Chapter 16: Particle Swarm Optimization - Computational Intelligence

Chapter 16. Particle Swarm Optimization

Chapter 16: Particle Swarm Optimization

Computational Intelligence: Second Edition

Computational Intelligence: Second Edition Chapter 16: Particle Swarm Optimization


IntroductionOverview of the basic PSOGlobal Best PSOLocal Best PSOAspects of Basic PSOBasic Variations of PSOPSO ParametersParticle TrajectoriesSingle-Solution Particle Swarm Optimizers

Contents


References102




Introduction

Particle swarm optimization (PSO):developed by Kennedy & Eberhart [3, 7],first published in 1995, andwith an exponential increase in the number of publicationssince then.

What is PSO?a simple, computationally efficient optimization methodpopulation-based, stochastic searchbased on a social-psychological model of social influence andsocial learning [8]individuals follow a very simple behavior: emulate the successof neighboring individualsemergent behavior: discovery of optimal regions in highdimensional search spaces




Introduction

Particle swarm optimization (PSO):developed by Kennedy & Eberhart [3, 7],first published in 1995, andwith an exponential increase in the number of publicationssince then.

What is PSO?a simple, computationally efficient optimization methodpopulation-based, stochastic searchbased on a social-psychological model of social influence andsocial learning [8]individuals follow a very simple behavior: emulate the successof neighboring individualsemergent behavior: discovery of optimal regions in highdimensional search spaces




Introduction: What are the origins of PSO?

In the work of Reynolds on “boids” [12]: Flocking is anemergent behavior which arises from the interaction of simplerules:

Collision avoidanceVelocity matchingFlock centering

The work of Heppner and Grenander on using a “rooster” asan attractor [5]

Simplified social model of determining nearest neighbors andvelocity matching




Origins of PSO (cont)

Initial objective: to simulate the graceful, unpredictablechoreography of collision-proof birds in a flock

At each iteration, each individual determines its nearestneighbor and replaces its velocity with that of its neighbor

Resulted in synchronous movement of the flock

Random adjustments to velocities prevented individuals tosettle too quickly on an unchanging direction

Adding roosters as attractors:

personal bestneighborhood best→ particle swarm optimization




Overview of basic PSO

What are the main components?A swarm of particlesEach particle represents a candidate solutionElements of a particle represent parameters to be optimized

The search process:Position updates

xi (t + 1) = xi (t) + vi (t + 1)

wherexij(0) ∼ U(xmin,j , xmax,j)

Velocitydrives the optimization processstep sizereflects experiential knowledge and socially exchangedinformation




Overview of basic PSO

What are the main components?A swarm of particlesEach particle represents a candidate solutionElements of a particle represent parameters to be optimized

The search process:Position updates

xi (t + 1) = xi (t) + vi (t + 1)

wherexij(0) ∼ U(xmin,j , xmax,j)

Velocitydrives the optimization processstep sizereflects experiential knowledge and socially exchangedinformation




Social network structures (Figure 16.4):

Social interaction based on neighborhoodsFirst used network structures: star and ring topologies

(a) Star (b) Ring

Figure: Social Network Structures




Other social structures (Figure 16.4):

(a) Von Neumann (b) Pyramid

Figure: Advanced Social Network Structures




Other social structures (cont...)

(a) Clusters (b) Wheel

Figure: Advanced Social Network Structures




Global Best (gbest) PSO

Uses the star social network

Velocity update per dimension:

vij(t + 1) = vij(t) + c1r1j(t)[yij(t)− xij(t)]

+c2r2j(t)[yj(t)− xij(t)]

vij(0) = 0 (usually, but can be random)

c1, c2 are positive acceleration coefficients

r1j(t), r2j(t) ∼ U(0, 1)




Global Best PSO (cont)

yi (t) is the personal best position calculated as (assuming

minimization):

yi(t+1) =

{yi(t) if f (xi(t + 1)) ≥ f (yi(t))xi(t + 1) if f (xi(t + 1)) < f (yi(t))




Global Best PSO (cont)

y(t) is the global best position calculated as

y(t) ∈ {y0(t), . . . , yns(t)}|f (y(t))

= min{f (y0(t)), . . . , f (yns(t))}

or

y(t) ∈ {x0(t), . . . , xns(t)}|f (y(t))

= min{f (x0(t)), . . . , f (xns(t))}

where ns is the number of particles in the swarm




gbest PSO Algorithm

Create and initialize an nx -dimensional swarm, S ;repeat

for each particle i = 1, . . . ,S .ns doif f (S .xi ) < f (S .yi ) then

S .yi = S .xi ;endif f (S .yi ) < f (S .y) then

S .y = S .yi ;end

endfor each particle i = 1, . . . ,S .ns do

update the velocity and then the position;end

until stopping condition is true ;




Local Best (lbest) PSO

Uses the ring social network


+c2r2j(t)[yij(t)− xij(t)]

yi is the neighborhood best, defined as:

yi (t + 1) ∈ {Ni |f (yi (t + 1)) = min{f (x)}, ∀x ∈ Ni}

with the neighborhood defined as

Ni = {yi−nNi(t), yi−nNi

+1(t), . . . ,

yi−1(t), yi (t), yi+1(t), . . . , yi+nNi(t)}

where nNiis the neighborhood size




Local Best PSO (cont...)

Neighborhoods:

Neighborhoods are based on particle indices, not spatialinformationNeighborhoods overlap to facilitate information exchange

The lbest PSO algorithm?

gbest PSO vs lbest PSO:

Speed of convergenceSusceptibility to local minima?




Velocity Components

Previous velocity, vi (t)

inertia componentmemory of previous flight directionprevents particle from drastically changing direction

Cognitive component, c1r1(yi − xi )

quantifies performance relative to past performancesmemory of previous best positionnostalgia

Social component, c2r2(yi − xi )

quantifies performance relative to neighborsenvy




Velocity Components










Velocity Components










Geometric Illustration Figure 16.1

social velocity

cognitive velocity

inertia velocity

new velocity

x(t)y(t)

x(t + 1)

y(t)

x1

x2

(a) Time Step t

new velocity

cognitive velocity

social velocity

inertiavelocity

x(t)

y(t + 1)

x1

x2

y(t + 1)

x(t + 1)

x(t + 2)

(b) Time Step t + 1

Figure: Geometrical Illustration of Velocity and Position Updates for aSingle Two-Dimensional Particle




Cumulative Effect of Position Updates (Figure 16.2)

x2

x1

y

(a) Time Step t

x2

x1

y

(b) Time Step t + 1

Figure: Multi-particle gbest PSO Illustration




Stopping Conditions

Terminate when a maximum number of iterations, or FEs, hasbeen exceeded

Terminate when an acceptable solution has been found, i.e.when (assuming minimization)

f (xi ) ≤ |f (x∗)− ε|

Terminate when no improvement is observed over a number ofiterations




Stopping Conditions (cont)

Terminate when the normalized swarm radius is close to zero,i.e.

Rnorm =Rmax

diameter(S(0))≈ 0

whereRmax = ||xm − y||, m = 1, . . . , ns

with||xm − y|| ≥ ||xi − y||, ∀i = 1, . . . , ns

Terminate when the objective function slope is approximatelyzero, i.e.

f′(t) =

f (y(t))− f (y(t − 1))

f (y(t))≈ 0




Basic PSO Variations: Introduction

Main objectives of these variations are to improve

Convergence speedQuality of solutionsAbility to converge




Exploration-Exploitation Tradeoff

Exploration: the ability to explore regions of the search space

Exploitation: the ability to concentrate the search around apromising area to refine a candidate solution

c1 vs c2 and the influence on the exploration–exploitationtradeoff






Exploration-Exploitation Tradeoff

Exploration: the ability to explore regions of the search space

Exploitation: the ability to concentrate the search around apromising area to refine a candidate solution

c1 vs c2 and the influence on the exploration–exploitationtradeoff






Velocity Clamping:

Problem with basic PSO:

Velocity quickly explodes to large values

Solution:

Limit step sizes

vij(t + 1) =

{vij(t + 1) if |vij(t + 1)| < Vmax,j

Vmax,j if |vij(t + 1)| ≥ Vmax,j

Controls global exploration of particles

Optimal value is problem-dependent

Does not confine the positions, only the step sizes




Velocity Clamping:



Solution:

Limit step sizes

vij(t + 1) =









Velocity Clamping:



Solution:

Limit step sizes

vij(t + 1) =









Velocity Clamping:



Solution:

Limit step sizes

vij(t + 1) =









Velocity Clamping:



Solution:

Limit step sizes

vij(t + 1) =









Velocity Clamping Problem (Figure 16.5)

position updatevelocity update

x1

v2(t + 1)

v1(t + 1) xi(t)

v′

2(t + 1)

x

′

i(t + 1)

xi(t + 1)

x2

Figure: Effects of Velocity Clamping




More Velocity Clamping Problems

What happens when, for a dimension,

xmax ,j − xmin,j << Vmax?

What happens if all velocities are equal to the maximumvelocity?

Particles search on the boundaries of the hypercube,

[xi (t)− Vmax , xi (t) + Vmax ]




Velocity Clamping:

Dynamically changing Vmax when gbest does not improveover τ iterations [13]

Vmax ,j(t + 1) =

βVmax ,j(t) if f (y(t)) ≥ f (y(t − t

′))

∀ t′= 1, . . . , τ

Vmax ,j(t) otherwise

Exponentially decaying Vmax [4]

Vmax ,j(t + 1) = (1− (t/nt)α)Vmax ,j(t)




Velocity Clamping:

Dynamically changing Vmax when gbest does not improveover τ iterations [13]

Vmax ,j(t + 1) =

βVmax ,j(t) if f (y(t)) ≥ f (y(t − t

′))

∀ t′= 1, . . . , τ

Vmax ,j(t) otherwise

Exponentially decaying Vmax [4]

Vmax ,j(t + 1) = (1− (t/nt)α)Vmax ,j(t)




Inertia Weight[15]

Developed to control exploration and exploitationControls the momentumVelocity update changes to

vij(t + 1) = wvij(t) + c1r1j(t)[yij(t)− xij(t)]


For w ≥ 1velocities increase over time and swarm divergesparticles fail to change direction towards more promisingregions

For 0 < w < 1particles decelerateconvergence also dependent on values of c1 and c2




Inertia Weight (cont...)

Exploration–exploitation trade-offlarge values – favor explorationsmall values – promote exploitation

Best value is problem-dependent

Dynamically changing inertia weightsw ∼ N(0.72, σ)linear decreasing [17]

w(t) = (w(0)− w(nt))(nt − t)

nt+ w(nt)

non-linear decreasing [23]

w(t + 1) = αw(t′)

with w(t) = 1.4




Inertia Weight (cont...)

Exploration–exploitation trade-offlarge values – favor explorationsmall values – promote exploitation

Best value is problem-dependentDynamically changing inertia weights

w ∼ N(0.72, σ)linear decreasing [17]

w(t) = (w(0)− w(nt))(nt − t)

nt+ w(nt)

non-linear decreasing [23]

w(t + 1) = αw(t′)

with w(t) = 1.4




Constriction Coefficient [2]

Developed to ensure convergence to a stable point withoutthe need for velocity clamping

vij(t + 1) = χ[vij(t) + φ1(yij(t)− xij(t))

+φ2(yj(t)− xij(t))]

where

χ =2κ

|2− φ−√

φ(φ− 4)|with

φ = φ1 + φ2

φ1 = c1r1, φ2 = c2r2




Constriction Coefficient (cont...)

If φ ≥ 4 and κ ∈ [0, 1], then the swarm is guaranteed toconverge to a stable point

χ ∈ [0, 1]

κ controls exploration–exploitationκ ≈ 0: fast convergence, exploitationκ ≈ 1: slow convergence, exploration

Effectively equivalent to inertia weight for specific χ:w = χ, φ1 = χc1r1 and φ2 = χc2r2




Synchronous vs asynchronous updates

Synchronous:

Personal best and neighborhood bests updated separately fromposition and velocity vectorsslower feedbackbetter for gbest

Asynchronous:

new best positions updated after each particle position updateimmediate feedback about best regions of the search spacebetter for lbest




Synchronous vs asynchronous updates

Synchronous:

Personal best and neighborhood bests updated separately fromposition and velocity vectorsslower feedbackbetter for gbest

Asynchronous:

new best positions updated after each particle position updateimmediate feedback about best regions of the search spacebetter for lbest




Velocity Models

Cognition-only model:

vij(t + 1) = vij(t) + c1r1j(t)(yij(t)− xij(t))

particles are independent hill-climberslocal search by each particle

Social-only model:

vij(t + 1) = vij(t) + c2r2j(t)(yj(t)− xij(t))

swarm is one stochastic hill-climberSelfless model:

Social model, but nbest solution chosen from neighbors onlyParticle itself is not allowed to become the nbest




Basic Parameters

Swarm size, ns

Particle dimension, nx

Neighborhood size, nNi

Number of iterations, nt

Inertia weight, w

Acceleration coefficients, c1 and c2




Acceleration Coefficients

c1 = c2 = 0?

c1 > 0, c2 = 0: cognition-only model

c1 = 0, c2 > 0: social-only model

c1 = c2 > 0:

particles are attracted towards the average of yi and yi

c2 > c1:

more beneficial for unimodal problems




Acceleration Coefficients (cont)

c1 < c2:more beneficial for multimodal problems

Low c1 and c2:smooth particle trajectories

High c1 and c2:more acceleration, abrupt movements

Adaptive acceleration coefficients

c1(t) = (c1,min − c1,max)t

nt+ c1,max

c2(t) = (c2,max − c2,min)t

nt+ c2,min




Acceleration Coefficients (cont)

c1 < c2:more beneficial for multimodal problems

Low c1 and c2:smooth particle trajectories

High c1 and c2:more acceleration, abrupt movements

Adaptive acceleration coefficients

c1(t) = (c1,min − c1,max)t

nt+ c1,max

c2(t) = (c2,max − c2,min)t

nt+ c2,min




Simplified Particle Trajectories [19, 22]

No stochastic component

Single, one-dimensional particle

Using w

Personal best and global best are fixed:y = 1.0, y = 0.0

φ1 = c1r1 and φ2 = c2r2




Example trajectories:

-40

-30

-20

-10

0

10

20

30

0 50 100 150 200 250

xt

t

(a) Time domain

-6e-16-40 -30 -20 -10 0 10 20 30

im

re

(b) Phase space

Figure: Convergent Trajectory for Simplified System, with w = 0.5 andφ1 = φ2 = 1.4





-500

-400

-300

-200

-100

0

100

200

300

400

500

0 50 100 150 200 250

xt

t

(a) Time domain

-6e-16-500 -400 -300 -200 -100 0 100 200 300 400 500

im

re

(b) Phase space

Figure: Cyclic Trajectory for Simplified System, with w = 1.0 andφ1 = φ2 = 1.999





-1e+07

-8e+06

-6e+06

-4e+06

-2e+06

0

2e+06

4e+06

6e+06

0 50 100 150 200 250

xt

t

(a) Time domain

-1.2e+100

-1e+100

-8e+99

-6e+99

-4e+99

-2e+99

0

2e+99

4e+99

6e+99

8e+99

-6e+99 -4e+99 -2e+99 0 2e+99 4e+99 6e+99 8e+99 1e+100

im

re

(b) Phase space

Figure: Divergent Trajectory for Simplified System, with w = 0.7 andφ1 = φ2 = 1.9




Convergence Conditions:

What do we mean by the term convergence?

0 4.0

1

2.0z1phi

wz2

Figure: Convergence Map for Values of w and φ = φ1 + φ2




Convergence Conditions (cont)

Convergence behavior is sensitive to the values of w and c1, c2

Theoreticall derived heuristics for setting these values:Constriction coefficientThe trajectory of a particle converges if

1 > w >1

2(φ1 + φ2)− 1 ≥ 0

Since φ1 = c1U(0, 1) and φ2 = c2U(0, 1), the accelerationcoefficients, c1 and c2 serve as upper bounds of φ1 and φ2

So, particles converge if

1 > w >1

2(c1 + c2)− 1 ≥ 0




Convergence Conditions (cont)

It can happen that, for stochastic φ1 and φ2 and a w that violatesthe condition above, the swarm may still converge

-30

-20

-10

0

10

20

30

40

0 50 100 150 200 250

x(t)

t

Figure: Stochastic Particle Trajectory for w = 0.9 and c1 = c2 = 2.0




Guaranteed Convergence PSO

For the basic PSO, what happens when

xi = yi = y

and if this condition persists for a number of iterations?

The problem:wvi → 0

which leads to stagnation, where particles have converged onthe best position found by the swarm




Guaranteed Convergence PSO

For the basic PSO, what happens when

xi = yi = y

and if this condition persists for a number of iterations?

The problem:wvi → 0

which leads to stagnation, where particles have converged onthe best position found by the swarm




Guaranteed Convergence PSO (cont)

How can we address this problem?

Force the global best position to change when

xi = yi = y

This is what happens in the GCPSO, where the global bestparticle is forced to search in a bounding box around thecurrent position for a better position




Guaranteed Convergence PSO (cont)

How can we address this problem?

Force the global best position to change when

xi = yi = y

This is what happens in the GCPSO, where the global bestparticle is forced to search in a bounding box around thecurrent position for a better position




GCPSO Equations

The position update of the global best particle changes to

xτ j(t + 1) = yj(t) + wvτ j(t) + ρ(t)(1− 2r2(t))

where τ is the index of the global best particle

The velocity update of the global best particle then has tochange to

vτ j(t + 1) = −xτ j(t) + yj(t) + wvτ j(t) + ρ(t)(1− 2r2j(t))

where where ρ(t) is a scaling factor




GCPSO Equations (cont)

The scaling factor is defined as

ρ(t + 1) =

2ρ(t) if #successes(t) > εs0.5ρ(t) if #failures(t) > εfρ(t) otherwise

where #successes and #failures respectively denote thenumber of consecutive successes and failures

A failure is defined as f (y(t)) ≤ f (y(t + 1))




Social-Based Particle Swarm Optimization

Spatial neighborhoods

Fitness-Based spatial neighborhoods

Growing neighborhoods

Hypercube structure

Fully informed PSO

Barebones PSO




Spatial Social Networks: Algorithm 16.4

Neighborhoods usually formed on the basis of particle indices

Based on spatial neighborhoods:

Calculate the Euclidean distance E(xi1 , xi2),∀i1, i2 = 1, . . . , ns ;S = {i : i = 1, . . . , ns};for i = 1, . . . , ns do

S′= S ;

for i′= 1, . . . , nN

i′ do

Ni = Ni ∪ {xi ′′ : E(xi , xi ′′ ) = min{E(xi , xi ′′′ ),∀xi ′′′ ∈ S′};

S′= S

′ \ {xi ′′};end

end




Fitness-Based Spatial Neighborhoods

The neighborhood of particle i is defined as the nN particleswith the smallest value of

E(xi , xi ′ )× f (xi ′ )

where E(xi , xi ′ ) is the Euclidean distance between the particles

Overlapping neighborhoods are allowed




Growing Neighborhoods

Start with a ring topology having smallest connectivity, andgrow towards a star topology

Particle position xi2(t) is added to the neighborhood ofparticle position xi1(t) if

||xi1(t)− xi2(t)||2dmax

< ε

where dmax is the largest distance between any two particles

ε =3t + 0.6nt

nt

with nt the maximum number of iterations.




Hypercube Structure

Neighborhood structure for binary-valued problems

Particles are defined as neighbors if the Hamming distancebetween the bit representation of their indices is one

Total number of particles must be a power of two, whereparticles have indices from 0 to 2nN − 1

Hypercube has the properties [1]:

Each neighborhood has exactly nN particles.The maximum distance between any two particles is exactlynN .If particles i1 and i2 are neighbors, then i1 and i2 will have noother neighbors in common.




Fully Informed PSO

Velocity equation is changed such that each particle isinfluenced by the successes of all its neighbors, and not on theperformance of only one individual

Each particle in the neighborhood, Ni , of particle i is regardedequally:

vi (t + 1) = χ

(vi (t) +

nNi∑m=1

r(t)(ym(t)− xi (t))

nNi

)

where nNi= |Ni |, and r(t) ∼ U(0, c1 + c2)

nx




Fully Informed PSO (cont)

A weight is assigned to the contribution of each particle basedon the performance of that particle:

vi (t + 1) = χ

vi (t) +

∑nNim=1

(φmpm(t)f (xm(t))

)∑nNi

m=1

(φm

f (xm(t))

)

where φm ∼ U(0, c1+c2nNi

), and

pm(t) =φ1ym(t) + φ2ym(t)

φ1 + φ2




Fully Informed PSO (cont)

Advantages:

More information is used to decide on the best direction toseachInfluence of a particle on the step size is proportional to theparticle’s fitness

Disadvantages

Influences of multiple particles may cancel each otherWhat happens when all particles are positioned symmetricallyaround the particle being updated?




Barebones PSO

Formal proofs [18, 19, 22] have shown that each particleconverges to a point that is a weighted average between thepersonal best and neighborhood best positions:

yij(t) + yij(t)

2

This behavior supports Kennedy’s proposal to replace theentire velocity by

vij(t + 1) ∼ N

(yij(t) + yij(t)

2, σ

)where

σ = |yij(t)− yij(t)|




Barebones PSO (cont)

The position update is simply

xij(t + 1) = vij(t + 1)

Alternative formulation:

vij(t + 1) =

{yij(t) if U(0, 1) < 0.5

N(yij (t)+yij (t)

2 , σ) otherwise

There is a 50% chance that the j-th dimension of the particledimension changes to the corresponding personal best position




Hybrid Algorithms

Selection-based PSO

Reproduction in PSO

Mutation in PSO

Differential evolution based PSO




Selection-Based PSO: Algorithm 16.5

Calculate the fitness of all particles;for each particle i = 1, . . . , ns do

Randomly select nts particles;Score the performance of particle i against the nts randomlyselected particles;

endSort the swarm based on performance scores;Replace the worst half of the swarm with the top half, withoutchanging the personal best positions;

What are the problems with this?

Any advantages?




Global Best Spawning: Algorithm 16.6

if y(t) is in a potential minimum thenrepeat

y = y(t);for NumberOfSpawns=1 to 10 do

for a = 1 to NumberOfSpawns doya = y(t) + N(0, σ);if f (ya) < f (y) then

y = ya;end

end

enduntil f (y) ≥ f (y(t)) ;y(t) = y;

end




Using Arithmetic Crossover

An arithmetic crossover operator to produce offspring fromtwo randomly selected particles [10, 11]:

xi1(t + 1) = r(t)xi1(t) + (1− r(t))xi2(t)

xi2(t + 1) = r(t)xi2(t) + (1− r(t))xi1(t)

with the corresponding velocities,

vi1(t + 1) =vi1(t) + vi2(t)

||vi1(t) + vi2(t)||||vi1(t)||

vi2(t + 1) =vi1(t) + vi2(t)

||vi1(t) + vi2(t)||||vi2(t)||

where r1(t) ∼ U(0, 1)nx




Using Arithmetic Crossover (cont)

Personal best position of an offspring is initialized to itscurrent position:

yi1(t + 1) = xi1(t + 1)

Particles are selected for breeding at a user-specified breedingprobability

Random selection of parents prevents the best particles fromdominating the breeding process

Breeding process is done for each iteration after the velocityand position updates have been done




Using Arithmetic Crossover (cont)

Disadvantage:

Parent particles are replaced even if the offspring is worse off infitnessIf f (xi1(t + 1)) > f (xi1(t)) (assuming a minimization problem),replacement of the personal best with xi1(t + 1) losesimportant information about previous personal best positions

Solutions:

Replace parent with its offspring only if fitness of offspring isbetter than that of the parent.




Mutation PSO

Mutate the global best position:

y(t + 1) = y′(t + 1) + η

′N(0, 1)

where y′(t + 1) represents the unmutated global best position,

and η′is referred to as a learning parameter

Mutate the componets of position vectors: For each j , ifU(0, 1) < Pm, then component x

′ij(t + 1) is mutated using [6]

xij(t + 1) = x′ij(t + 1) + N(0, σ)x

′ij(t + 1)

whereσ = 0.1(xmax ,j − xmin,j)




Mutation PSO (cont)

Each xij can have its own deviation:

σij(t) = σij(t − 1) eτ′N(0,1)+τNj (0,1)

with

τ′

=1√

2√

nx

τ =1√2nx




Mutation PSO (cont)

Secrest and Lamont [14] adjust particle positions as follows

xi (t + 1) =

{yi (t) + vi (t + 1) if U(0, 1) > c1

y(t) + vi (t + 1) otherwise

wherevi (t + 1) = |vi (t + 1)|rθ

rθ is a random vector with magnitude of one and angleuniformly distributed from 0 to 2π and

vi (t+1)| ={

N(0, (1− c2)||yi (t)− y(t)||2) if U(0, 1) > c1

N(0, c2||yi (t)− y(t)||2) otherwise




Differential Evolution PSO

After the normal velocity and position updates,

Select x1(t) 6= x2(t) 6= x3(t)Compute offspring:

x′

ij(t + 1) =

x1j(t) + β(x2j(t)− x3j(t)) if U(0, 1) ≤ Pc

or j = U(1, nx)xij(t) otherwise

where Pc ∈ (0, 1) is the probability of crossover, and β > 0 isa scaling factorReplace position of the particle if the offspring is better, i.e.xi (t + 1) = x

′

i (t + 1) only if f (x′

i (t + 1)) < f (xi (t)), otherwisexi (t + 1) = xi (t)




Differential Evolution PSO (cont)

Apply DE crossover on personal best positions only [24]:

y′ij(t + 1) =

{yij(t) + δj if U(0, 1) < Pc and j = U(1, nx)yij(t) otherwise

where δ is the general difference vector defined as,

δj =y1j(t)− y2j(t)

2

and y1j(t) and y2j(t) randomly selected personal bestpositions

yij(t + 1) is set to y′ij(t + 1) only if the new personal best has

a better fitness




Sub-Swarm Based PSO

Cooperative Split PSO

Hybrid Cooperative Split PSO

Predator-Prey PSO

Life-cycle PSO

Attractive and Repulsive PSO




Cooperative Split PSO

Each particle is split into K separate parts of smallerdimension [19, 20, 21]

Each part is then optimized using a separate sub-swarm

If K = nx , each dimension is optimized by a separatesub-swarm, using any PSO algorithm

What are the issues?

Problem if there are strong dependencies among variablesHow should the fitness of sub-swarm particles be evaluated?




Cooperative Split PSO: Fitness Evaluation




Cooperative Split PSO: Algorithm 16.8

K1 = nx mod K and K2 = K − (nx mod K );Initialize K1 dnx/Ke-dimensional and K2 bnx/Kc-dimensional swarms;repeat

for each sub-swarm Sk ,k = 1, . . . ,K dofor each particle i = 1, . . . ,Sk .ns do

if f (b(k,Sk .xi )) < f (b(k,Sk .yi )) thenSk .yi = Sk .xi ;

endif f (b(k,Sk .yi )) < f (b(k,Sk .y)) then

Sk .y = Sk .yi ;end

endApply velocity and position updates;

enduntil stopping condition is true ;




Cooperative Split PSO (cont)

Advantages:

Instead of solving one large dimensional problem, severalsmaller dimensional problems are now solvedFitness function is evaluated after each subpart of the contextvector is updated, allowing a finer-grained searchImproved accuracies have been obtained for many optimizationproblems




Hybrid Cooperative Split PSO

Hybrid Cooperative Split PSO:

Have subswarm and a main swarmMain swarm solves the complete problemAfter one iteration of the cooperative algoritihm, replace arandomly selected particle from the main swarm with thecontext vectorRandomly selected particle’s personal best should not be aneighborhood bestAfter a GCPSO update of the main swarm, replace a randomlyselected particle from each subswarm with the correspondingelement in the global best position of the main swarmRandomly selected subswarm particle’s personal best shouldnot be the global best for that particle.




Predator-Prey PSO

Competition introduced to balance exploration–exploitationUses a second swarm of predator particles:

Prey particles scatter (explore) by being repelled by thepresence of predator particlesSilva et al. [16] use only one predator to pursue the global bestprey particleThe velocity update for the predator particle is defined as

vp(t + 1) = r(y(t)− xp(t))

where vp and xp are respectively the velocity and positionvectors of the predator particle, pr ∼ U(0,Vmax,p)

nx

Vmax,p controls the speed at which the predator catches thebest prey




Predator-Prey PSO (cont)

The prey particles update their velocity using

vij(t + 1) = wvij(t) + c1r1j(t)(yij(t)− xij(t))

+ c2r2j(t)(yj(t)− xij(t))

+c3r3j(t)D(d)

d is Euclidean distance between prey, i , and the predatorr3j(t) ∼ U(0, 1), and D(d) = α e−βd

D(d) quantifies the influence that the predator has on theprey, growing exponentially with proximityPosition update of prey particles: If U(0, 1) < Pf , then theprey velocity update above is used, otherwise the normalvelocity update is used




Life-Cycle PSO

Life-cylce PSO is used to change the behavior of individuals[9, 10]

An individual can be in any of three phases:

a PSO particle,a GA individual, ora stochastic hill-climber

All individuals start as PSO particles

If an individual does not show an acceptable improvement infitness, it changes to the next life-cycle

First start with PSO, then GA, then hill-climbing, to havemore exploration initially, moving to more explotation




Attractive and Repulsive PSO (ARPSO)

A single swarm is used, which switch between two phasesdepending on swarm diversity

Diversity is measured using

diversity(S(t)) =1

ns

ns∑i=1

√√√√ nx∑j=1

(xij(t)− x j(t))2

where x j(t) is the average of the j-th dimension over allparticles, i.e.

x j(t) =

∑nsi=1 xij(t)

ns




Attractive and Repulsive PSO (cont)

If diversity(S(t)) > ϕmin, then switch to attraction phase

Otherwise, swarm switches to repulsion phase until athreshold diversity, ϕmax is reached

Attraction phase uses basic velocity update

Repulsion phase changes velocity update to

vij(t+1) = wvij(t)−c1r1j(t)(yij(t)−xij(t))−c2r2j(t)(yij(t)−xij(t))




Multi-Start PSO

Major problems with the basic PSO is lack of diversity whenparticles start to converge to the same point

Multi-start methods have as their main objective to increasediversity, whereby larger parts of the search space are explored

This is done by continually inject randomness, or chaos, intothe swarm

Note that continual injection of random positions will causethe swarm never to reach an equilibrium state

Methods should reduce chaos over time to ensure convergence




Craziness PSO

Craziness is the process of randomly initializing some particles[7]

What are the issues in reinitialization?

What should be randomized?When should randomization occur?How should it be done?Which members of the swarm will be affected?What should be done with personal best positions of affectedparticles?




Repelling Methods

The focus of repelling methods is to improve explorationabilities

Charged PSO:

Changes the velocity equation by adding a particleacceleration, ai :

vij(t + 1) = wvij(t) + c1r1(t)[yij(t)− xij(t)]

+ c2r2(t)[yj(t)− xij(t)]

+ aij(t)




Charged PSO (cont)

The acceleration determines the magnitude of inter-particlerepulsion:

ai (t) =ns∑

l=1,i 6=l

ail(t)

The repulsion force between particles i and l defined as

ail(t) =

(

QiQl ((xi (t)−xl (t))||xi (t)−xl (t)||3

)if Rc ≤ ||xi (t)− xl(t)|| ≤ Rp(

QiQl (xi (t)−xl (t))R2

c ||xi (t)−xl (t)||

)if ||xi (t)− xl(t)|| < Rc

0 if ||xi (t)− xl(t)|| > Rp

where Qi is the charged magnitude of particle i




Charged PSO (cont)

Neutral particles have a zero charged magnitude, i.e. Qi = 0

Inter-particle repulsion occurs only when the separationbetween two particles is within the range [Rc ,Rp]

Rc is the core radiusRp is the perception limit

The smaller the separation, the larger the repulsion betweenthe corresponding particles

The acceleration, ai (t), is determined for each particle beforethe velocity update




Binary PSO

PSO was originally developed for continuous-valued searchspaces

Binary PSO was developed for binary-valued domains

Particles represent positions in binary space

xi ∈ Bnx , xij ∈ {0, 1}

Changes in a particle’s position then basically implies amutation of bits, by flipping a bit from one value to the other




Binary PSO

Interpretation of velocity vector:

Velocity may be described by the number of bits that changeper iteration, which is the Hamming distance between xi (t)and xi (t + 1), denoted by H(xi (t), xi (t + 1))If H(xi (t), xi (t + 1)) = 0, zero bits are flipped and the particledoes not move; ||vi (t)|| = 0||vi (t)|| = nx is the maximum velocity, meaning that all bitsare flipped




Binary PSO

Interpretation of velocity for a single dimension:

Velocity is used to calculate a probability of a bit flip:

v′

ij(t) = sig(vij(t)) =1

1 + e−vij (t)

Position update changes to

xij(t + 1) =

{1 if r3j(t) < sig(vij(t + 1))0 otherwise


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Chapter 16: Particle Swarm Optimization - Computational Intelligence