An Overview of Particle Swarm Optimization

Jagdish Chand BansalMathematics Group

Birla Institute of Technology and Science, PilaniEmail: jcbansal@gmail.com, bits-pilani.ac.in

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Overview

IntroductionAn ExampleSome DevelopmentsResearch Issues

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Optimization Methods

Deterministic and

Probabilistic

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Deterministic Method

MeritsGive exact solutionsDo not use any stochastic techniqueRely on the thorough search of the feasible domain.

DemeritsNot Robust- can only be applied to restricted class of problems.Often too time consuming or sometimes unable to solve real world problems.

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Merits• Applicable to wider set of problems i.e. function need not be

convex, continuous or explicitly defined• Use the stochastic or probabilistic approach i.e. random

approach

Probabilistic Method

DemeritsConverges to the global optima probabilisticallySome times get stuck at local optima.

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Some Existing Probabilistic Methods

Simulated Annealing (SA)Random Search Technique (RST)Genetic Algorithm (GA)Memetic Algorithm (MA)Ant Colony Optimization (ACO)Differential Evolution (DE)Particle Swarm Optimization (PSO)

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Why PSO for Optimization ?

Continuous Optimization ProblemNon-differentiable,Non-Convex Highly nonlinear Many local-optima

Discrete Optimization ProblemNP-Complete problems: Nobody has found so far any good algorithm for any problem in this class

Search speed

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Artificial Life

The term artificial life is used to describe research into human made systems that possess some of the essential properties of life. A-life includes two folded research:

A-life studies how computational techniques can help studying biological phenomena

Particle Swarm Optimization Inspiration

A-life studies how biological techniques can help out with computational problem

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Inspiration cont..

Based on bird flocking or fish schooling and swarming theory of A-Life.

About fish schooling: “In theory at least, individual members of the school can profit from the discoveries and previous experience of all other members of the school during the search for food “.(a sociobiologist E. O. Wilson)

This is the basic concept behind PSO.

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Inventors

Developed in 1995 by

Prof. James Kennedy (Right)

Prof. Russel Eberhart (Left)

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PSO uses a population of individuals, to search feasible region of the function space. In this context, the population is called swarm and the individuals are called particles.

Though the PSO algorithm has been shown to perform well, researchers have not been able to explain fully how it works yet.

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Each particle tries to modify its current position and velocity according to the distance between its current position and pbest, and the distance between its current position and gbest.

Update Equations

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)()( 2211 currentgbestrccurrentpbestrcvv −+−+=

Current Velocity

Updated Velocity

Velocity Update Equation (Rate of Change in Particle’s Position)

rand (0,1), to stop the swarm converging too quickly

Acceleration factors, can be used to change the weighting between personal and population experience

This is the cognitive component which draws individuals back to their previous best situations.

This is the social component where individuals compare themselves to others in their group.

vcurrentcurrent += Position Update Equation

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PSO Parameters

1. The number of particles :

20 – 40 particles. For most of the problems 10 particles are large enough to get good results.

2. Dimension of particles :It is determined by the problem to be optimized.

3. Range of particles :It is also determined by the problem to be optimized, we can specify different ranges for different dimension of particles.

4. Vmax :This is done to help keep the swarm under control. we set the range of the particle as the Vmax. e.g. X belongs [-10, 10], then Vmax = 20.One another approach is Vmax= ⎣(UpBound – LoBound)/5⎦

5. Learning/Acceleration factors :c1 and c2 usually equal to 2. However, other settings were also used in different papers. But usually c1equals to c2 and ranges from [0, 4].

6. The stopping criteria :The maximum number of iterations the PSO execute and the minimum error requirement.

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Basic Flow of PSO

1. Initialize the swarm from the solution space2. Evaluate fitness of individual particles3. Modify gbest, pbest and velocity4. Move each particle to a new position.5. Go to step 2, and repeat until convergence or a

stopping condition is satisfied.

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An Example

Understanding of Step by step Procedure of PSO

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gbest PSO - global version is faster but might converge to local optimum for some problems.

lbest PSO - local version is a little bit slower but not easy to be trapped into local optimum.

One can use global version to get quick result and use local version to refine the search

Two Versions of PSO

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BINARY PSO

This version has attracted much lesser attention as compared to PSO

Particle position is not a real value, but either 0 or 1

Velocity represents the probability of a bit to take the value 0 or 1 not the rate of change in particle’s position as in PSO for continuous optimization

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BINARY PSO

The particle’s position in a dimension is randomly generated using sigmoid function

)exp(11)(

xxsigm

−+=

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6

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Velocity and Position Update

)()( 2211 idgdidididid xprcxprcvv −+−+=

⎩⎨⎧ <

=otherwise

vsigmrandifx id

id 0)(()1

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No Free Lunch Theorem

• In a controversial paper in 1997 (available at AUC library), Wolpert and Macready proved that “averaged over all possible problems or cost functions, the performance of all search algorithms is exactly the same”

• No algorithm is better on average than blind guessing

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Important Developments

Almost all modifications vary in some way the velocity update equation.

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PSO-W : With Inertia WeightPSO-C : With Constriction FactorFIPSO : Fully Informed PSOHPSOM : Hybrid PSO with Mutation MeanPSO : Mean PSOqPSO : Quadratic approximation PSO

A Brief Review

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Inertia Weight

Shi and Eberhart introduced the inertia weight w in thealgorithm (PSO-W).Then the iterative expression becomes:

w represents the inertia weight, which enhances the exploration ability of particles

)()(* 2211 currentgbestrccurrentpbestrcvwv −+−+=vcurrentcurrent +=

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Why Inertia Weight

When using PSO, it is possible for the magnitude of the velocities to become very large.

Performance can suffer if Vmax is inappropriately set.

For controlling the growth of velocities a dynamically adjusted or constant inertia weight were introduced.

Larger w - greater global search ability

Smaller w - greater local search ability.

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Constriction Factor

Clerc and Kennedy proposed that the constriction factor is effective for the algorithm to converge (PSO-C)

))()((* 2211 currentgbestrccurrentpbestrcvv −+−+= χ

vcurrentcurrent +=

421 >+= ccφ( )42

2−−−

=φφφ

χ

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Fully Informed PSO

A particle is attracted by every other particle in its neighborhood.

{ }⎥⎦

⎤⎢⎣

⎡−+= ∑

∈ iNiii icurrentiprcvv )()(*χ

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)()(* 2211 currentgbestrccurrentpbestrcvwv −+−+=

)()(* 2211 currentgbestrccurrentpbestrcvv −+−+= χ

PSO algorithm performs well in the early stage, but easily becomes premature in the local optima area.

The velocity is only related with inertia weight and constriction factor

If the current position of a particle is identical with the global best position and if the current velocity is a small value, the velocity in next iteration will be smaller. Then the particle will be trapped in this area which leads to premature convergence.

This phenomenon is known as stagnation

Stagnation

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Hybrid Particle Swarm Optimizer with Mutation (HPSOM).

HPSOM has the potential to escape from the local optimum and search in a new position. The mutation scheme randomly chooses a particle and then move to a different position in search area. The operation shows as follows:

5.0(),)(

5.0(),)(

>∆−=

<∆+=

randxxxmut

randxxxmutdi

di

di

di

x∆ is randomly obtained from

))](min)((max1.0,0[ drangedrange −×

This mutation operation is governed by a constant called probability of mutation

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MeanPSO

gbest Solution0 current pbest

PSO

47.1pbestgbest −

47.1pbestgbest +

MeanPSO

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R1 Particle with best fitness value

R2 and R3 Randomly chosen distinct particles

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛−+−+−−+−+−

=321213132

322

212

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231

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22*

)()()()(*5.0R

RfRRRfRRRfRRRfRRRfRRRfRR

Where f(Ri) is the objective function value at Ri , for i=1, 2, and 3.

The calculations are to be done component wise to obtain R* ..

qPSO:Quadratic Approximation (QA)

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The Process of Hybridization

Figure 4.1: Transition from ith iteration to i+1th iteration

s1s2---sp

sp+1sp+2--sm

s'1s'2---s'p

s'p+1s'p+2--s'm

qPSO

PSO

QA

qPSO

PSO

QA

ith iteration i+1th iteration

Particle Index

The percentage of swarm which is to be updated by QA is called Coefficient of Hybridization (CH)

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Random Swarm

ITER =0

Yes

End

Evaluate Objective Function Value of all Particles and Determine GBEST

Stopping Criterion Satisfied?

ITER =ITER + 1

Split Swarm S into subswarms S1 and S2

Is it possible to Determine R1, R2 and

R3 such that atleast two of them are distinct?

No

Determine pbest and gbest (=

GBEST) No

Yes

Velocity Update

Position Update using PSO

Determine R1 (= GBEST), R2 and R3

Position Update using QA

Report Best Particle

Evaluate Objective Function Value of all Particles and Determine

GBEST

Start

For S1 For S2

Flowchart of qPSO Process

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Research Issues

Hybridization

Parallel Implementation

New Variants modification in Velocity Update EquationIntroduce some new operators in PSO

Discrete Particle Swarm Optimization

Interaction with biological intelligence

Convergence Analysis 36

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Some Unsolved Issues

• Convergence analysis.• Dealing with discrete variables.• Combination of various PSO techniques

to deal with complex problems.• Interaction with biological intelligence.• Cryptanalysis.