A Chinese version is also available.
Particle swarm optimization (PSO) is a population based stochastic optimization technique
developed by Dr. Eberhart and
Dr. Kennedy in 1995,
inspired by social behavior of bird flocking or fish schooling.
PSO shares many similarities with evolutionary computation techniques such as
Genetic Algorithms (GA).
The system is initialized with a population of random solutions and searches for
optima by updating generations. However, unlike GA, PSO has no evolution
operators such as crossover and mutation. In PSO, the potential solutions,
called particles, fly through the problem space by following the current
optimum particles. The detailed information will be given in following sections.
Compared to GA, the advantages of PSO are that PSO is easy to implement and
there are few parameters to adjust. PSO has been successfully applied in many
areas: function optimization, artificial neural network training, fuzzy system
control, and other areas where GA can be applied.
The remaining of the report includes six sections:
Background: artificial life.
Comparisons between Genetic algorithm and PSO
Artificial neural network and PSO
PSO parameter control
Online resources of PSO
2. Background: Artificial life
The term "Artificial Life" (ALife) is used to describe research into human-made
systems that possess some of the essential properties of life. ALife includes
two-folded research topic: (http://www.alife.org)
1. ALife studies how computational techniques can help when studying biological
2. ALife studies how biological techniques can help out with computational
The focus of this report is on the second topic. Actually, there are already
lots of computational techniques inspired by biological systems. For example,
artificial neural network is a simplified model of human brain; genetic
algorithm is inspired by the human evolution.
Here we discuss another type of biological system - social system, more
specifically, the collective behaviors of simple individuals interacting with
their environment and each other. Someone called it as swarm intelligence. All
of the simulations utilized local processes, such as those modeled by cellular
automata, and might underlie the unpredictable group dynamics of social
Some popular examples are floys and
boids. Both of the simulations
were created to interpret the movement of organisms in a bird flock or fish
school. These simulations are normally used in computer animation or computer
There are two popular swarm inspired methods in computational intelligence
areas: Ant colony optimization (ACO) and particle swarm optimization (PSO). ACO
was inspired by the behaviors of ants and has many successful applications in
discrete optimization problems. (http://iridia.ulb.ac.be/~mdorigo/ACO/ACO.html)
The particle swarm concept originated as a simulation of simplified social
system. The original intent was to graphically simulate the choreography of bird
of a bird block or fish school. However, it was found that particle swarm model
can be used as an optimizer. (http://www.engr.iupui.edu/~shi/Coference/psopap4.html)
3. The algorithm
As stated before, PSO simulates the behaviors of bird flocking. Suppose the
following scenario: a group of birds are randomly searching food in an area.
There is only one piece of food in the area being searched. All the birds do not
know where the food is. But they know how far the food is in each iteration. So
what's the best strategy to find the food? The effective one is to follow the
bird which is nearest to the food.
PSO learned from the scenario and used it to solve the optimization problems. In
PSO, each single solution is a "bird" in the search space. We call it
"particle". All of particles have fitness values which are evaluated by the
fitness function to be optimized, and have velocities which direct the flying of
the particles. The particles fly through the problem space by following
the current optimum particles.
PSO is initialized with a group of random particles (solutions) and then
searches for optima by updating generations. In every iteration, each particle
is updated by following two "best" values. The first one is the best solution
(fitness) it has achieved so far. (The fitness value is also stored.) This value
is called pbest. Another "best" value that is tracked by the particle swarm
optimizer is the best value, obtained so far by any particle in the population.
This best value is a global best and called gbest. When a particle takes part of
the population as its topological neighbors, the best value is a local best and
is called lbest.
After finding the two best values, the particle updates its velocity and
positions with following equation (a) and (b).
v = v + c1 * rand() * (pbest - present) + c2 * rand() * (gbest -
present = persent + v (b)
v is the particle velocity, persent is the current particle (solution).
pbest and gbest are defined as stated before. rand () is a random number
between (0,1). c1, c2 are learning factors. usually c1 = c2 = 2.
The pseudo code of the procedure is as follows
For each particle
For each particle
Calculate fitness value
If the fitness value is better than
the best fitness value (pBest) in history
value as the new pBest
Choose the particle with the best fitness value of all the
particles as the gBest
For each particle
Calculate particle velocity according
Update particle position according
While maximum iterations or minimum error criteria is not attained
Particles' velocities on each dimension are clamped to a maximum velocity Vmax.
If the sum of accelerations would cause the velocity on that dimension to exceed
Vmax, which is a parameter specified by the user. Then the velocity on that
dimension is limited to Vmax.
4. Comparisons between Genetic Algorithm and PSO
Most of evolutionary techniques have the following procedure:
1. Random generation of an initial population
2. Reckoning of a fitness value for each subject. It will directly depend on the
distance to the optimum.
3. Reproduction of the population based on fitness values.
4. If requirements are met, then stop. Otherwise go back to 2.
From the procedure, we can learn that PSO shares many common points with GA.
Both algorithms start with a group of a randomly generated population, both have
fitness values to evaluate the population. Both update the population and search
for the optimium with random techniques. Both systems do not guarantee success.
However, PSO does not have genetic operators like crossover and mutation.
Particles update themselves with the internal velocity. They also have memory,
which is important to the algorithm.
Compared with genetic algorithms (GAs), the information sharing mechanism in PSO
is significantly different. In GAs, chromosomes share information with each
other. So the whole population moves like a one group towards an optimal area.
In PSO, only gBest (or lBest) gives out the information to others. It is a one
-way information sharing mechanism. The evolution only looks for the best
solution. Compared with GA, all the particles tend to converge to the best
solution quickly even in the local version in most cases.
5. Artificial neural network and PSO
An artificial neural network (ANN) is an analysis paradigm that is a simple
model of the brain and the back-propagation algorithm is the one of the most
popular method to train the artificial neural network. Recently there have been
significant research efforts to apply evolutionary computation (EC) techniques
for the purposes of evolving one or more aspects of artificial neural networks.
Evolutionary computation methodologies have been applied to three main
attributes of neural networks: network connection weights, network architecture
(network topology, transfer function), and network learning algorithms.
Most of the work involving the evolution of ANN has focused on the network
weights and topological structure. Usually the weights and/or topological
structure are encoded as a chromosome in GA. The selection of fitness function
depends on the research goals. For a classification problem, the rate of mis-classified
patterns can be viewed as the fitness value.
The advantage of the EC is that EC can be used in cases with non-differentiable
PE transfer functions and no gradient information available. The disadvantages
are 1. The performance is not competitive in some problems. 2. representation of
the weights is difficult and the genetic operators have to be carefully selected
There are several papers reported using PSO to replace the back-propagation
learning algorithm in ANN in the past several years. It showed PSO is a
promising method to train ANN. It is faster and gets better results in most
cases. It also avoids some of the problems GA met.
Here we show a simple example of evolving ANN with PSO. The problem is a
benchmark function of classification problem: iris data set. Measurements of
four attributes of iris flowers are provided in each data set record: sepal
length, sepal width, petal length, and petal width. Fifty sets of measurements
are present for each of three varieties of iris flowers, for a total of 150
records, or patterns.
A 3-layer ANN is used to do the classification. There are 4 inputs and 3
outputs. So the input layer has 4 neurons and the output layer has 3 neurons.
One can evolve the number of hidden neurons. However, for demonstration only,
here we suppose the hidden layer has 6 neurons. We can evolve other parameters
in the feed-forward network. Here we only evolve the network weights. So the
particle will be a group of weights, there are 4*6+6*3 = 42 weights, so the
particle consists of 42 real numbers. The range of weights can be set to [-100,
100] (this is just a example, in real cases, one might try different ranges).
After encoding the particles, we need to determine the fitness function. For the
classification problem, we feed all the patterns to the network whose weights is
determined by the particle, get the outputs and compare it the standard outputs.
Then we record the number of misclassified patterns as the fitness value of that
particle. Now we can apply PSO to train the ANN to get lower number of
misclassified patterns as possible. There are not many parameters in PSO need to
be adjusted. We only need to adjust the number of hidden layers and the range of
the weights to get better results in different trials.
6. PSO parameter control
From the above case, we can learn that there are two key steps when applying PSO
to optimization problems: the representation of the solution and the fitness
function. One of the advantages of PSO is that PSO take real numbers as
particles. It is not like GA, which needs to change to binary encoding, or
special genetic operators have to be used. For example, we try to find the
solution for f(x) = x1^2 + x2^2+x3^2, the particle can be set as (x1, x2, x3),
and fitness function is f(x). Then we can use the standard procedure to find the
optimum. The searching is a repeat process, and the stop criteria are that the
maximum iteration number is reached or the minimum error condition is satisfied.
There are not many parameter need to be tuned in PSO. Here is a list of the
parameters and their typical values.
The number of particles: the typical range is 20 - 40. Actually for most of the
problems 10 particles is large enough to get good results. For some difficult or
special problems, one can try 100 or 200 particles as well.
Dimension of particles: It is determined by the problem to be optimized,
Range of particles: It is also determined by the problem to be optimized, you
can specify different ranges for different dimension of particles.
Vmax: it determines the maximum change one particle can take during one
iteration. Usually we set the range of the particle as the Vmax for example, the
particle (x1, x2, x3)
X1 belongs [-10, 10], then Vmax = 20
Learning factors: c1 and c2 usually equal to 2. However, other settings were
also used in different papers. But usually c1 equals to c2 and ranges from [0,
The stop condition: the maximum number of iterations the PSO execute and the
minimum error requirement. for example, for ANN training in previous section, we
can set the minimum error requirement is one mis-classified pattern. the maximum
number of iterations is set to 2000. this stop condition depends on the problem
to be optimized.
Global version vs. local version: we introduced two versions of PSO. global and
local version. global version is faster but might converge to local optimum for
some problems. local version is a little bit slower but not easy to be trapped
into local optimim. One can use global version to get quick result and use local
version to refine the search.
Another factor is inertia weight, which is introduced by Shi and Eberhart. If
you are interested in it, please refer to their paper in 1998. (Title: A
modified particle swarm optimizer)
7. Online Resources of PSO
The development of PSO is still ongoing. And there are still many unknown areas
in PSO research such as the mathematical validation of particle swarm theory.
One can find much information from the internet. Following are some information
you can get online:
http://www.particleswarm.net lots of
information about Particle Swarms and, particularly, Particle Swarm
Optimization. Lots of Particle Swarm Links.
http://icdweb.cc.purdue.edu/~hux/PSO.shtml lists an updated bibliography of
particle swarm optimization and some online paper links
can search particle swarm related papers and references.
Kennedy, J. and Eberhart, R. C. Particle swarm optimization. Proc. IEEE int'l
conf. on neural networks Vol. IV, pp. 1942-1948. IEEE service center,
Piscataway, NJ, 1995.
Eberhart, R. C. and Kennedy, J. A new optimizer using particle swarm theory.
Proceedings of the sixth international symposium on micro machine and human
science pp. 39-43. IEEE service center, Piscataway, NJ, Nagoya, Japan, 1995.
Eberhart, R. C. and Shi, Y. Particle swarm optimization: developments,
applications and resources. Proc. congress on evolutionary computation 2001 IEEE
service center, Piscataway, NJ., Seoul, Korea., 2001.
Eberhart, R. C. and Shi, Y. Evolving artificial neural networks. Proc. 1998
Int'l Conf. on neural networks and brain pp. PL5-PL13. Beijing, P. R. China,
Eberhart, R. C. and Shi, Y. Comparison between genetic algorithms and
particle swarm optimization. Evolutionary programming vii: proc. 7th ann. conf.
on evolutionary conf., Springer-Verlag, Berlin, San Diego, CA., 1998.
Shi, Y. and Eberhart, R. C. Parameter selection in particle swarm
optimization. Evolutionary Programming VII: Proc. EP 98 pp. 591-600. Springer-Verlag,
New York, 1998.
Shi, Y. and Eberhart, R. C. A modified particle swarm optimizer. Proceedings
of the IEEE International Conference on Evolutionary Computation pp. 69-73. IEEE
Press, Piscataway, NJ, 1998