PARTICLE SWARM INTELLIGENCE: A PARTICLE SWARM OPTIMIZER WITH ENHANCED GLOBAL SEARCH QUALITIES AND GUARANTEED CONVERGENCE

A4 Model- Based Process Optimisation and Control

PARTICLE SWARM INTELLIGENCE: A PARTICLE SWARM OPTIMIZER
WITH ENHANCED GLOBAL SEARCH QUALITIES AND
GUARANTEED CONVERGENCE

Hendrik Große-Löscher

Postgraduate, Faculty of Computer Science and Automation,
Institute for Automation and Systems Engineering, Ilmenau University of Technology

ABSTRACT continuous nonlinear functions through the simulation
of a simplified social milieu. PSO does not
A new particle swarm optimizer is presented. Particle incorporate the principle ‘survival of the fittest’ like it
swarm optimization PSO is a relatively young is used in genetic algorithms. From the beginning to
optimization algorithm related to evolutionary the end of an optimization run all particles ‘survive’
computation. This metaheuristic is best suited to and actively search the function area for optima. A
optimize nonlinear functions. After a brief particle is a tuple of all parameters and thus contains a
introduction to particle swarm optimization giving possibility for solving a given function. After an initial
insights to the underlying paradigms, advantages and distribution within the search area which is often
drawbacks are highlighted. The operation of the random, the particles move through the search space
canonical PSO in a synthetically generated 2D test with a direction-dependent velocity. After each
environment with three different dynamic landscapes movement the fitness of a particle is evaluated. Fitness
shows the failure of the original version in transient equals the numerical function value where the particle
conditions, leading to a curious phenomenon of ‘linear is located. As the swarm members interact with each
collapse’. Consecutively, an advanced algorithm other based on the information of the best solution of
demonstrates the potential of PSO in dynamic the entire swarm found so far (global best) and the
applications. Based on the experience of two best individual location determined (personal best),
engineering optimization tasks, a refined optimizer cognitive/perception-based and social patterns can be
using constriction coefficient strategy is introduced derived. The particles are attracted by the best
and compared to the classic algorithm and a later positions and move towards the global best. During
introduced version with implemented inertia weight. the optimization process the particles may encounter
The new optimizer for static optimization problems positions with higher fitness values. Depending on the
incorporates superior global search characteristics and chosen PSO variant and the adjusted parameters, the
guarantees final convergence. global search ability, swarm behaviour, convergence
rate and the quality of the final solution are basically
Index Terms – Particle swarm optimization, influenced. The algorithm proved to be very simple,
dynamic environments, linear collapse, constriction robust and highly efficient. Regarding the
coefficient, inertia weight, global search, convergence dimensionality and the complexity of the optimization
task no limitations or restrictions occur.
1. INTRODUCTION The outline of the paper is as follows. Section 2
briefly explains the basic principles of PSO.
Particle swarm optimization - PSO - was introduced in Following the scheme for static optimization tasks, a
1995 by Russell Eberhart and James Kennedy, an special focus is put on dynamic environments with
electrical engineer and a social psychologist. In recent three different kind of optimization problems (height
years particle swarm intelligence has gained a lot of change of peaks, location change of peaks with
recognition as PSO proved to be an effective method constant heights and location change of peaks with
to handle different kind of optimization problems. changing heights). The application of the canonical
Meanwhile, the application areas span from PSO to a synthetically generated dynamic 2D test case
engineering tasks to economics. Related to swarm demonstrates the failure of the algorithm to track
intelligence and evolutionary computation, PSO is a changing peaks, leading to a remarkable appearance
metaheuristic whose paradigms are inspired by bird of ‘linear collapse’. An implementation of an
flocking and fish schooling. The precursor of the advanced PSO technique denotes its feasibility in
canonical PSO algorithm was originally intended as a changing conditions. A distinguished PSO variant is
graphical simulator of the graceful but unpredictable described in Section 3. After the recapitulation of the
choreography of a flock of birds. The originators commonly used constriction coefficient limiting the
discovered the potential of the method to optimize particles’ velocities, a refined algorithm with a simple

186

but effective new strategy is introduced. The xid : potential solution, location of particle i in
implementation leads to a significant improvement of dimension d
the global search ability and enables an enhanced vid : velocity of particle i in dimension d
control of the swarm’s convergence behaviour at the pid : best location so far of particle i in dimension d
end of an optimization run. Consequently, the pgd : best location so far of the best particle g of all
excellence of the final result is influenced and the neighbours of particle i in dimension d
swarm can be adapted to the optimization task. c1 : cognitive parameter/acceleration constant
Experimental investigations are illustrated and c2 : social parameter/acceleration constant
indicate the outstanding performance of the new
rand(), Rand(): equally dispersed random numbers
approach. Section 4 briefly specifies the application of
from [0,1]
PSO to two engineering tasks with 11 and 361
parameters respectively. Finalizing the work, section 5
In equation (1), t is the current iteration step. The
summarizes the results.
movement of the particles following equation (1) is
based on a so called cognitive term (term 2) and a
2. PARTICLE SWARM OPTIMIZATION
social term (term 3) along with the velocity vid(t-1)
(term 1) which is equal to a momentum. The cognitive
Optimization aims to find the minimum or maximum
constant c1 influences the individual particle
of an objective function within a predefined search
behaviour regarding its own best position. The social
area. Beside other mathematical scopes, this work is
constant c2 controls the movement towards the
focused on nonlinear function optimization including
direction of the particle which currently has the best
constraints. The objective function can either be
position in the swarm. c2 influences the behaviour of
single-objective or multi-objective, e.g. a function
particle i with reference to the fitness of the
with standardized weighted objectives where
neighbours and thus describes a component of social
constraints are implemented by penalty terms and the
behaviour. In each iteration, the cognitive and social
problem is converted into a single objective function.
part of the movement is varied by random to maintain
diversity. During their move, particles can find
2.1. Static environments
locations which are characterized by a higher fitness
PSO is a stochastic, gradient-less and derivative-free
than the best optimum found so far. The behaviour of
nature-analogous algorithm that is based on a set of
the swarm is defined by the experience of each
particles. A particle includes a tuple of all parameters
individual swarm member, its current position and the
of the optimization task and thus provides a potential
exchange of information by the individual swarm
solution. After the initial distribution of the particles
members among themselves (orientation of the
within the search space, all particles attain direction-
individuals to group orders). Thereby, PSO
dependent and dimension-individual velocities. The
successfully imitates the natural behaviour of animals
operational sequence for static applications is as
in groups or swarms.
follows [4, 5, 6].
The swarm size, neighbourhood size and topology
affect the swarm behaviour and thus influence the
1. Stochastical initialization of the particle population
search characteristics. Beside several approaches, two
within the function/search area.
main formulations concerning the particles’
2. Calculation of the fitness of each particle.
communication topology exist, namely gbest and
3. Modification of the individual velocity based on
lbest. In the gbest model, each particle is influenced
the best individual and best global position so far
by the best particle of the entire swarm while in the
(neighbourhood).
lbest model each particle is influenced by the particles
4. Determination of the new positions of the particles.
of the local neighbourhood. In many applications, the
5. Fitness evaluation (2.) of each particle, with
gbest model tends to converge faster.
convergence/termination criterion: END, otherwise
Another mentionable topic is the information
go to 3.
analysis of the best position. In a so called
The referring equation for the velocity update is synchronous PSO which the original version is related
to, the best positions are updated after all particle
movements in one iteration step. The asynchronous
vid (t ) = vid (t − 1) + c1 ⋅ rand ( ) ⋅ ( pid − xid (t − 1) ) + ...
PSO updates the best positions after each particle
movement which allows an immediate feedback about
... + c 2 ⋅ Rand ( ) ⋅ ( p gd − xid (t − 1) ) (1)
the best regions and leads to a higher convergence
rate. Based on application experiences with the
The new positions are updated by canonical PSO, numerous optimization runs with
small populations are more effective in finding an
xid (t ) = xid (t − 1) + vid (t ) (2) admissible solution than few runs with large
populations. This surprising effect is based on the fast
Equations (1) and (2) employ the denotations convergence of the PSO.

187

2.2. Dynamic environments
Based on the belief that PSO converges fast, the
dynamic tracking of peaks seemed to be possible [10].
Today, with deeper insights into PSO and many
applications at hand, the conventional PSO is
inapplicable to dynamic environments. Due to its
immanent paradigms, PSO fails to most dynamic
environments.
A synthetical 2D test environment mainly consisting
of trigonometrical expressions is generated to provide
three different dynamic test cases:
x1
y (x1 , x2 ) = [[(− x12 − x2 ) ⋅ sin(
2
) ⋅ 2 ⋅ x1 ⋅ x2 ⋅ ... (3)
4
... ⋅ csc h( x1 ) ⋅ csc h( x2 ) + sin(3 ⋅ x1 ) + sin(3 ⋅ x2 )] ⋅ ...
... ⋅ sin(3 ⋅ x2 ) + sin(3 ⋅ x1 )] ⋅ cos(x1 ) ⋅ sin(x2 ) ⋅ tan(x1 )
with x1 , x2 ∈ [−5.5;5.5]
This unsimplified equation provides the following
search area.

Figure 1: 2D test environment

To develop a proper test suite, three different dynamic
tasks are included:
- height change of peaks (α)
- location change of peaks with constant heights (β)
- location change of peaks with changing heights (δ)

During all calculations, 155 iterations are performed
in one way and another155 iterations in the other way
so that the search area moves forward and backward.
A sidewise slide with constant heights (β) equals
∆x1=0.05 while ∆x1=0.02 when the peaks also
encounter a change in heights (δ). ∆x2 is constantly
kept to zero. Two additional terms in equation (3)
ensure the increasing and decreasing heights of the
peaks.
The application of the conventional algorithm to the
three different dynamic tasks indicate the failure of the
PSO technique to identify and trace single peaks as
well as to find and trace the global optimum. The
main deficit of the classic PSO version in dynamic
environments is the outdated memory of the particles
when a change in the function occurs additionally to a
fatal loss of diversity. Attuned with standard
parameter settings, the PSO rapidly converges to a
Figure 2: flow diagram of PSO for dynamic tasks

188

single peak and only performs a weak local search the species’ seed. These particles are initialised when
around the peak (test case α and δ). The parameters the species is converged to achieve a balanced ratio of
are c1=c2=2.05, as commonly applied [4, 6], vmax (the convergence and diversity. The subsequent results
maximum move of a particle in one dimension) is set indicate the performance of the quoted algorithm.
to 0.5 and if a particle tends to leave the search space,
it is placed to the limit value of the dimension
restricting the search area. Soon after starting test
instance β, all particles are collapsing to a single line,
showing a curious phenomenon called ‘linear
collapse’ [2]. The length of the line is almost constant
for approximately two thirds of the iterations and
amounts to ∆x2=1.44.

Figure 5: PSO applied to test case α

Figure 3: ‘linear collapse’ as seen from above

Figure 6: PSO in dynamic test case β

Figure 4: ‘linear collapse’ laterally seen

As the phenomenon appears, all velocities in direction Figure 7: PSO in dynamic environment δ
of x1 permanently receive the maximum velocity
-vmax,1=-0.5, while the velocities in direction of x2 are All procedures employ 200 particles. The radius of a
alternating between ±vmax,2=±0.5. Animations and the species is set to rspecies=4 around a seed, the minimum
analysis of the velocities indicate the establishment of amount of particles per species is 15 while 35
alternating linear particle trajectories. While particles are the maximum quantity. To initialize the
v1=const., all particles are moving between the best neutral and quantum particles around a species’ seed,
particles/attractors which are mostly at the both ends a radius of rcloud=0.2 is applied. The convergence
of the path. threshold determining that a species is converged is
The utilisation of PSO for dynamic environments still set to ∆=0.0001. ∆ correlates to a mean distance
is a young field of activity. Figure 2 shows the flow between a species and the species’ seed. Neutral
diagram of a proposed algorithm for dynamic tasks particles provide convergence while the randomly
which is mainly based on [3]. Without going into initialized quantum particles within rcloud ensure
detail, the foremost principles are the permanent diversity of the species. The current PSO version is
update of the best positions, the identification of capable of detecting and tracing all major dynamic
species seeds and their members with a predefined peaks. The algorithm and its success seriously depend
amount of minimum and maximum particles and the on the problem-specific parameter settings. Future
initialisation of neutral and quantum particles around work is encouraged to generalize this tuning.

189

3. PSO WITH CONSTRICTION COEFFICIENT followed by an intensive local search (exploitation)
with controlled convergence, suitable for many tasks.
Numerous works are dedicated to improve PSO and
its premature convergence. Most versions study the 3.3. Experimental results
swarm behaviour by employing different inertia The sequences below demonstrate the various swarm
weight (adding a factor to term 1 in equation (1)) and behaviours of different PSO versions in the static 2D
constriction coefficient approaches. The aim is to test case of equation (3).
control both, exploration and exploitation, global and
local search.

3.1. Common constriction coefficient strategy
Mathematically, the implementation of a constriction
coefficient is a special case of the inertia weight
version. The standard constriction coefficient
algorithm [1, 4, 6, 10] is an extension of equation (1).

vid (t ) = χ ⋅ [vid (t − 1) + c1 ⋅ rand () ⋅ ( pid − xid (t − 1) ) + ...

... + c2 ⋅ Rand () ⋅ ( pgd − xid (t − 1) )] (4)

According to [4], χ is defined as

2
χ= with ϕ=c1+c2, ϕ>4 (5)
2 − ϕ − ϕ 2 − 4 ⋅ϕ

Commonly, c1=c2=2.05 resulting in χ=0.72984.

3.2. Refined constriction coefficient strategy
Based on application experiences [7, 8, 9], the refined
utilisation of χ can significantly improve the swarm Figure 9: positions visited by conventional PSO
behaviour.

Figure 8: exemplary trend of χ

The simple but effective procedure consists of
uniformly distributed random numbers within the first
75% of the iterations in a proposed range of at least
[0.1;1.3]. Between 75% and 95% of the iterations, χ
decreases linearly from 1 to 0.20871 which results of
equation (5) with c1=2 and c2=5. Within the last 5% of Figure 10: positions visited by PSO with constriction
the iterations, χ is kept constant at 0.20871. An
innovative implementation of a random χ>1 leads to a The figures show all visited positions of the PSO with
significant improvement of the global search ability 75 particles on their way of 200 iterations at 1 (1), 10
(exploration). The idea is to provide an exhaustive (2), 30 (3), 50 (4), 100 (5) and 200 iterations (6), seen
global search that maintains diversity which is from above. The conventional PSO and the version

190

with inertia weight (not presented) have a quite similar (nozzle) as well as up to 361 parameters, 7
performance and show premature convergence with a constraints, 86 equations and 6 objective criterions
weak global search. The conventional constriction (cam) [7, 8, 9].
version keeps χ constant and incorporates an early
convergence after one third of the iterations with no 5. CONCLUSIONS
further global search. The refined constriction
algorithm has a superior global search capability A new PSO algorithm with a refined constriction
followed by a controlled local search. The coefficient strategy is presented. The algorithm
conventional version as well as the algorithm with a provides superior global search quality and guarantees
linear decreasing inertia weight from 0.9 to 0.4 [6, 10] convergence to balance exploration and exploitation.
utilize c1=c2=2.05, vmax=0.5 and set the particles to the
border line when they tend to leave the search area. 6. REFERENCES
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[9] Große-Löscher, H.: Application of PSO for the
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