Modified Vortex Search Algorithm for Real Parameter Optimization

Jan Zizka et al. (Eds) : CCSEIT, AIAP, DMDB, MoWiN, CoSIT, CRIS, SIGL, ICBB, CNSA-2016
pp. 113–126, 2016. © CS & IT-CSCP 2016 DOI : 10.5121/csit.2016.60610
MODIFIED VORTEX SEARCH ALGORITHM
FOR REAL PARAMETER OPTIMIZATION
Berat Doğan
Department of Biomedical Engineering, Inonu University, Malatya, Turkey
berat.dogan@inonu.edu.tr
ABSTRACT
The Vortex Search (VS) algorithm is one of the recently proposed metaheuristic algorithms
which was inspired from the vortical flow of the stirred fluids. Although the VS algorithm is
shown to be a good candidate for the solution of certain optimization problems, it also has some
drawbacks. In the VS algorithm, candidate solutions are generated around the current best
solution by using a Gaussian distribution at each iteration pass. This provides simplicity to the
algorithm but it also leads to some problems along. Especially, for the functions those have a
number of local minimum points, to select a single point to generate candidate solutions leads
the algorithm to being trapped into a local minimum point. Due to the adaptive step-size
adjustment scheme used in the VS algorithm, the locality of the created candidate solutions is
increased at each iteration pass. Therefore, if the algorithm cannot escape a local point as
quickly as possible, it becomes much more difficult for the algorithm to escape from that point
in the latter iterations. In this study, a modified Vortex Search algorithm (MVS) is proposed to
overcome above mentioned drawback of the existing VS algorithm. In the MVS algorithm, the
candidate solutions are generated around a number of points at each iteration pass.
Computational results showed that with the help of this modification the global search ability of
the existing VS algorithm is improved and the MVS algorithm outperformed the existing VS
algorithm, PSO2011 and ABC algorithms for the benchmark numerical function set.
KEYWORDS
Metaheuristics, Numerical Function Optimization, Vortex Search Algorithm, Modified Vortex
Search Algorithm.
1. INTRODUCTION
In the past two decades, a number of metaheuristic algorithms have been proposed to solve
complex real-world optimization problems. Most of these algorithms are nature inspired methods
and therefore mimic natural metaphors such as, evolution of species (GA [1] and DE [2-3]),
annealing process (SA [4-5]), ant behaviour (ACO [6]), swarm behaviour (PSO [7] and ABC [8-
9]) etc. These algorithms make few or no assumptions for the problem at hand and provide fast
and robust solutions. Although, the solutions provided by metaheuristics may not be optimal
solutions, they are highly preferred because of their simplicity and flexibility.
Despite the high number of available metaheuristics, developing new metaheuristic algorithms is
still an active research area. In [10-15], a number of recently proposed metaheuristics can be

114 Computer Science & Information Technology (CS & IT)
found. All of these metaheuristics have certain characteristics and thus each one may be more
successful on a certain optimization problem when compared to the others. The Vortex Search
(VS) algorithm [16] is one of these recently proposed metaheuristic algorithms which was
inspired from the vortical flow of the stirred fluids. The search behaviour of the VS algorithm is
modelled as a vortex pattern by using an adaptive step-size adjustment scheme. By this way, it is
aimed to have a good balance between the explorative and exploitative behaviour of the search.
The proposed VS algorithm was tested over 50 benchmark mathematical functions and the
obtained results compared to the single-solution based (Simulated Annealing, SA and Pattern
Search, PS) and population-based (Particle Swarm Optimization, PSO2011 and Artificial Bee
Colony, ABC) algorithms. A Wilcoxon-Signed Rank Test was performed to measure the pair-
wise statistical performances of the algorithms, the results of which indicated that the proposed
VS algorithm outperforms the SA, PS and ABC algorithms while being competitive with the
PSO2011 algorithm. Because of the simplicity of the proposed VS algorithm, a significant
decrease in the computational time of the 50 benchmark numerical functions was also achieved
when compared to the population-based algorithms. In some other studies [17-20], the VS
algorithm has also been successfully used for the solution of some real-world optimization
problems.
Although the proposed VS algorithm is a good candidate for the solution of optimization
problems, it also has some drawbacks. In the VS algorithm, candidate solutions are generated
around the current best solution by using a Gaussian distribution at each iteration pass. This
provides simplicity to the algorithm but it also leads to some problems along. Especially, for the
functions those have a number of local minimum points, to select a single point to generate
candidate solutions leads the algorithm to being trapped into a local minimum point. Due to the
adaptive step-size adjustment scheme used in the VS algorithm, the locality of the created
candidate solutions is increased at each iteration pass. Therefore, if the algorithm cannot escape a
local point as quickly as possible, it becomes much more difficult for the algorithm to escape
from that point in the latter iterations.
In this study, a modified Vortex Search algorithm (MVS) is proposed to overcome above
mentioned drawback of the existing VS algorithm. In the MVS algorithm, the candidate solutions
are generated around different points at each iteration pass. These points are iteratively updated
during the search process, details of which are given in the following section. The MVS
algorithm is tested with 7 benchmark functions that was used earlier in [16]. These 7 functions
are selected from the benchmark set of 50 functions for which the VS algorithm trapped into the
local minimum points. Because the SA and PS algorithms showed poor performances in [16], in
this study these two algorithms are excluded and the results are compared to the results those
obtained by the VS algorithm, PSO2011 and ABC algorithms. It is shown that, the MVS
algorithm outperforms all of these algorithms and can successfully escape from the local
minimum points of the functions that the VS algorithm was being trapped earlier.
The remaining part of this paper is organized as follows. In the following section, first a brief
description of the VS algorithm is given. Then, the modification performed on the VS algorithm
is detailed and the MVS algorithm is introduced. Section 3 covers the experimental results and
discussion. Finally, Section 4 concludes the work.

Computer Science & Information Technology (CS & IT) 115
2. METHODOLOGY
2.1. A Brief Description of the Vortex Search Algorithm
Let us consider a two-dimensional optimization problem. In a two dimensional space a vortex
pattern can be modelled by a number of nested circles. Here, the outer (largest) circle of the
vortex is first centered on the search space, where the initial center can be calculated using Eq. 1.
In Eq.1, upperlimit and lowerlimit are 1×d vectors that define the bound constraints of the
problem in d dimensional space. Then, a number of neighbor solutions )(sCt , (t represents the
iteration index and initially 0=t ) are randomly generated around the initial center 0µ in the d -
dimensional space by using a Gaussian distribution. Here, { } nkssssC k ,...,2,1,...,,)( 210 ==
represents the solutions, and n represents the total number of candidate solutions. In Eq. 2, the
general form of the multivariate Gaussian distribution is given.
In Eq.2, d represents the dimension, x is the 1×d vector of a random variable, µ is the 1×d
vector of sample mean (center) and Σ is the covariance matrix. If the diagonal elements
(variances) of the values of Σ are equal and if the off-diagonal elements (covariance) are zero
(uncorrelated), then the resulting shape of the distribution will be spherical (which can be
considered circular for a two-dimensional problem, as in our case). Thus, the value of Σ can be
computed by using equal variances with zero covariance by using Eq. 3.
In Eq. 3,
2
σ represents the variance of the distribution and I represents the dd × identity
matrix. The initial standard deviation ( 0σ ) of the distribution can be calculated by using Eq. 4.
Here, 0σ can also be considered as the initial radius ( 0r ) of the outer circle for a two
dimensional optimization problem. Because a weak locality is required in the initial phases, 0r is
chosen to be a large value. Thus, a full coverage of the search space by the outer circle is
provided in the initial step. This process provides a bird's-eye view for the problem at hand.
In the selection phase, a solution (which is the best one) )(0
'
sCs ∈ is selected and memorized
from )(0 sC to replace the current circle center 0µ . Prior to the selection phase, the candidate
solutions must be ensured to be inside the search boundaries. For this purpose, the solutions that
exceed the boundaries are shifted into the boundaries, as in Eq. 5.

In Eq.5, nk ,...2,1= and di ,...,2,1= and rand is a uniformly distributed random number. Next,
the memorized best solution
'
s is assigned to be the center of the second circle (the inner one). In
the generation phase of the second step, the effective radius ( 1r ) of this new circle is reduced, and
then, a new set of solutions )(1 sC is generated around the new center. Note that in the second
step, the locality of the generated neighbors increased with the decreased radius. In the selection
phase of the second step, the new set of solutions )(1 sC is evaluated to select a solution
)(1
'
sCs ∈ . If the selected solution is better than the best solution found so far, then this solution
is assigned to be the new best solution and it is memorized. Next, the center of the third circle is
assigned to be the memorized best solution found so far. This process iterates until the
termination condition is met. An illustrative sketch of the process is given in Figure 1. In this
manner, once the algorithm is terminated, the resulting pattern appears as a vortex-like structure,
where the center of the smallest circle is the optimum point found by the algorithm. A
representative pattern is sketched in Figure 2 for a two-dimensional optimization problem for
which the upper and lower limits are between the [-10,10] interval. A description of the VS
algorithm is also provided in Figure 3.
The radius decrement process given in Figure 3 can be considered as a type of adaptive step-size
adjustment process which has critical importance on the performance of the VS algorithm. This
process should be performed in such a way that allows the algorithm to behave in an explorative
manner in the initial steps and in an exploitative manner in the latter steps. To achieve this type
of process, the value of the radius must be tuned properly during the search process. In the VS
algorithm, the inverse incomplete gamma function is used to decrease the value of the radius
during each iteration pass.
Figure 1. An illustrative sketch of the search process

Figure 2. A representative pattern showing the search boundaries (circles) of the VS algorithm after a
search process, which has a vortex-like structure.
The incomplete gamma function given in Eq. 6 most commonly arises in probability theory,
particularly in those applications involving the chi-square distribution [21].
In Eq.6, 0>a is known as the shape parameter and 0≥x is a random variable. In conjunction
with the incomplete gamma function, its complementary ),( axΓ is usually also introduced (Eq.
7).
Thus, it follows that,
where )(aΓ is known as the gamma function. There exist many studies in the literature on
different proposed methods for the numerical calculation of the incomplete gamma function [22-
24]. MATLAB® also provides some tools for the calculation of the inverse incomplete gamma
(gammaincinv) function. The inverse incomplete gamma function (gammaincinv), computes the
inverse of the incomplete gamma function with respect to the integration limit x and represented
as gammaincinv(x,a) in MATLAB®.
In Figure 4, the inverse incomplete gamma function is plotted for 1.0=x and [ ]1,0∈a . Here, for
our case the parameter a of the inverse incomplete gamma function defines the resolution of the
search. By equally sampling a values within [ ]1,0 interval at a certain step size, the resolution of
the search can be adjusted. For this purpose, at each iteration, a value of a is computed by using
the Eq.9

Figure 3. A description of the VS algorithm
where 0a is selected as 10 =a to ensure a full coverage of the search space at the first iteration,
t is the iteration index, and MaxItr represents the maximum number of iterations.
Let us consider an optimization problem defined within the [-10,10] region. The initial radius 0r
can be calculated with Eq. 10. Because 10 =a , the resulting function value is
1),()1( 0 ≈⋅ axvgammaincinx , which means 00 σ≈r as indicated before.
By means of Eq.4, the initial radius value 0r can be calculated as 100 ≈r . In Eq.11, a general
formula is also given to obtain the value of the radius at each iteration pass.
Here, t represents the iteration index.

Figure 4. ),()1( axvgammaincinx ⋅ where 1.0=x and [ ]1,0∈a
2.2. The Modified Vortex Search Algorithm
The VS algorithm creates candidate solutions around a single point at each iteration pass. At the
first iteration, this point is the initial center 0µ which is determined with the upper and lower
limits of the problem at hand while in the latter iterations the center is shifted to the current best
position found so far. As mentioned before, this mechanism leads the VS algorithm to being
trapped into local minimum points for a number of functions.
To overcome above mentioned drawback, in this study a modified VS algorithm (MVS) is
proposed. In the MVS algorithm, candidate solutions are generated around multiple centers at
each iterations pass. The search behavior of the MVS algorithm can be thought as a number of
parallel vortices that have different centers at each iteration pass. Initially, the centers of these
multiple vortices are selected as in the VS algorithm. Let us consider, the total number of centers
(or vortices) to be represented by m . Let us say, )(µtM represents the matrix that stores the
values of these m centers at each iteration pass and t represents the iteration index. Thus,
initially { } mlM l
,...,2,1,,...,,)( 0
2
0
1
00 == µµµµ and initial positions of these centers are computed
as in Eq. 12.
Next, a number of candidate solutions are generated with a Gaussian distribution around these
initial centers by using the initial radius value 0r . In this case the total number of candidate
solutions is again selected to be n. But note that, these n solutions are generated around m
centers. Thus, one should select mn solutions around each center.
Let us say, { } mnkssssCS k
l
t
,...,2,1,...,,)( 21 == represents the subset of solutions generated
around the center ml ,...,2,1= for the iteration t . Then, the total solution set generated for the

iteration 0=t can be represented by
phase, for each subset of solutions, a solution (which is the best one)
Prior to the selection phase it must be ensured that the candid
the search boundaries. For this purpose, the solutions that exceed the boundaries are shifted into
the boundaries, as in Eq. 5. Let us say, the best solution of each subset is stored in a matrix
)( '
sPBestt at each iteration pass. Thus, for
that, the best solution of this matrix (
solution set )(0 sC for the current iteration, which is represented as
In the VS algorithm, at each iterations pass, the center is always shifted to the best solution found
so far, bests . However, in the MVS algorithm, there exist
updated for the next iteration. The most important difference between the VS and MVS algorithm
arises from here. In the MVS algorithm, one of these centers is again shifted to the best solution
found so far, bests . But, the remaining
the best positions generated around the each center at the iteration
so far, bests as shown in Eq. 13.
In Eq. 13, rand is a uniformly distributed random number,
)( '
1
'
sPBests tl −∈ . Thus, for t
using the )( '
0
'
sPBestsl ∈ positions and the best position found so far,
illustrative sketch of the center update process is given for a two
5, only one center is considered.
Figure 5. An illustrative sketch of the center updating process for the MVS algorithm (only one center is
In the MVS algorithm, the radius decrement process is held totally in the same way as it is done
in the VS algorithm. At each iteration pass, the radius is decreased by u
Computer Science & Information Technology (CS & IT)
can be represented by { } mlCSCSCSsC l
,...,2,1,,...,,)( 0
2
0
1
00 == . In the selection
phase, for each subset of solutions, a solution (which is the best one) (0
'
sCSs l
l ∈
Prior to the selection phase it must be ensured that the candidate subsets of solutions are inside
t each iteration pass. Thus, for 0=t , { }lssssPBest l ,1,,...,,)( ''
2
'
1
'
0 ==
that, the best solution of this matrix ( )( '
0 sPBest ) is also the best solution of the total candidate
for the current iteration, which is represented as bestItr .
. However, in the MVS algorithm, there exist m centers which positions need to be
. But, the remaining 1−m centers are shifted to a new position determined by
the best positions generated around the each center at the iteration t and the best position found
is a uniformly distributed random number, ,...,2,1=l
1= , { } 1,...,2,1,,...,,)( 1
2
1
1
11 −== mlM l
µµµµ is determined by
positions and the best position found so far, bests . In Figure
illustrative sketch of the center update process is given for a two-dimensional problem. In Figure
5, only one center is considered.
of the center updating process for the MVS algorithm (only one center is
considered)
in the VS algorithm. At each iteration pass, the radius is decreased by utilizing the inverse
. In the selection
)s is selected.
ate subsets of solutions are inside
m,...,2, . Note
) is also the best solution of the total candidate
centers which positions need to be
centers are shifted to a new position determined by
and the best position found
1,..., −m and
is determined by
. In Figure-5, an
dimensional problem. In Figure
of the center updating process for the MVS algorithm (only one center is
tilizing the inverse

incomplete gamma function and thus, the locality of the generated solutions is increased. In
Figure 6, a description of the MVS algorithm is provided.
Figure 6. A description of the MVS algorithm
3. RESULTS
The proposed MVS algorithm is tested on 7 benchmark functions for which the VS algorithm
was being trapped into a local minimum point. By using these functions, in this study, the
Figure 6, a description of the MVS algorithm is provided.
Figure 6. A description of the MVS algorithm
121

performance of the MVS algorithm is compared to the VS, PSO2011 and ABC algorithms.
PSO2011 [25-26] is an extension of the standard PSO algorithm and the ABC algorithm is a
well-known optimization algorithm which was inspired from the collective behaviours of honey
bees.
The functions used in the experiments are listed in Table 1. For the formulations of the functions
listed in Table 1, please refer to the reference [16].
3.1. Algorithm Settings
The ABC and PSO2011 algorithms are selected to have a population size of 50, which is also the
number of neighborhood solutions of the proposed VS algorithm. The acceleration coefficients (
1c and 2c ) of the PSO2011 algorithm are both set to 1.8, and the inertia coefficient is set to 0.6,
as in [27]. The limit value for the ABC algorithm is determined as limit = SN * D, where SN
represents the number of food sources and D represents the dimension. VS algorithm does not
have any additional parameters. Different from the VS algorithm, the MVS algorithm has the
parameter m , which represents the total number of centers.
3.2. Experimental Results
For each algorithm, 30 different runs are performed, and the mean and the best values are
recorded. The maximum number of iterations is selected to be 500,000. For the MATLAB®
codes of the PSO2011, ABC, VS and MVS algorithms please refer to [25], [28], [29] and [30].
For each algorithm, all of the functions are run in parallel using a 32 core Intel® CPU 32 GB
RAM workstation. For the first set of experiments, results are given in Table 2
Table 1. Benchmark function set that is used in the experiments
No Function Characteristics Range Dim. Min.
F1 Powell Unimodal Non-Separable [-4,5] 24 0
F2 Rosenbrock Unimodal Non-Separable [-30, 30] 30 0
F3 Dixon-Price Unimodal Non-Separable [-10, 10] 30 0
F4 Rastrigin Multimodal Separable [-5.12, 5.12] 30 0
F5 Schwefel Multimodal Separable [-500, 500] 30 -12569.5
F6 Griewank Multimodal Non-Separable [-600, 600] 30 0
F7 Penalized Multimodal Non-Separable [-50, 50] 30 0
As shown in Table 2, for the MVS algorithm two different cases are considered. In the first case,
the total number of candidate solutions is selected to be 50, which means 10 candidate solutions
are generated around each center for 5=m . In this case, the MVS algorithm can avoid from the
local minimum points of the functions which is not the case for the VS algorithm. However, poor
sampling of the search space for this case (10 points around each center) may lead the MVS
algorithm to show a correspondingly poor performance on the improvement of the found near
optimal solutions (exploitation). Therefore, another case in which the total number of candidate
solutions is selected to be 250 is considered for the MVS algorithm. In this case, 50 candidate
solutions are generated around each center for 5=m . As can be shown in Table 2, the MVS
algorithm with 250 candidate solutions performs better than the MVS algorithm with 50
candidate solutions.

In [31], authors stated that after a sufficient value for colony size, any increment in the value does
not improve the performance of the ABC algorithm significantly. For the test problems carried
out in [31] colony sizes of 10, 50 and 100 are used for the ABC algorithm. It is shown that
although from 10 to 50 the performance of the ABC algorithm significantly increased, there is
not any significant difference between the performances achieved by 50 and 100 colony sizes.
Similarly, for the PSO algorithm it is reported that, PSO with different population sizes has
almost the similar performance which means the performance of PSO is not sensitive to the
population size [32]. Based on the above considerations, in this study a comparison of the MVS
algorithm to the ABC and PSO2011 algorithms with a different population size is not performed.
For the VS algorithm it is expected to achieve better exploitation ability with an increased
number of candidate solutions. But the problem with the VS algorithm is with its global search
ability rather than the local search ability for some of the functions listed above. Therefore, a
comparison of the MVS (m = 5, n = 50) to VS algorithm with 50 candidate solutions is thought to
be enough to show the improvement achieved by the modification performed on the VS
algorithm.
In Figure 7, average computational time of 30 runs for 500,000 iterations is also provided for the
MVS (m = 5, n = 50), MVS (m = 5, n = 250), VS, PSO2011 and ABC algorithms. As shown in
this figure, the required computational time to perform 500,000 iterations with the MVS
algorithm is slightly increased when compared to the VS algorithm. However, even for the MVS
(m = 5, n = 250) algorithm the required computational time to perform 500,000 iterations is still
lower than the PSO2011 and ABC algorithms.
Table 2. Statistical results of 30 runs obtained by PSO2011, ABC, VS and MVS algorithms (values <
16
10−
are considered as 0).

Figure 7. Average computational time of 30 runs for 50 benchmark functions (500,000 iterations)
4. CONCLUSIONS
This paper presents a modified VS algorithm in which the global search ability of the existing VS
algorithm is improved. This is achieved by using multiple centers during the candidate solution
generation phase of the algorithm at each iteration pass. In the VS algorithm, only one center is
used for this purpose and this usually leads the algorithm to being trapped into a local minimum
point for some of the benchmark functions. Computational experiments performed on the
benchmark functions showed that, the MVS algorithm outperforms the VS, PSO211 and ABC
algorithms and can successfully escapes from the local minimum points of the functions that the
VS algorithm was being trapped earlier. Although the complexity of the existing VS algorithm is
a bit increased with the performed modification, there is not any significant difference between
the computational time of the modified VS algorithm and the existing VS algorithm.
In the future studies, the proposed MVS algorithm will be used for the solution of some real
world optimization problems such as neural network optimization, optimum data partitioning,
and analog circuit parameters optimization.
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AUTHORS
Dr. Berat Doğan received his BSc. degree in Electronics Engineering from Erciyes
University, Turkey, 2006. He received his MSc. degree in Biomedical Engineering
from Istanbul Technical University, Turkey, 2009. He received his PhD. in
Electronics Engineering at Istanbul Technical University, Turkey, 2015. Between
2008-2009 he worked as a software engineer at Nortel Networks Netas
Telecommunication Inc. Then, from 2009 to July 2015 he worked as a Research
Assistant at Istanbul Technical University. Now he is working as an Assistant
Professor at Inonu University, Malatya, Turkey. His research interests include
optimization algorithms, pattern recognition, biomedical signal and image processing, and bioinformatics.

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