Particle Swarm Optimization is a class of stochastic, population based optimization techniques which are mostly suitable for static problems. However, real world optimization problems are time variant, i.e., the problem space changes over time. Several researches have been done to address this dynamic optimization problem using Particle Swarms. In this paper we probe the issues of tracking and optimizing Particle Swarms in a dynamic system where the problem-space drifts in a particular direction. Our assumption is that the approximate amount of drift is known, but the direction of the drift is unknown. We propose a Drift Predictive PSO (DriP-PSO) model which does not incur high computation cost, and is very fast and accurate. The main idea behind this technique is to determine the approximate direction using a small number of stagnant particles in which the problem-space is drifting so that the particle velocities may be adjusted accordingly in the subsequent iteration of the algorithm.

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A Fast and Inexpensive Particle Swarm Optimization
for Drifting Problem-Spaces
Zubin Bhuyan
Department of Computer Science and Engineering
Tezpur University,
Tezpur, India
zubin_csi11@agnee.tezu.ernet.in
Sourav Hazarika
Department of Computer Science and Engineering
Tezpur University,
Tezpur, India
sourav_csi11@agnee.tezu.ernet.in
Abstract— Particle Swarm Optimization is a class of stochastic,
population based optimization techniques which are mostly
suitable for static problems. However, real world optimization
problems are time variant, i.e., the problem space changes over
time. Several researches have been done to address this dynamic
optimization problem using Particle Swarms. In this paper we
probe the issues of tracking and optimizing Particle Swarms in a
dynamic system where the problem-space drifts in a particular
direction. Our assumption is that the approximate amount of
drift is known, but the direction of the drift is unknown. We
propose a Drift Predictive PSO (DriP-PSO) model which does not
incur high computation cost, and is very fast and accurate. The
main idea behind this technique is to use a few stagnant particles
to determine the approximate direction in which the problem-
space is drifting so that the particle velocities may be adjusted
accordingly in the subsequent iteration of the algorithm.
Keywords- pso, dynamic exploration, drifting problem-space
I. INTRODUCTION
Swarm intelligence may be defined as the collective
behavior of simple rule-following agents in a decentralized
system, where the overall behavior of the entire system appears
intelligent to an external observer. In nature, this kind of
behavior is seen in bird flocks, fish schools, ant colonies and
animal herds. Given a large space of possibilities, a population
of agents is often able solve difficult problems by finding
multivariate solutions or patterns through a simplified form of
social interaction [1].
Particle swarm optimization was first put forward by
Kennedy and Eberhart in 1995 [2, 3]. The PSO algorithm
exhibits all common evolutionary computation characteristics,
viz., initialization with a random population, searching for
optima by updating generations, and updating generations
based on previous ones. It has been implemented with different
approaches for a wide range of generic problems, as well as for
case-specific applications focused on a precise requirement [4,
5, 6, 7].
However, almost all practical problems are time-varying or
dynamic, i.e., the environment and the characteristics of the
global optimum changes over time. More formally, a dynamic
system is one where the system changes its state in a repeated
or non-repeated manner. In such cases a standard PSO might
not give the most optimal results. Also, there are several ways
in which a system may change over time. The changes may
occur periodically in some predefined sequence or in random
fashion. References [8, 9] define three kinds of dynamic
systems. First, the location of the optimum value in the
problem space may change. Second, the location can remain
constant but the optimum value may vary. Third, both the
location and the value of the optimum may vary.
II. BACKGROUND
A. Standard Particle Swarm Optimization
PSO is initialized with a population of random solutions
called particles. Each particle moves about, or flies, in the
given problem space with a velocity which keeps on varying
continuously according to its own flying experience and other
particles as well. In a D-dimension space the location of the ith
particle is represented as Xi = (xi1,…, xid,…, xiD), and velocity
for the ith
particle is represented as Vi = (vi1,…, vid, …, viD).
The best previous position of the ith
particle is called the
pbesti. The best pbest among all the particles is called the
gbest.
Equations (1a) and (1b) are used to update the particles‟
position and velocity.
𝑣𝑖𝑑 = 𝑤 × 𝑣𝑖𝑑 + 𝑐1 × 𝑟𝑎𝑛𝑑() × 𝑝𝑖𝑑 − 𝑥𝑖𝑑 + 𝑐2 ×
𝑟𝑎𝑛𝑑() × (𝑝 𝑔𝑑 − 𝑥𝑖𝑑 ) (1a)
𝑥𝑖𝑑 = 𝑥𝑖𝑑 + 𝑣𝑖𝑑 (1b)
Equation (1a) calculates the new velocity of the particles
based on its previous velocity (𝑣𝑖𝑑 ), location where the particle
has achieved its best value (pbesti or 𝑝𝑖𝑑 ), location where the
highest of all pbest has been achieved (gbest or 𝑝 𝑔𝑑 ), w is
inertia weight, c1 and c2 are cognitive and social acceleration
constants, and rand() is a random number generator function
The new position of each particle is then updated using
equation (1b). In both the equations subscript d indicates the
dth
dimension.
If the current fitness of any particle is better than its pbest,
then the value of the pbest will be replaced by the current
solution. Again, if that pbest is better than the existing gbest,

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then the pbest will become the new gbest. This process is
repeated until a satisfactory result is obtained.
B. PSO for Dynamic Systems
Several propositions have been made regarding the
modification of the PSO algorithm to address the dynamic
optimization problem, i.e., scenarios where the problem space
changes over time. In such situation the particles might lose its
global exploration ability due to changing position of global
optimum. This usually leads to unsatisfactory, unacceptable
and sub-optimal results.
Eberhart and Hu in [8] use a “fixed gBest-value method”
where the gbest value and the second-best gbest value are
monitored. If these two values do not change for certain
number of iterations then a possible optimum change is
declared. Actually the gbest value and second-best gbest value
are monitored to increase the accuracy and prevent false
alarms.
Another very successful PSO algorithm for dynamic
systems is the charged-PSO developed by Blackwell and
Bentley [10]. The driving principle behind charged PSO is a
good balance between exploration and exploitation, which in
turn results in continuous search for better solution while
refining the current soluiton. Rakitianskaia and Engelbrecht in
[11] have further modified the charged-PSO (CPSO) by
incorporating within it the concept of cooperative split PSO
(CSPSO). CSPSO is an approach where search space is
divided into smaller subspaces, with each subspace being
optimised by a separate swarm [12].
Hashemi and Meybodi introduced cellular PSO [13]. This is
a hybrid model of particle swarm optimization and cellular
automata where the population of particles is split into
different groups across cells of cellular automata by imposing
a restriction on number of particles in each cell. This was
further modified for dynamic systems by introducing
temporary quantam particles [14].
III. PROPOSED DRIFT PREDICTIVE PSO MODEL
(DRIP-PSO)
In this paper we propose a cost-effective and accurate PSO
model, DriP-PSO, which has been specifically designed for
the scenario where the problem-space drifts in an unknown
direction over time and an approximate amount of drift is
known. The algorithm determines the approximate direction in
which the problem-space is drifting so that the particle
velocities may be adjusted accordingly in the subsequent
iteration of the algorithm. This is achieved by selecting a few
stagnant particles which try to detect the direction of drift.
In each iteration of the DriP-PSO algorithm, a small
number of stagnant particles are selected randomly. The
stagnant particles do not change their positions for that
particular round. These stagnant particles would then compare
its previous fitness value to its current fitness value. If a
change is detected, the stagnant particles will generate four
sub-particles which will rest on a circular orbit of radius ρ.
Every stagnant particle will be the centre of its circular orbit,
and the sub-particles will be placed at right angle to one
another.
For example, let us consider a particle Pi, such that for a
particular round it has been selected as a stagnant particle. In
order to determine the direction of drift we select two sub-
particles,Sj,Pi
and Sk,Pi
from among the four sub-particles of
particle Pi such that the previous fitness of the particle Pi lies
between the fitness values of the two selected sub-particles.
The approximate direction of drift, i.e. the direction in which
the adjustment ξ is required, is calculated by equation (2).
θ(ξ) = θSj,Pi
+ (θSk,Pi
− θSj,Pi
) ×
α−Sj,Pi
Sk,Pi
−Sj,Pi
(2)
In Equation (2), θ(ξ) is the angle representing the direction
in which adjustment ξ has to be made, α is the previous fitness
value of the particle Pi, Sk,Pi
and Sj,Pi
are fitness of the selected
sub-particles between which the value α lies, θSj,Pi
and θSk,Pi
are the angles at which the selected sub-particles are oriented.
If the previous fitness of the particle is greater than the
fitness value of all the sub-particles, then the direction along
the sub-particle with highest value is chosen. And if the
previous fitness of the particle is smaller than the fitness value
of all the sub-particles, then the direction along the sub-
particle with smallest value is chosen. A graphical
representation is shown in Fig. 1.
Figure 1. Graphical representation of drift evaluation using sub-particles
Sj,Pi
and Sj,Pi
of particle Pi. Orbit radius is ρ.
Then, for all stagnant particles, their values of ξ are
averaged with weights and added as an extra term to the
velocity equation as shown in 3(a). The weights are evaluated

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using the occurrence frequency of adjustment values. Weight
𝑤𝑖 for adjustment ξ i is calculated using the equation (3).
𝑤𝑖 =
𝑓ξ 𝑖
𝑛
(3)
Here 𝑓ξ 𝑖
is the number of times the value ξi occurs, and 𝑛 is
the total number of stagnant particles. The significance of the
weight is that values of ξ which are found to occur more
frequently are given more importance.
The assumption here is that the drift rate is near about ρ, i.e,
the sub-particle orbit radius. The value of ρ is chosen in such a
way that any change in the particle‟s vicinity, due to problem
space drift, is contained in close proximity to the sub-particles.
Algorithm 1 Drift Predictive PSO:
1. Initialize a population of particles scattered randomly
over the problem space. These particles have arbitrary
initial velocities.
2. Select randomly 𝑛 number of stagnant particles.
3. For each particle
a. Evaluate fitness of particle.
b. Evaluation of drift: For each stagnant particle,
evaluate probable drift using equation (2)
c. Calculate weights wi corresponding to each ξ.
d. Change velocity according to the equation (4a):
𝑣𝑖𝑑 = 𝑤 × 𝑣𝑖𝑑 + 𝑐1 × 𝑟𝑎𝑛𝑑() × 𝑝𝑖𝑑 − 𝑥𝑖𝑑 + 𝑐2 ×
𝑟𝑎𝑛𝑑() × 𝑝 𝑔𝑑 − 𝑥𝑖𝑑 +
𝜉 𝑖𝑑 𝑤 𝑖𝑑
𝑤 𝑖𝑑
(4a)
Change the position according to equation (4b):
𝑥𝑖𝑑 = 𝑥𝑖𝑑 + 𝑣𝑖𝑑 (4b)
Here w is inertia weight, c1 and c2 are
cognitive and social acceleration constants, and
rand() is a random number generator function. i is
the particle index, g represents index of particle with
best fitness, 𝜉𝑖 is the adjustment required due to the
dynamic change in the problem space and wid is the
accuracy with which the drift is predicted. Subscript
d indicates the dth
dimension.
e. If current fitness of particle is better than pbest,
then set pbest value equal to current fitness. Set
pbest location to current location.
f. If current fitness is better than gbest, reset gbest
to current fitness value. Set gbest location to
current location of particle.
g. Loop back to Step 2 until end criterion is
satisfied, or maximum number of iterations is
completed.
Algorithm 1 illustrates the step-by-step working of the
proposed DriP-PSO for drifting problem-spaces.
IV. SIMULATION AND EXPERIMENTL RESULTS
We designed and implemented a test tool in WPF (.Net
Framework 4.0) for testing and comparing the proposed DriP-
PSO model with the standard PSO. Fig. 2 shows a screenshot
of the PSO test tool. Testing for the proposed PSO models
have been done for five functions, viz. the Sphere, Step,
Rastrigin, Rosenbrock and an arbitrary peak function as shown
in Table 1.
Figure 2. PSO TEST TOOL screenshot showing the arbitrary
peaks function
In order to simulate a dynamic system, we designed the test
tool to drift the problem space in any direction, by applying an
offset, λ, in every dimension, as given by equation (5)
fn+1 = fn(x - λ, y - λ) (5)
TABLE I. FUNCTIONS USED FOR TESTING
Functions Formula
Sphere f(x, y) = x2
+ y2
Step f(x, y) = |x| + |y|
Rastrigin f(x,y) = 20 + x2
+ y2
– 10(cos(2πx) + cos(2πy))
Rosenbrock f(x,y) = (1-x)2
+ 100(y - x2
)2
Arbitrary
Peaks
f(x, y) = 1 – [3(1-x)2
e-x2
– (y+1)2
+ 10(x/5 – x3
– y5
)
e-(x2+y2)
– 1/3e-(x+1)2-y2

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For all functions the offset is varied in the range [0.01,
0.09]. Based on [15], in all experiments, the inertia weight w
was set to 0.729844, and c1 and c2 were set to 1.49618 to
increase convergent behavior.
The stagnant particles are selected randomly at run time.
Table 2 and 3 shows a comparison among standard PSO
and Drift Predictive PSO in dynamic environment. The error
percentage is calculated on the basis of actual minima and the
minima detected by the PSO. All results are averages of 20
different runs.
TABLE II. RESULTS OF DIFFERENT PSOS IN DYNAMIC SCENARIO USING 25
PARTICLES
Percent error in finding global minima
Function Standard PSO Drift Predictive PSO
Sphere 6.799% 2.571%
Step 9.847% 2.091%
Rastrigin 29.900% 9.143%
Rosenbrock 24.616% 3.592%
Arbitrary Peaks 27.629% 5.126%
TABLE III. RESULTS OF DIFFERENT PSOS IN DYNAMIC SCENARIO USING 35
PARTICLES
Percent error in finding global minima
Function Standard PSO Drift Predictive PSO
Sphere 5.021% 1.871%
Step 8.268% 1.438%
Rastrigin 25.728% 7.895%
Rosenbrock 21.616% 2.332%
Arbitrary Peaks 25.744% 4.661%
V. CONCLUSION
The experimental results presented in this paper clearly
show that the proposed Drift Predictive PSO gives accurate
results for drifting problem spaces. It is stable and incurs less
computational cost.
ACKNOWLEDGEMENTS
We are grateful to Tuhin Bhuyan of Jorhat Engineering
College, India, who helped us in designing the class structure
of the PSO Test Tool.
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