Particle Swarm Optimization (PSO) 3 Particle Swarm Optimization (PSO) 3.1 Introduction Swarm Intelligence is a kind of Artificial Intelligence based on the behavior of the animals living in groups and having some ability to interact with one other and with the environment in which they are inserted [1]. Particle swarm optimization, abbreviated as PSO, is based on the behavior of a colony or swarm of insects, such as ants, termites, bees, and wasps; a flock of birds; or a school of fish . The particle
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Particle Swarm Optimization (PSO)
3
Particle Swarm Optimization (PSO)
3.1 Introduction
Swarm Intelligence is a kind of Artificial Intelligence based on the behavior
of the animals living in groups and having some ability to interact with one
other and with the environment in which they are inserted [1].
Particle swarm optimization, abbreviated as PSO, is based on the behavior
of a colony or swarm of insects, such as ants, termites, bees, and wasps; a flock
of birds; or a school of fish . The particle swarm optimization algorithm mimics the behavior of these social organisms[2]. The word particle denotes, for example, a bee in a colony or a bird in a flock. Each individual or particle in a swarm behaves in a distributed way using its own intelligence and the collective or group intelligence of the swarm. As such, if one particle discovers a good path to food, the rest of the swarm
Particle Swarm Optimization (PSO)
will also be able to follow the good path instantly even if their location is far away in the swarm [2][3][4]. The PSO algorithm was originally proposed by James Kennedy and Russell Eberhart in 1995 [3] . Each particle is assumed to have two characteristics: a position and a
velocity. Each particle wanders around in the design space and remembers the
best position (in terms of the food source or objective function value) it has
discovered. The particles communicate information or good positions to each
other and adjust their individual positions and velocities based on the
information received on the good positions [2][4] .
As an example, consider the behavior of birds in a flock. Although each bird
has a limited intelligence by itself, it follows the following simple rules:
1. It tries not to come too close to other birds.
2. It steers toward the average direction of other birds.
3. It tries to fit the “average position” between other birds with no wide gaps in
the flock.
Thus the behavior of the flock or swarm is based on a combination of three
simple factors:
1. Cohesion—stick together.
2. Separation—don’t come too close.
3. Alignment—follow the general heading of the flock.
Cohesion, Separation and Alignment are explained in figure (3.1).
The PSO is developed based on the following model:
1. When on one bird locates a target or food (or minimum of objective function) ,
it instantaneously transmits the information to all other birds. 2. All other birds gravitate to the target or food (or minimum of the objective
function), but not directly.
Particle Swarm Optimization (PSO)
3. There is a component of each bird’s own independent thinking as well as its
past memory. Thus the model simulates a random search in the design space for the minimum value of the objective function. As such, gradually over many iterations, the birds go to the target (or minimum of the objective function) [2][3][4].
3.2 Basic elements of PSO technique The basic elements of PSO technique are briefly stated and defined as
follows:
·Particle, X i : It is a candidate solution represented by an m-dimensional vector,
where m is the number of optimized parameters. At time i, the jth particle X ji can be
described as X ji = [x j , 1
i , ..., x j , mi ], where xs are the optimized parameters and x j , k
i is the
position of the jth particle with respect to the kth dimension, i.e. the value of the kth
optimized parameter in the jth candidate solution.
·Population, pop(t),: It is a set of n particles at time i ,i.e. pop(i) = [X1i , ..., X n
i ]T.
Particle Swarm Optimization (PSO)
The number of particles in population between (20 to 30) .
·Swarm: It is an apparently disorganized population of moving particles that tend
to cluster together while each particle seems to be moving in a random
direction[5].
·Particle velocity, V i: It is the velocity of the moving particles represented by an m-
dimensional vector. At time i , the jth particle velocity V ji can be described as
V ji = [ V j ,1
i , ..., V j ,mi ], where V j ,k
i is the velocity component of the jth particle with
respect to the kth dimension.
· Inertia weight, w i : It is a control parameter that is used to control the impact of the
previous velocities on the current velocity. Hence, it influences the trade-off
between the global and local exploration abilities of the particles . For initial stages
of the search process, large inertia weight to enhance the global exploration is
recommended while, for last stages, the inertia weight is reduced for better local
exploration.
•Individual best, pi : As a particle moves through the search space, it compares its
fitness value at the current position to the best fitness value it has ever attained at any time
up to the current time. The best position that is associated with the best fitness
encountered so far is called the individual best, p ji . For each particle in the swarm, p j
i
can be determined and updated during the search. In a minimization problem with
objective function f, the individual best of the jth particle p ji is determined such that
f(p ji ) ≤ f(X e
i ) , e ≤ j . For simplicity, assume that f* = f(p ji ). For the jth particle,
individual best can be expressed as
p ji = [ p j ,1
i , ..., p j ,mi ].
· Global best, p jg : It is the best position among all individual best positions achieved
so far. Hence, the global best can be determined such that f (p j
g) ≤ f (p ji ), j=1……,n . for simplicity, assume that , f** = f )p j
g. (
·Stopping criteria: these are the conditions under which the search process will
terminate. The search will terminate if one of the following criteria is satisfied : (a)
Particle Swarm Optimization (PSO)
the number of iterations since the last change of the best solution is greater than a pre-
specified number or (b) the number of iterations reaches the maximum allowable
number [2][5].
3.3 PSO Methods There are several methods of PSO dependent on the shape of updated
velocity equation of the particle :
3.3.1 Original PSO Basic algorithm as proposed by Kennedy and Eberhart (1995)
xki : Particle position
vki : Particle velocity
pki : Best position found by jth particle (personal best)
pkg : Best position found by swarm (global best, best of personal bests)
c1,c2 : Cognitive and social parameters
r1,r2 :Random numbers between 0 and 1
Position of individual particles updated as follows:
xk+1i =xk
i + vk+1i k=1,…,n (3-1)
with the velocity calculated as follows:
vk+1i = vk
i + c1 r1 (pki - xk
i ) +c2r2 (pkg - xk
i ) k=1,…,n (3-2)
where c1 and c2 are the cognitive (individual) and social (group) learning
rates, respectively, and r1 and r2 are uniformly distributed random numbers in
the range 0 and 1. The parameters c1 and c2 denote the relative importance of
the memory (position) of the particle itself to the memory (position) of the
swarm. The values of c1 and c2 are usually assumed to be 2 [2][3][4][5][9].
Particle Swarm Optimization (PSO)
3.3.2 Inertia Weighted PSO
It is found that usually the particle velocities build up too fast and the
maximum of the objective function is skipped. Hence an inertia term, θ, is
added to reduce the velocity. Usually, the value of θ is assumed to vary linearly
from 0.9 to 0.4 as the iterative process progresses. The velocity of the jth
particle, with the inertia term, is assumed as :
vk+1i = w i vk
i + c1 r1 (pki - xk
i ) +c2r2 (pkg - xk
i ) k=1,…,n (3-3)
The inertia weight θ was originally introduced by Shi and Eberhart in 1998 [2]
[5] to dampen the velocities over time (or iterations), enabling the swarm to
converge more accurately and efficiently compared to the original PSO
algorithm with Eq. (3-2).
Equation (3.3) denotes an adapting velocity formulation, which improves its
fine tuning ability in solution search. Equation (3.3) shows that a larger value of
θ promotes global exploration and a smaller value promotes a local search. Thus
a large value of θ makes the algorithm constantly explore new areas without
much local search and hence fails to find the true optimum.
To achieve a balance between global and local exploration to speed up
convergence to the true optimum, an inertia weight whose value decreases
linearly with the iteration number has been used:
w i = wmax - ( wmax−wmin
imax ) * i (3-4)
y xk+1
vk
vk+1 vgbest
Figure (3.2) xk vpbest
Particle Swarm Optimization (PSO)
Concept of modification of a search point by PSO x
vgbest : velocity based on gbest vpbest : velocity based on pbest
where wmax and wmin are the initial and final values of the inertia weight,
respectively, and imax is the maximum number of iterations used in PSO. The
values of wmax = 0.9 and wmin = 0.4 are commonly used [2][5][9].
Figure (3.3)
Searching concept with agents in a solution space by PSO.
Particle Swarm Optimization (PSO)
Figure (3.4) The linearly decreasing of inertia weight (w) with the number of
iteration
3.3.3 Constriction factor approach
Though the earliest researchers recognized that some form of damping of
the dynamics of a particles (e.g., Vmax) was necessary, the reason for this was
not understood. But when the particle swarm algorithm is run without
restraining velocities in some way, these rapidly increase to unacceptable levels
within a few iterations. Kennedy (1998) noted that the trajectories of non
stochastic one-dimensional particles contained interesting regularities when
c1+c2 was between 0.0 and 4.0. Clerc’s analysis of the iterative system led him
to propose a strategy for the placement of “constriction coefficients” on the
terms of the formulas; these coefficients controlled the convergence of the
particle and allowed an elegant and well-explained method for preventing
explosion ,ensuring convergence, and eliminating the arbitrary Vmax parameter.
The analysis also takes the guesswork out of setting the values of c1 and c2.
Clerc and Kennedy (2002) noted that there can be many ways to
implement the constriction coefficient. One of the simplest methods of
incorporating it is the following:
(3-5)
Where φ = c1 + c2 , φ ≥ 4
The velocity of the jth particle is assumed as:
vk+1i = k ( vk
i + c1 r1 (pki - xk
i ) +c2r2 (pkg - xk
i )) (3-6)
The position of the jth particle assumed as :
xk+1i = xk
i + vk+1i k= 1,…….,n (3-7)
Particle Swarm Optimization (PSO)
When Clerc’s constriction method is used, φ is commonly set to 4.1, c1 = c2,
and the constant multiplier k is approximately 0.7298. This results in the
previous velocity being multiplied by 0.7298 and each of the two (pk-xk)terms
being multiplied by a random number limited by 0.7298×2.05 ≈ 1.49618.[9]
Note that a PSO with constriction is algebraically equivalent to a PSO with
inertia. Indeed, eq.(3-3) and eq.(3-6) can be transformed into one another via
the mapping ω ↔ k and ci ↔ k.ci . So, the optimal settings suggested by Clerc
correspond to ω = 0.7298 and c1 = c2 = 1.49618 for a PSO with inertia weight.
[6][7][8].
3.3.4 Fully informed particle swarm (FIPS) In the standard version of PSO, the effective sources of influence are in
fact only two: self and best neighbor. Information from the remaining neighbors
is unused. Mendes has revised the way particles interact with their neighbors
[10]. Whereas in the traditional algorithm each particle is affected by its own
previous performance and the single best success found in its neighborhood, in
Mendes’ fully informed particle swarm (FIPS), the particle is affected by all its
neighbors, sometimes with no influence from its own previous success. FIPS
can be depicted as follows:
vk+1i = k { vk
i + 1Ki ∑n=1
Ki
c1 r1.( pnbrn−xk
i )} (3-8)
xk+1i = xk
i + vk+1i (3-9)
where Ki is the number of neighbors for particle i, and nbrn is i’s nth
neighbor. It can be seen that this is the same as the traditional particle swarm if
only the self and neighborhood best in a Ki = 2 model are considered. With
good parameters, FIPS appears to find better solutions in fewer iterations than
the canonical algorithm, but it is much more dependent on the population
topology [9][10].
Particle Swarm Optimization (PSO)
3.4 Computational implementation of PSO The computational flow of PSO technique can be described in the following
steps:
Step 1 ( Number of Particles):
Assume the size of the swarm (number of particles) is N. To reduce the
total number of function evaluations needed to find a solution, we must assume
a smaller size of the swarm. But with too small a swarm size it is likely to take
us longer to find a solution or, in some cases, we may not be able to find a
solution at all. Usually a size of 20 to 30 particles is assumed for the swarm as
a compromise.
Step 2 (Initial position):
Generate the initial population of X in the range X (l) and X (u) randomly asX1,X2, . . . ,X N. Hereafter, for convenience, the particle (position of) j and
its velocity in iteration i are denoted as X ji and V j
i , respectively. Thus the
particles generated initially are denoted X1(0),X2
(0) , . . . ,X N(0). The vectors
X j(0))j = 1, 2, . . . ,N (are called particles or vectors of coordinates of
particles(similar to chromosomes in genetic algorithms) .
Step 3 (Initial velocity):
Find the velocities of particles. All particles will be moving to the optimal
point with a velocity. Initially, all particle velocities are assumed to be zero.
Step 4
Each particle in the initial population is evaluated using the objective function, f. For
each particle, set p j(0 )
= X j(0)
and f j
¿
= f j , j = 1,...,n. Search for the best value of the
Particle Swarm Optimization (PSO)
objective function f best Set the particle associated with f best as the global best, p jg , with
an objective function of f¿∗¿ ¿
. Set the initial value of the inertia weight w(0).
Step 5
Update the iteration number as i = i + 1.
Step 6 (Weight updating):
Update the inertia weight
w(i) = wmax - ( wmax−wmin
imax ) * i
Step 7 (velocity updating):
Using the global best and individual best of each particle, the jth particle velocity
in the kth dimension is updated according to the following equation :
v j+1i = w i v j
i + c1 r1 (p ji - x j
i) +c2r2 (p jg - x j
i)
where c1 and c2 are positive constants and r1 and r2 are uniformly distributed
random numbers in [0,1]. It is worth mentioning that the second term represents the
cognitive part of PSO where the particle changes its velocity based on its own thinking
and memory. The third term represents the social part of PSO where the particle changes
its velocity based on the social-psychological adaptation of knowledge. If a particle
violates the velocity limits, set its velocity equal to the limit.
Step 8 (Position updating):
Based on the updated velocities, each particle changes its position according to the
following equation:
x j+1i =x j
i + v j+1i
If a particle violates the its position limits in any dimension, set its position at the
proper limit.
Step 9 (individual best updating):
Particle Swarm Optimization (PSO)
Each particle is evaluated according to its updated position.
If , f j< f j¿ j=1,…….,N
then update individual best as:
p ji = X j
i and
f j¿ = f j
and go to step 10; else go to step 10.
Step 10 (Global best updating):
Search for the minimum value f min among f j¿ , where min is the index of the
particle with minimum objective function.
If f min < f best
then update global best
p jg = X min
(i) and
f best = f min
and go to step 11; else go to step 11.
Step 11 (Stopping criteria)
If one of the stopping criteria is satisfied then stop; else go to step 5.
Particle Swarm Optimization (PSO)
3.5 Flowchart of PSO algorithm
unsatisfied
Satisfied
Figure (3.5)
Flowchart of PSO
START
Create an initial swarm
Evaluate the fitness of each particle
Check and update personal best and global best
Update each individual velocity and position
Check stopping criteria
Success
END
Set new iteration
i= i+1
Particle Swarm Optimization (PSO)
3.6 The Categories of Optimization Techniques
Techniques used to solve the optimization problems (let's can be placed into
two categories : Local and Global optimization algorithm .
3.6.1 Local Optimization
A local minimizer ,X B¿ , of the region B , is defined so that
(3-10)
Where ,and S denotes the search space . More importantly, note that B
is a proper subset os S. A given search space S can contain multiple regions Bi
such that Bi∩B j =Ø when i ≠ j . It then follows that X Bi
¿ ≠ X B j
¿ , so that the
minimizer of each region Bi is unique . Any of the X Bi
¿ can be considered a
minimizer of B , although they are merely local minimizers. There is no
restriction on the value that the function can assume in minimizer, so that
f (X Bi
¿ ) = f (X B j
¿ ) is allowed . the value f (X Bi
¿ ) will be called the local minimum.
Most optimization algorithms require a starting point z0S . A local
optimization algorithm should guarantee that it will be able to find the
minimizer X Bi
¿ of the set B if z0B . Some algorithms satisfy a slightly weaker
constraint, namely that they guarantee to find a minimizer X Bi
¿ of some set Bi ,
not necessarily the one closest to z0 .[11]
3.6.2 Global Optimization
The global minimize ,X ¿ , is defined so that
(3-11)
Where S is the search space .The term global minimum will refer to the value
f (X ¿), and X ¿ will be called the global minimizer .
Particle Swarm Optimization (PSO)
A global optimization algorithm , like the local optimization algorithm
described above, also starts by choosing an initial starting position z0 S .
Contrary to the definition above in eq. (3.10), some texts define a global
optimization algorithm differently, namely an algorithm that is able to find a
(local) minimizer of B S, regardless of the actual position of z0. These
algorithms consist of two processes : "global" steps and "local" steps. Their local
steps are usually the application of a local minimization algorithm, and their
"global" steps are designed to ensure that the algorithm will move into a region Bi , from where the "local" step will be able to find the minimizer of Bi .
These methods will be referred to as globally convergent algorithms,
meaning that they are able to converge to a local minimizer regardless of their
starting position z0 . These methods are also capable of finding the global
minimizer, given that the starting position z0 is chosen correctly. There is no
known reliable, general way of doing this.
Figure (3.6) illustrates the difference between the local minimizer X B¿ and the
global minimizer X ¿ . A true global optimization algorithm will find X ¿
regardless of the choice of starting position z0 . [11][12]
Particle Swarm Optimization (PSO)
Figure (3- 6)
3.7 PSO Updates
Particle Swarm Optimization (PSO)
There are two types of particle swarm optimization updates:
1- Synchronous update2- Asynchronous update
3.7.1 Synchronous update
A synchronous method is obtained if all birds stop after flying in the same
time interval, and communicate each other to get the current global best
position. Once this communication is completed, the birds can update their
directions and speeds based on their own history and the globally best
information available, and fly again. This scheme will be called synchronous, as
all of the birds update their information at the same time. This type is better for
global optimization PSO .
The synchronous PSO algorithm updates all particle velocities and positions
at the end of every optimization iteration. The figure(3.8) bellow shows the
synchronous scheme.[13][14][15][16]
3.7.2 Asynchronous update
As mentioned above, PSO was inspired by flocks of birds. Suppose that a
group of birds is randomly searching food in an area where only one piece of
bread exists. Although none of the birds knows the exact location of the bread,
if they each have some idea of their proximity to the bread, they can search the
area cooperatively by communicating their positions and proximity to one
another.
The birds may fly in different ways to search in the area. In the asynchronous
scheme, the first bird flies a certain distance and in a certain direction based on
his own experience and the known location of the best position found so far.
When he is done flying, if his location is closer to the bread than the previous
globally optimal position, he communicates this information to the second bird.
(Otherwise, the global optimal location remains the same.) The second bird then
uses this information to update his speed and position, and so on. Since in this
Particle Swarm Optimization (PSO)
method of searching an area the birds all update their speeds and directions at
different times, it will be called asynchronous. The figure(3.9) bellow shows the
asynchronous scheme.[13][14][15][16]
Figure (3.7)
Pseudo-code listings for the (a) Asynchronous and (b) Synchronous PSO.
