pso mini tutorial - · 2004−12−15 [email protected] swarm optimisation...

2004−12−15 [email protected] Swarm optimisation

Particle Swarm optimisation: A mini tutorial


The “inventors” (1)

RussellEberhart

[email protected]


The “inventors” (2)

James Kennedy

[email protected]

Jim at work


Part 1: United we stand


Cooperation example


Memory and informers

It is thanks to these eccentrics, whose behaviour is not conform to the one of the other bees, that all fruits sources around the colony are so quickly found.Karl von Frish 1927

DPNP


Initialisation. Positions and velocities


Neighbourhoods

geographicalsocial


Psychosocial compromise

Here I am!

The best perf. of my

neighbours

My bestperf.

x g

p

v

i−proximity

g−proximity


The historical algorithm

( )( )

( ) ( )( )( ) ( )( )

1

2

1

0,

0,

d

d

d d

d d

v t

v t

rand p x t

rand g x t

α

β ϕ

β ϕ

+ =⎧⎪⎪⎨+ −⎪⎪+ −⎩

for each particle

update the

velocity

( ) ( )1)1( ++=+ tvtxtxthen move

for each component d

At each time step tRandomness

inside the

loop


Oscillations

432

1

Search space

Fitness

Initial v


The circular neighbourhood

Virtual circle

1

5

7

6 4

3

8 2Particle 1’s

3−neighbourhood


Random proximity

xg

p

v

i−proximity

g−proximity

Hyperparallelepiped => Biased

DPNP=Mayan pyra

mid


Animated illustration

Global optimum


Maths and parametersThe right way

This way

Or this way


Neighbourhoods (topologies)

“Small world” graph

or ?

(informers)


Type 1” form

2121 ''),0(),0( ϕϕϕϕϕ +=+= randrand

1 2

1 2

' '' 'p gq ϕ ϕ

ϕ ϕ+

=+

( 1) ( ( ) ( ( )))( 1) ( 1) ( )

v t v t q x tx t v t x t

χ ϕ+ = + −⎧⎨ + = + +⎩with

⎪⎪⎩

⎪⎪⎨

⎧ >−+−=

4for 42

22

κ

ϕϕϕϕ

κ

χ

else

Usual values:κ=1ϕ=4.1=> χ=0.73swarm size=20hood size=3

Non divergence c

riterion

Global constriction coefficie

nt


5D complex space

4

2

0

-2

-4

2

1

0

-1

-2

6

5

4

3

2

1

0

}

}Convergence

Non divergence

A 3D section ϕ

Re(y)Re(v)


Move in a 2D section (attractor)

10.50-0.5-1

0.8

0.6

0.4

0.2

0-0.2

-0.4

-0.6

-0.8

Im(v)

Re(v)


Beyond real numbers

0

1

2

3

4

0

1

2

3

0

1

2

3

4

1

2

3

4

5

6 0

1

2

0

1

2

3

4

8 8Bingo!


Minimun requirements

( ) ( )( ) ( ) ( )( )'',)',( xfxfxfxfHHxx ≥∨<×∈∀

( )( )( )( ) positionvelocityposition

velocitypositionposition

velocityvelocityvelocity

velocityvelocitytcoefficien

⎯→⎯

⎯→⎯

⎯→⎯

⎯→⎯

⊕

Θ

⊗

,

,

,

,o

Comparing positions in the search space H

Algebraic operators


velocity = pos_minus_pos(position1, position2)

velocity = linear_combin(α,velocity1,β,velocity2)

position = pos_plus_vel(position, velocity)

(position,velocity) = confinement(positiont+1,positiont)

Pseudo code form

} algebraic operators


Confinements

=>

=>

Frontiers (ex. : interval)

Granularity


End of Part 1


Part 2: When the algo mutates


The PSO Family

EmpiricalEmpirical CanonicalCanonical

ConstraintsConstraints

Multi−objectiveMulti−objective

WeightedWeighted

DiscreteDiscrete

HybrideHybrideCombinatorialCombinatorial

AdaptiveAdaptive

Multi−swarmMulti−swarm

Spherical,

Gaussian

Spherical,

Gaussian

PivotPivot

TRIBES

1995 1998 2000 2002 2004

OEP 0, 1, …


0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Unbiased random proximity

xg

p

v

i−proximity

g−proximity

Hyperparallelepiped => Biased

DimensionVolume

Hypersphere vs hypercube


The three balls

x

i−proximity

g−proximity

g

p


Think locally, act locally

narchy


Adaptive coefficients

The better I am the more I follow my own way

The better is my best neighbour the more I tend to go towards him

αvrand(0…b)(p-x)

Crisp or fuzzy rules


Adaptive swarm size

There has been enough improvement

but there has been not enough improvement

although I'm the worst

I'm the best

I try to kill myself

I try to generate a new particle

Crisp or fuzzy rules


TRIBES and strategies

Adaptive

information links

TRIBES Adaptive proximity

distributions (DPNP)


Energies: classical process

Rosenbrock 2D. Swarm size=20, constant coefficients.

0,00E+00

1,00E+03

2,00E+03

3,00E+03

4,00E+03

5,00E+03

6,00E+03

7,00E+03

8,00E+03

9,00E+03

1,00E+04

Number of evaluations

0

5

10

15

20

25

Potential energyKinetic energySwarm size

ε=0.00001 after 2240 evaluations

Rosenbrock 2D. Swarm size=20, constant coefficients


Energies: adaptive process

Rosenbrock 2D. Adaptive swarm size, adaptive coefficients.

0,00E+00

1,00E+03

2,00E+03

3,00E+03

4,00E+03

5,00E+03

6,00E+03

7,00E+03

8,00E+03

9,00E+03

1,00E+04

Number of evaluations

0

5

10

15

20

25

Potential energyKinetic energySwarm size

ε=0.00001 after 1414 evaluations

Rosenbrock 2D. Adaptive swarm size, adaptive coefficients


The simplest PSORandom informers Pivot method

g−proximity

gx

1

5

7

6 4

3

8 2

K=3


End of Part 2


Part 3:Story of Optimisation


Classical results

30D function PSO Type 1" Evolutionaryalgo.(Angeline 98)

Griewank [±300] 0.003944 0.4033

Rastrigin [±5] 82.95618 46.4689

Rosenbrock [±10] 50.193877 1610.359

Optimum=0, dimension=30Best result after 40 000 evaluations


Some small problems


Fifty−fifty

xi ∈ 1...N{ }i ≠ j ⇒ xi ≠ xj

xi = xiD / 2+1

D

∑1

D / 2

∑

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

N=100, D=20. Search space: [1,N]D

105 evaluations:63+90+16+54+71+20+23+60+38+15=12+48+13+51+36+42+86+26+57+79 (=450)

granularity=1


Knapsack

xi ∈ 1...N{ }i ≠ j ⇒ xi ≠ xj

xi = Si∈I , I = D, I ⊂ 1, N{ }

∑

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

N=100, D=10, S=100, 870 evaluations: run 1 => (9, 14, 18, 1, 16, 5, 6, 2, 12, 17)run 2 => (29, 3, 16, 4, 1, 2, 6, 8, 26, 5)

granularity=1


n1

n3n2

Apple trees

( ) ( )232

221 nnnnf −+−=

Evaluation n1 n2 n30 1 2 3

Swarm size=3Best position

7 7 63 11 66 4 103 0 17


Graph Coloring Problem

1

45

2

11

55

3

22

1

1

5

5

5 0

2

0

-1

4-3

-1

-1+

=

pos−plus−v

el


The Tireless Traveller

Example of position: X=(5,3,4,1,2,6)Example of velocity: v=((5,3),(2,5),(3,1))


BR17, the movie ?Structured

search space


Ecological niche

Non linear Multiobjective

Fuzzy data

Heterogeneous

t1 t2 t3 t4 t5 t6 t7

Dynamical, real time


End of Part 3


Part 4: Real applications

Medical diagnosis Industrial mixer

Electrical generator Electrical vehicle


Applications (1)

Salerno, J. Using the particle swarm optimization technique to train a recurrent neural model. IEEE International Conference on Tools with Artificial Intelligence, 1997, p. 45-49, 1997.

He Z., Wei C., Yang L., Gao X., Yao S., Eberhart R. C., Shi Y., "Extracting Rules from Fuzzy Neural Network by PSO", IEEE IEC, Anchorage, Alaska, USA, 1998.

Secrest B. R., Traveling Salesman Problem for Surveillance Mission using PSO, AFIT/GCE/ENG/01M-03, Air Force Institute of Technology, 2001.

Yoshida H., Kawata K., Fukuyama Y., "A PSO for Reactive Power and Voltage Control considering Voltage Security Assessment", IEEE TPS, vol. 15, 2001, p. 1232-1239.

Krohling, R. A., Knidel, H., and Shi, Y. Solving numerical equations of hydraulic problems using PSO. Proceedings of the IEEE CEC, Honolulu, Hawaii USA. 2002.


Applications (2)Kadrovach, B.A., and Lamont G., A particle swarm model for swarm-based networked sensor systems, ACM symposium on Applied computing, Madrid, Spain, p. 918-924, 2002

Omran, M., Salman, A., and Engelbrecht, A. P. Image classification using PSO. Proceedings of the 4th Asia-Pacific Conference on Simulated Evolution and Learning 2002 (SEAL 2002), Singapore. p. 370-374, 2002.

Coello Coello, C. A., Luna, E. H., and Aguirre, A. H. Use of PSO to designcombinational logic circuits. LNCS No. 2606, p. 398-409, 2003.

Onwubolu, G. C. and Clerc, M., "Optimal path for automated drilling operations by a new heuristic approach using particle swarm optimization," International Journal of

Production Research, vol. 4, p. 473-491, 2004.

Onwubolu G.C., TRIBES application to the flowshop scheduling problem, NewOptimization Techniques in Engineering. Heidelberg, Germany, Springer: p. 517-536, 2004.


Neuronal network

ei,1

ei,2

ei,N

.

.

.

si,1

si,P

.

.

.

s'i,1(t)

s'i,P(t)

. .

.

Transfer functions

Test Ei

Wanted output Si

Real outputS'i(t)

E = E1...Enb _tests( )X t( ) = xk, m t( )( )S = S1...Snb _ tests( )S ' t( )= S1

' t( )...Snb _tests' t( )( )

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

f X( )= S − S'Function to minimise

( ), ,,k m k mentry xτ

,

11 k mx entrye+


To know more

Clerc M., Kennedy J., "The Particle Swarm-Explosion, Stability, and Convergence in a Multidimensional Complex space", IEEE Transaction on Evolutionary Computation, 2002,vol. 6, p. 58-73

Particle Swarm Central, http://www.particleswarm.infoTHE site:

Self advert

Kennedy, J., R. Eberhart, et al. (2001). SwarmIntelligence, Morgan Kaufmann Academic Press.

2005 IEEE TEC award


More self ad.

Clerc M., "L'optimisation par essaim particulaire.Principes et pratique", Hermès, Techniques et Science de l'Informatique, 2002. Article de 25 p.

My PSO site: http://clerc.maurice.free.fr/pso/index.htm

If you read French

Clerc M., L ’optimisation par essaims particulaires. http://www.editions-hermes.fr/fr/. Parution février 2005


PSO in the world

eXtended Particle Swarms (XPS) project


Some open questions

New mathematical id

eas to

model particle intera

ctions

Metamodel

Better combinator

ial approaches

Adaptive we

ighted

relationships


Beat the swarm!

Your current posit

ion

Your best perf.

Best perf. of

the swarm


APPENDIX


Canonical form

( ) ( ) ( )( )( ) ( ) ( )

1

1 1

v t v t q x t

x t x t v t

ϕ⎧ + = + −⎪⎨

+ = + +⎪⎩

( ) ( )y t q x t= −

v t +1( )= αv t( )+ βϕy t( )y t + 1( ) = −γv t( )+ δ − ηϕ( )y t( )

⎧ ⎨ ⎩

α β

γ δ−ηϕ

( )( )

( )( )⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡++

tytv

tytv

ϕϕ

111

11

M

Eigen values e1 and e2


Constriction

( ) ( ) ( )

( ) ( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−+−

−+−−+−−+=

−+−

−+−−++−+=

ϕϕϕ

δαβγδηαηϕηϕηϕδαχ

ϕϕϕ

δαβγδηαηϕηϕηϕδαχ

42

22

42

22

2

22

2

2

22

1

Constriction coefficients


Convergence criterion

ϕ654321

3.5

3

2.5

2

1.5

1

0.5

χ1e1 < 1

χ2e2 < 1

⎧ ⎨ ⎪

⎩ ⎪ ⇐

χ1 < 1

χ2e2 < 1

⎧ ⎨ ⎪

⎩ ⎪ κ = χ2e2


Robustness

( )( ) ( )2

1212

21

1100

,

xxx

xxf

−+−

=

Performance map :

NeededIterations(κ,ϕ)

Iter


Clusters and queens

Each particle is weighted by its perf.

Dynamic clustering

Centroids = queens = temporary new “particles”


Magic Square (1)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

DDD

ji

D

mmm

mm

,1,

,

,11,1

K

MM

K

( ){ }

⎪⎪⎩

⎪⎪⎨

⎧

≠

∈

= −+

lkji

ji

Dijji

mm

Nm

xm

,,

,

1,

1L

( )

( )

0

1

1

2

11,,

1

1

2

1,1,

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

∑ ∑

∑ ∑−

= =+

−

= =+

D

j

D

ijiji

D

i

D

jjiji

mm

mm


Magic Square (2)

D=3x3, N=10010 runs13430 evaluations

10 solutions

55 30 68 42 49 62 56 74 23

30 61 53 89 32 23 25 51 68

27 96 39 73 40 49 62 26 74

22 70 58 40 75 35 88 5 57

50 43 67 58 55 47 52 62 4

43 51 78 75 33 64 54 88 30

50 65 68 69 42 72 64 76 43

18 25 59 32 53 17 52 24 26

80 3 30 22 72 19 11 38 64

65 28 64 63 55 39 29 74 54


Non linear system

( )⎪⎩

⎪⎨⎧

=−=−+

010sin01

21

22

21

xxxx

Search space[0,1]2

1 run143 evaluations

1 solution

10 runs1430 evaluations

3 solutions


Model fitting (ARMA +AIC)

Autoregressive Moving Average + Akaike's Information Criterion

∑ ∑= =

−− =n

i

m

jjtjiti ay

0 0θφ

( )N

yyN

itt ARMAdata∑

=

−= 1

2

2σ

( )( )mnnf

++=2

log 2σ

heterogenuous


A binary PSO C code

// Pivot method -----------------------------------------// Works pretty well on some problems .. and pretty bad on some othersP[s]=P_m[g]; // Initialise the new position of particle s

// at the position of the best known around

dist=log(D); // We suppose here D>=2

r=alea(1,dist); // Radius for DPNP

for (k=0;k<r;k++)// Switch at random some bits{

d=alea_integer(0,D-1);P[s][d]=1- P[s][d][d]; // Around g

}

Information links are modified at random if there has been no improvement


End of ANNEXE

pso mini tutorial - · 2004−12−15 [email protected] swarm optimisation...

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