personal best
;calculate global best
if (global best was improved)
noImprovmentSteps=0
else
noImprovmentSteps ++
if (noImprovmentSteps==MAX)
;generate new links
noImprovmentSteps=0
For the second and third strategies the swarm is
divided into subswarms. The subswarms are disjoint
meaning there are no connections between particles
belonging to different subswarms. Inside a
subswarm the particles are fully connected, one
particle has as neighbors all the other particles
(Figure 2).
Just as for the first strategy, the subswarms are
P1
P2
P3
P4
P5
SS
1
(126)
SS
2
(345)
P6
P1
P2
P3
P4
P5
P6
N
1
(1236)
N
2
(12346)
N
3
(3456)
N
4
(1245)
N
5
(125)
N
6
(3456)
ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications
150
dynamically changed during the optimization
process if the global best is not improved for several
iterations. The particles are assigned to subswarms
randomly, each subswarm having a fixed number of
particles.
The second strategy will compute the attractor of
each particle using the local best of the belonging
subswarm (or particles inside it, in case of QPSO-
RO), while the third strategy will use the global best.
For both strategies the mean will be evaluated
(generated in case of QPSO-RM) only using
particles inside the corresponding subswarm. The
second strategy will be reffered as SS-LB while the
third strategy will be reffered as SS-GB (from
subswarm with local/global best).
The pseudocode for the second strategy (similar
for third strategy) is the following:
;generate subswarms
noImprovmentSteps=0
foreach iteration
foreach subswarm SSj
;calculate localbest
;calculate medium
foreach particle i in SSj
;calculate attractor
;calculate new position
;evaluate
foreach particle i
;update position
;update personal best
;calculate global best
if (global best was improved)
noImprovmentSteps=0
else
noImprovmentSteps ++
if (noImprovmentSteps==MAX)
;generate subswarms
noImprovmentSteps=0
4 ELECTROMAGNETIC
PROBLEMS
The QPSO based algorithms have been tested on
two electromagnetic benchmark problems defined
by the COMPUMAG community.
4.1 The TEAM22 Problem
Two coaxial coils carry current with opposite
directions (Figure 3), operate under superconducting
conditions and offer the opportunity to store a
significant amount of energy in their magnetic
fields, while keeping within certain limits the stray
field (Ioan et al., 1999). An optimal design should
couple the energy to be stored by the system with a
minimum stray field into one objective function.
Figure 3: TEAM22 problem configuration.
The objective function has as parameters, the
radii (R), the heights (h) , the thicknesses (d) and the
current densities (J). Besides domain restrictions, the
problem must take into account that the solenoids do
not overlap each other, and the superconducting
material should not violate the quench condition that
links together the value of the current density and
the maximum value of magnetic flux density.
The evaluation method of the objective function
is based on the Biot-Savart-Laplace formula in
which the elliptic integrals are computed by using
the King algorithm and numerical integration as in
(TEAM22, 2015).
4.2 The Loney’s Solenoid
The Loney's solenoid benchmark problem,
formulated in (Di Barba et al., 1995) consists of a
main coil and two identical correction coils, having
fixed dimensions (Figure 4).
Figure 4: Loney’s solenoid problem configuration.
main coil correcting coils
-
z
0
z
0
r
1
r
2
r
3
r
4
s
l
s
h
r
R
1
R
2
h
2
h
1
d
1
d
2
line a 11
line b 11
points
(0,10)
(10,0)
J
1
J
2
z (axis of rotation)
Neighborhood Strategies for QPSO Algorithms to Solve Benchmark Electromagnetic Problems
151
A constant current flows through the coils such
that their current density is the same. The aim is to
produce a constant magnetic flux density in the
middle of the main coil. The parameters to be
optimized are the length of the correction coils (s)
and the axial distance between them (l).
The objective function is of minmax type, i.e.
minimize the maximum difference between the
values of the magnetic flux density along a straight
segment in the middle of the main solenoid, i.e.
minimize (B
max
- B
min
)/B
0
, where B
0
is the magnetic
field density in the middle of the main coil. The
maximum and minimum values are sought along the
segment [-z
0
,z
0
]. Tests done by the authors of this
benchmark revealed that the problem is non convex
and ill conditioned (Di Barba, Savini, 1995). The
electromagnetic field problem is easily solved, in a
magnetostatic regime, by discretizing the coils in
elementary coils without thickness and by applying
well known analytical formulas for the field along
the solenoid axis.
5 RESULTS
To solve the electromagnetic optimization problems
four QPSO based algorithms have been considered
QPSO-WM, QPSO-Gauss, QPSO-RM and QPSO-
RO. After a preliminary study, QPSO-Gauss and
QPSO-RM, together with the basic QPSO, have
been chosen for further testing. Each of the three
mentioned QPSO algorithms have been adapted and
combined with each of the described neighborhood
strategies, INF, SS-LB and SS-GB. Further more,
for each combination have been analyzed the
influence over the performances of the structure
change frequency.
Tables 1 and 2 (see Appendix section) present
the solution fitness values for 30 independent tests
(runs), each run having different random values for
the initial population. For each test the swarm size
was 32, and the stop criteria was the maximum
number of iterations equivalent to 2560 objective
function evaluations. Mean-best is the average of the
best solutions (minimum values) obtained at each of
the 30 runs, while Min-best (Max-best) is the
minimum (maximum) of the minimum values
obtained at each run. The number of informants for
INF strategy was 3, and the number of subswarms
for SS-LB(GB) was 4. Two different frequencies
were tested, a low frequency (LF) of 10 iterations,
and a high frequency (HF) which meant change at
each iteration if the global best was not improved.
Mean - best OF value (QPSO)
1.E-08 1.E-07 1.E-06 1.E-05 1.E-04
1 1019283746556473
classic
SS-LB-LF
INF-HF
Figure 5: QPSO mean for Loney.
Mean - best OF value (QPSO-RM)
1.E-08 1.E-07 1.E-06 1.E-05 1.E-04
1 1019283746556473
classic
SS-LB-LF
SS-GB-HF
Figure 6: QPSO-RM mean for Loney.
Mean - best OF value (QPSO-Gauss)
1.E-08 1.E-07 1.E-06 1.E-05 1.E-04
1 1019283746556473
classic
SS-LB-LF
INF-HF
Figure 7: QPSO-Gauss mean for Loney.
ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications
152
Mean - best OF value (QPSO)
1.E-03 1.E-02 1.E-01 1.E+00 1.E+01
1 1019283746556473
classic
SS-LB-HF
SS-GB-LF
Figure 8: QPSO mean for TEAM22.
Mean - best OF value (QPSO-RM)
1.E-03 1.E-02 1.E-01 1.E+00 1.E+01
1 1019283746556473
classic
SS-LB-LF
INF-HF
Figure 9: QPSO-RM mean for TEAM22.
Mean - best OF value (QPSO-Gauss)
1.E-03 1.E-02 1.E-01 1.E+00 1.E+01
1 1019283746556473
classic
SS-LB-LF
SS-GB-HF
Figure 10: QPSO-Gauss mean for TEAM22.
For the Loney’s solenoid benchmark, the classic
algorithms performances are always improved when
the algorithms are enhanced with neighborhood
strategies. The most stable combinations (smallest
mean-best, and standard deviation) are QPSO-RM
with SS-LB-LF, and QPSO-Gauss with INF-HF.
The overall best solution, which is among the best
found in the literature (by our knowledge), was
obtained with QPSO-RM with SS-LB-LF. In terms
of frequency, while for the SS-LB strategy the low
frequency is better, for SS-GB and INF strategies the
high frequency provides better results.
For the TEAM22 benchmark the algorithms with
neighborhood strategies provide most of the time
better mean and standard deviation values, but the
improvement for the best solution is not as
significant as in the case of Loney’s solenoid.
Surprisingly, the best solution is obtained with the
classic version of the basic QPSO, which offers a
solution close to the well known best from the
literature (1.8 E-3) (TEAM22, 2015). The most
stable combinations are QPSO-Gauss with SS-GB-
HF (LF), QPSO-RM with SS-LB-LF(HF), and
QPSO with SS-GB-LF. Regarding the frequency
change of particles connections, the small
frequencies are suitable for obtaining better mean
values while high frequencies lead to better standard
deviations.
The improvements obtained with the algorithms
enhanced with neighborhood strategies can also be
seen from mean-best evolution during the
optimization process. Besides the fact that statistical
mean values are smaller, the neighborhood enhanced
algorithms are more stable having a smoother
evolution while the classic algorithms evolve (with
some exceptions) in slopes.
6 CONCLUSIONS
The present paper studied the efficiency of
neighborhood strategies when applied to QPSO
based algorithms to solve benchmarks
electromagnetic problems.
Three different neighborhood have been
proposed and analyzed, one with unidirectional and
two with bidirectional particle connections. In the
first strategy, inspired from Clerc’s SPSO, each link
has an informant and an informed particle, thus each
particle has its own neighborhood containing the
informants. The other two strategies divide the
swarm into disjoint subswarms and use to calculate
the attractors the local best of the subswarm or the
global best. For all the strategies the connections are
Neighborhood Strategies for QPSO Algorithms to Solve Benchmark Electromagnetic Problems
153
dynamically changed, reset and randomly
regenerated, if the solution is not improved for
several iterations.
These strategies have been applied to the best
QPSO algorithms available, such as QPSO-RM,
QPSO-Gauss or basic QPSO, and were tested on two
problems from electromagnetism, namely TEAM22
and Loney’s solenoid.
In case of Loney’s solenoid benchmark the
QPSO algorithms enhanced with neighborhood
strategies significantly improve the results for each
of the combinations. The enhanced QPSO
algorithms provide much small mean and standard
deviation values. In the same time the overall best
solution obtained with a QPSO-RM with SS-LB is
one of the best solutions available in the literature.
In case of TEAM 22 problem the enhanced
QPSO algorithms performed better in terms of
stability providing smaller mean and standard
deviation values. However, the best solution is given
surprisingly by the basic QPSO.
For both testing problems the frequency of
structure (connections) change has also been studied.
A low frequency was more suitable for the SS-LB
strategy. For the other two strategies a higher
frequency leads most of the times to better results,
but the optimal frequency also depends on the QPSO
algorithm.
ACKNOWLEDGEMENTS
This work has been supported by the Politehnica
University of Bucharest in the frame of the project
UPB Grant of Excellence, no. 254/2016, the
Romanian Government in the frame of the PN-II-
PT-PCCA-2011-3 program, grant no. 5/2012
(managed by CNDI– UEFISCDI, ANCS), and in the
frame of RO-BE bilateral project, grant no xx/2016.
REFERENCES
Bratton, Kennedy, 2007. Defining a standard for particle
swarm optimization. Proceedings of the IEEE Swarm
Intelligence Symposium, 2007.
Ciuprina, Ioan, Munteanu, 2002. Use of intelligent-particle
swarm optimization in electromagnetics. IEEE
Transactions on Magnetics, vol. 38 (2), pp. 1037-
1040.
Clerc, 2012. Standard particle swarm optimization. Open
access archive HAL (http://clerc.maurice.free.fr/pso/
SPSO_descriptions.pdf).
Coelho, 2007. A novel Gaussian quantum-behaved
particle swarm optimizer applied to electromagnetic
design, IET Science, Measurement & Technology 1,
pp. 290–294.
Coelho, Alotto, 2008. Global optimization of
electromagnetic devices using an exponential
quantum-behaved particle swarm optimizer, IEEE
Transactions on Magenetics 44, pp. 1074–1077.
Di Barba, Gottvald, Savini, 1995. Global optimization of
Loney’s solenoid: A benchmark problem. Int. J. Appl.
Electromagn. Mech., vol. 6, no. 4, pp. 273–276.
Di Barba, Savini, 1995. Global optimization of Loney’s
solenoid by means of a deterministic approach. Int. J.
Appl. Electromagn. Mech., vol. 6, no. 4, pp. 247–254.
Duca, Duca, Ciuprina, Yilmaz, Altinoz, 2014, PSO
Algorithms and GPGPU Technique for
Electromagnetic Problems, in the International
Workshops on Optimization and Inverse Problems in
Electromagnetism (OIPE), Delft, The Netherlands.
(under review process, to be published by an ISI
indexed journal).
Duca, Rebican, Janousek, Smetana, Strapacova, 2014.
PSO Based Techniques for NDT-ECT Inverse
Problems. In Electromagnetic Nondestructive
Evaluation (XVII), vol. 39, pp. 323 - 330. Capova, K.,
Udpa, L., Janousek, L., and Rao, B.P.C. (Eds.), IOS
Press, Amsterdam.
Ioan, Ciuprina, Szigeti, 1999. Embedded stochastic-
deterministic optimization method with accuracy
control. IEEE Transactions on Magnetics, vol. 35 , pp.
1702-1705.
Kennedy, Eberhart, 1995. Particle swarm optimization.
Proceedings of IEEE International Conference on
Neural Networks, pp. 1942-1948.
Li, Wang, Hu, Sun, 2007. A new QPSO based BP neural
network for face detection, Fuzzy Information and
Engineering, Advances in Soft Computing 40,
Springer.
Mikki, Kishk, 2006. Quantum particle swarm optimization
for electromagnetics, IEEE Transactions on Antennas
and Propagation 54, pp. 2764–2775.
Sun, Feng, Xu, 2004, Particle swarm optimization with
particles having quantum behavior, in: IEEE
Proceedings of Congress on Evolutionary
Computation, pp. 325–331.
Sun, Fang, Palade, Wua, Xu, 2011. Quantum-behaved
particle swarm optimization with Gaussian distributed
local attractor point, Applied Mathematics and
Computation 218, pp. 3763–3775.
Sun, Wua, Palade, Fang, Lai, Xu, 2012.
Convergence
analysis and improvements of quantum-behaved
particle swarm optimization, Information Sciences
193, pp. 81–103.
TEAM22 benchmark problem, 2015.
http://www.compumag.org/jsite/team.html.
Xi, Sun, Xu, 2008. An improved quantum-behaved
particle swarm optimization algorithm with weighted
mean best position, Applied Mathematics and
Computation 205, pp. 751–759.
Zhang, Zuo, 2013. Deadline Constrained Task Scheduling
Based on Standard-PSO in a Hybrid Cloud, Advances
in Swarm Intelligence, Springer.
ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications
154
APPENDIX
Table 1: Objective function values and standard deviation for Loney’s solenoid.
Algorithm
Min-best
OF value
Max - best
OF value
Mean - best
OF value
Standard
deviation
QPSO
classic 9.73E-09 1.13E-06 7.72E-08 2.03E-07
SS–LB–LF 4.09E-09 1.78E-07
3.04E-08 3.47E-08
SS–LB–HF 9.89E-09 2.05E-07 3.87E-08 3.63E-08
SS–GB–LF 1.02E-08 5.99E-07 6.47E-08 1.19E-07
SS–GB–HF
3.87E-09
2.38E-06 1.21E-07 4.30E-07
INF–LF 9.29E-09 3.33E-07 4.07E-08 5.99E-08
INF–HF 8.64E-09
1.71E-07
3.33E-08 3.47E-08
QPSO
RM
classic 6.02E-09 4.04E-07 3.99E-08 7.33E-08
SS–LB–LF
1.05E-10 1.80E-08 1.07E-08 5.33E-09
SS–LB–HF 2.89E-09 8.23E-08 2.42E-08 1.52E-08
SS–GB–LF 1.17E-10 4.13E-07 3.84E-08 9.68E-08
SS–GB–HF 3.93E-09 2.55E-07 2.64E-08 4.85E-08
INF–LF 1.32E-08 3.78E-07 6.77E-08 8.62E-08
INF–HF 7.41E-09 2.21E-07 4.37E-08 5.15E-08
QPSO
Gauss
classic 8.03E-09 2.17E-07 3.01E-08 4.08E-08
SS–LB–LF 4.33E-09 5.28E-08 1.93E-08 1.04E-08
SS–LB–HF
1.88E-09
7.95E-08 2.18E-08 1.40E-08
SS–GB–LF 7.09E-09 4.29E-07 4.03E-08 7.91E-08
SS–GB–HF 1.23E-08 1.39E-07 3.43E-08 3.17E-08
INF–LF 1.02E-08 8.76E-08 2.44E-08 1.73E-08
INF–HF 1.25E-08
3.47E-08 1.92E-08 5.34E-09
Table 2: Objective function values and standard deviation for TEAM 22.
Algorithm
Min - best
OF value
Max - best
OF value
Mean - best
OF value
Standard
deviation
QPSO
classic
2.23E-03
2.76E-02 8.68E-03 6.46E-03
SS–LB–LF 4.69E-03 3.17E-02 8.78E-03 5.84E-03
SS–LB–HF 4.48E-03 1.65E-02 7.71E-03 3.15E-03
SS–GB–LF 3.85E-03
1.06E-02 6.76E-03 1.93E-03
SS–GB–HF 3.65E-03 2.71E-02 7.46E-03 5.08E-03
INF–LF 5.21e-03 3.38e-01 3.46e-02 7.82e-02
INF–HF 4.12e-03 3.56e-02 1.17e-02 8.71e-03
QPSO
RM
classic 3.09E-03 2.17E-02 7.31E-03 4.85E-03
SS–LB–LF 3.50E-03 1.40E-02
5.33E-03
2.55E-03
SS–LB–HF 4.53E-03
9.31E-03
6.11E-03
1.26E-03
SS–GB–LF
2.78E-03
1.77E-02 6.37E-03 4.01E-03
SS–GB–HF 3.64E-03 1.01E-02 6.16E-03 1.80E-03
INF–LF 3.21e-03 9.48e-03 5.93e-03 1.81e-03
INF–HF 3.65e-03 1.02e-01 1.24e-02 2.15e-02
QPSO
Gauss
classic 2.66E-03 9.54E-03 5.53E-03 1.54E-03
SS–LB–LF
2.58E-03
8.95E-03
4.27E-03
1.56E-03
SS–LB–HF 4.10E-03 9.25E-03 6.35E-03 1.54E-03
SS–GB–LF 4.54E-03 1.17E-02 6.76E-03 2.18E-03
SS–GB–HF 4.61E-03
7.65E-03
5.95E-03
8.79E-04
INF–LF 4.40e-03 9.86e-03 6.71e-03 1.73e-03
INF–HF 4.08e-03 1.69e-02 7.16e-03 3.08e-03
Neighborhood Strategies for QPSO Algorithms to Solve Benchmark Electromagnetic Problems
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