1. Particle Swarm Optimization
Fahimeh Fooladgar

2. Outline Swarm Intelligence Introduction to PSO Original PSO algorithms
Global Best PSO
Local Best PSO
Algorithm Aspects
Basic Variations
PSO Parameters
Application

3. Swarm Intelligence Example : benefits of cooperation Swarm  group
agents that communicate with each other
either directly or indirectly
acting on their local environment
Swarm Intelligence (SI) or collective intelligence
emerges from the interaction of such agents
Computational Swarm Intelligence(CSI)
algorithmic models of such behavior

4. Swarm Intelligence(cont.)
computational models of swarm intelligence
social animals and social insects
ants, termites, bees, spiders, fish schools, and bird flocks
individuals relatively simple in structure
but their collective behavior usually very complex
pattern of interactions between the individuals of the swarm over time

5. Swarm Intelligence(cont.)
objective of computational swarm intelligence models
simple behaviors of individuals
local interactions with the environment and neighboring
to obtain more complex behaviors
solve complex problems (optimization problems)

6. Introduction
First introduced by James Kennedy and Russell Eberhart in 1995
population-based search algorithm
simulation of the social behavior of birds within a flock
Individuals are particles
Individuals follow a very simple behavior
emulate the success of neighboring
emulate their own successes

7. Introduction (cont.) swarm of particles : population of individuals
particle have its own velocity
xi (t): position of particle i at t

8. Introduction (cont.) velocity vector drives the optimization process
reflects experiential knowledge and socially exchanged information
The experiential knowledge of a particle: cognitive component
distance of the particle from its own best position
particle’s personal best position
socially exchanged information :social component

9. Original PSO algorithms
Two PSO algorithms
Differ in the size of their neighborhoods
gbest PSO and lbest PSO

10. Global Best PSO Neighborhood for each particle is entire swarm
Social network : star topology
Velocity update statement

11. Global Best PSO (cont.)
vij(t) :velocity of particle i in dimension j = 1, , nx
yij(t) : personal best position
y^j (t) : best position found by the swarm
xij(t) : position of particle i in dimension j
c1 and c2 : positive acceleration constants
scale the contribution of the cognitive and social components
r1j(t), r2j(t) ∼ U(0, 1)
stochastic element to the algorithm

12. Global Best PSO (cont.) fitness function
personal best position at the next time step

13. Global Best PSO (cont.) global best position or
ns : total number of particles in the swarm

14. Global Best PSO (cont.)

15. Local Best PSO smaller neighborhoods are defined for each particle
network topology : ring social
Velocity update statement

16. Local Best PSO(cont.)
y^ij : best position, found by the neighborhood of particle i in dimension j
best position found in the neighborhood Ni

17. Local Best PSO(cont.) neighborhood defined
gbest PSO is a special case of the lbest PSO with nNi = ns

18. lbest PSO versus gbest PSO
Two main differences
gbest PSO converges faster than lbest PSO  less diversity
lbest PSO less susceptible to being trapped in local minima

19. Velocity Components vi(t) : previous velocity
memory of the previous flight direction
prevents the particle from drastically changing direction
bias towards the current direction
referred as the inertia component

20. Velocity Components(cont.)
c1r1(yi −xi ) : cognitive component
drawn back particle to their own best positions,
individuals return to situations that satisfied them most in the past
referred to as the “nostalgia” of the particle

21. Velocity Components(cont.)
social component
In gbest PSO
In lbest PSO
each particle drawn towards the best position found by the particle’s neighborhood
referred to as the “envy”

22. Geometric Illustration
inertia velocity
cognitive velocity
social velocity
new velocity

23. Algorithm Aspects initialize the swarm Particle position
initial velocities
Initial personal best position

24. Stopping conditions Maximum number of iterations
Acceptable solution has been found
No improvement is observed over a number of iterations
if the average change in particle positions is small
if the average particle velocity over a number of iterations is approximately zero

25. Stopping conditions(cont.)
Objective function slope is approximately zero
If f ’(t) < Є ,the swarm is converged

26. Social Network Structures
Star
Ring
Wheel
Von Neumann
Four Clusters
Pyramid

27. Basic Variations Improve basic PSO Velocity clamping Inertia weight
speed of convergence
Quality of solutions
Velocity clamping
Inertia weight
Constriction Coefficient

28. Velocity Clamping exploration–exploitation trade-off
Exploration : explore different regions of the search space
Exploitation : concentrate the search around a promising area
good optimization algorithm: balances these contradictory objectives
velocity update equation

29. Velocity Clamping(cont.)
velocity quickly explodes to large values
Then particles have large position updates
particles diverge
Should control the global exploration of particles
velocities clamped to stay within boundary constraints
Vmax,j denote the maximum allowed velocity in dimension j

30. Velocity Clamping(cont.)
Large values of Vmax,j facilitate global exploration
smaller values encourage local exploitation

31. Velocity Clamping(cont.)
If Vmax,j is too small
swarm may not explore sufficiently beyond locally good regions
increase the number of time steps to reach an optimum
swarm may become trapped in a local optimum
If Vmax,j is too large
risk the possibility of missing a good region
particles may jump over good solutions
but particles are moving faster

32. Velocity Clamping(cont.)
Balance between
moving too fast or too slow
exploration and exploitation
value of δ is problem-dependent

33. Inertia Weight introduced by Shi and Eberhart
control the exploration and exploitation abilities of the swarm
eliminate the need for velocity clamping
controlling influence of previous flight direction to new velocity

34. Inertia Weight(cont.) value of w is extremely important
ensure convergent behavior
tradeoff exploration and exploitation
For w ≥ 1
velocities increase over time
the swarm diverges
Particles fail to change direction
For w < 1
particles decelerate until their velocities reach zero

35. Inertia Weight(cont.) guarantees convergent particle trajectories
If this condition is not satisfied, divergent or cyclic behavior may occur

36. Inertia Weight(cont.) Dynamic Inertia Weight approaches
Linear decreasing
Start with w(0)=0.9 and final inertia weight w(nt)=0.4
nt : maximum number of time steps
w(0) is the initial inertia weight
w(nt) is the final inertia weight
w(t) is the inertia at time step t

37. Inertia Weight(cont.)
Random adjustments
Nonlinear decreasing

38. Constriction Coefficient
similar to the inertia weight
balance the exploration–exploitation trade-off
velocities are constricted by a constant χ
referred to as the constriction coefficient

39. Constriction Coefficient(cont.)
Κ controls the exploration and exploitation
For κ ≈ 0
fast convergence
local exploitation
For κ ≈ 1
slow convergence
high degree of exploration
Usually, κ set to a constant value
First K set close to one, decreasing it to zero

40. Constriction Coefficient(cont.)
Constriction approach equivalent to inertia weight approach if

41. PSO Parameters Swarm size (ns) Neighborhood size Number of iterations
more particles in the swarm, larger the initial diversity of the swarm
general heuristic : ns ∈ [10, 30]
actually problem dependent
Neighborhood size
Smaller neighborhoods , slower in convergence, more reliable convergence to optimal solutions
Best solution : starting with small neighborhoods and increasing the neighborhood
Number of iterations
It depend on problem

42. PSO Parameters(cont.) Acceleration coefficients c1 , c2 , r1 and r2
control the stochastic influence of the cognitive and social components
c1 : how much confidence a particle in itself
c2 : how much confidence a particle in its neighbors

43. PSO Application
Percent
Paper#
Application
7.6 51
Image processing
7.0 47
Control
5.8 39
Electronics and electromagnetics
antenna design
Power systems and plants
5.6 38
Scheduling
4.4 30
Design
Communication networks
4.3 29
Biological and medical
Clustering and classification
3.8 26
Fuzzy and neuro fuzzy
Signal processing
Neural networks
3.5 24
Combinatorial optimization

44. PSO Application(cont.)

45. What makes PSO so attractive to practitioners?
Simplicity
Easy to implement
ns×nx array for particle’s position
ns×nx array particle’s velocity
ns×nx d array particle’s personal best
1×nx array for global best
1×nx array for Vmax
Can adapt to different application

46. What makes PSO so attractive to practitioners?
All operations are simple and easy to implement
It require low computational resources (Memory and CPU)
It has ability to quickly converge to a reasonably good solution
It can easily and effectively run in distributed environments

47. References A.P.Engelbrecht, “computational intelligence ”,2007
R.Poli, " Analysis of the Publications on the Applications of Particle Swarm Optimisation ", Journal of Artificial Evolution and Applications, Vol. 2008,10 pages, 2007
K.E. Parsopoulos and M.N. Vrahatis. Particle Swarm Optimizer in Noisy and Continuously Changing Environments. In Proceedings of the IASTED International Conference on Artificial Intelligence and Soft Computing, pages 289–294,2001

48. Thanks for your attention?