Multiple-Swarm Concept in PSO

1. Parameter Settings

x₂ My Personal Best Position (pbest_i)

Global Best Position (gbest)

Inertial weight, w is large

I’m here (x_i(t))

My Velocity (v (t))

This is my new position (x_i(t+1))

53

x₁ My Velocity (v_i(t))

x₂

This is my new position (x_i(t+1))

c

₁

>> c

₂

I’m here (x_i(t)) My Velocity (v_i(t))

54

x₁

x₂ This is my new position (x_i(t+1))

p ( _i( ))

c

₂

>> c

₁

I’m here (x_i(t)) My Velocity (v_i(t))

55

x₁

z

Random inertia weight

– Experiment indicates this strategy accelerate the

convergence of particle swarm in the early time of the convergence of particle swarm in the early time of the algorithm

( )

5 2 .

0 + ⋅

= rand

w

– rand() is uniformly distributed random number within [0,1]

*Y. H. Shi, R. C. Eberhart, “Empirical Study of Particle Swarm Optimization”, Proceeding Congress on Evolutionary Computation, Piscataway, pp.1945-1949, 1999

z

Linear decreasing the inertial weight

(

_{1 2}

)

^w₂

max t

t max w t

w

w − +

×

−

=

– w₁ and w₂ are initial and final values of inertia weight

– Larger value for w facilitates global search at the beginning of the run

max t

of the run

– Smaller w encourage more local search ability near the end of the run

– Experiment indicates good performance when inertia weightExperiment indicates good performance when inertia weight descend from 0.9 to 0.4

*R. C. Eberhart, Y. Shi, “Comparing inertia weight and constriction factors in particle swarm optimization”, Proceeding Congress on Evolutionary Computation, San Diego, pp. 84-88, 2000

z

Chaotic inertia weight

– Use chaotic mapping to set inertia weight coefficient L i ti i

^{z ( t + 1 ) μ × z ( t ) ×} ( ^{1 z ( t )} )

– Logistic mapping

^{z ( t + 1 ) = μ × z ( t ) ×} ( ^{1 − z ( t )} )

• Distribution of Logistic mapDistribution of Logistic map when µ = 4

• Logistic mapping is iterated 30,000 times

Times happen to both Intervals are very high

,

• Mean times happening to interval [0.1,0.9] is 200

0.1 0.9

–

Strategy of chaotic initial weight

1. Select a random number z in the interval of (0, 1)

2 Calculate Logistic mapping z with µ = 4

2. Calculate Logistic mapping, z with µ = 4

3. Apply to either linear decreasing the inertial weight or random inertia weight

( )

E i t G d i i i k

( )

5 2 .

0 ⋅

+

×

= rand

z

( ) ^{w z}

w

max t

t max w t

w

w − + ×

×

−

=

_{1 2 2}

or

z

Experiment: Good convergence precision, quick

convergence velocity, and better global search ability

Yong Feng, Gui-Fa Teng, Ai-Xin Wang, and Yong-Mei Yao, “Chaotic inertia weight in particle swarm optimization,”

Proceeding 2^ndInternational Conference on Innovative Computing, Information and Control, Kumamoto, Japan, pp. 475-475, 2007

z

Time varying acceleration coefficients (c

₁

, c

₂

)

– Large c₁ and small c₂ in the early stage, to encourage particles to explore the search space

particles to explore the search space

– Promoted quick convergence to the optimum solution in the later stage with larger c₂ and smaller c₁

(

_{f i}

) ^c

_i

tmax c t

c

c

₁

=

₁

−

₁

+

₁

( )

^t

(

_{f i}

)

^c _i

tmax c t

c

c₂ = ₂ − ₂ + ₂

Ratnaweera A, Halgamuge S.K., and Watson H. C., “SELF-ORGANIZING HIERARCHICAL PARTICLE SWARM OPTIMIZER,” IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 8, NO. 3, JUNE 2004

2. Modifications of PSO Equation

z

Canonical PSO by Maurice Clerc

– Studied the swarm behavior using the second order differential equations

– The study shows that it is possible to determine under which conditions that the swarm will converge

– Introduces a constriction factor , to ensure convergence by restraining the velocity to guarantee convergence of the

χ

by restraining the velocity to guarantee convergence of the particles

– Observation: The amplitude of the particle’s oscillations decreases and increase depend on the distance between pbest and gbest

pbest and gbest

M. Clerc, and J. Kennedy, “The Particle Swarm Explosion, Stability,and Convergence in a Multidimensional Complex Space,” IEEE Transactionson Evolutionary Computation, Vol. 6, No. 1, February, (2002): 58-73

– Particle will oscillate around the weighted mean of pbest and gbest

– If pbest and gbest are near each other, particle will performIf pbest and gbest are near each other, particle will perform local search

– If pbest and gbest are far apart from each other, particle will perform global search

perform global search

– During the search process, particle will shift from local

search back to global search depending on pbest and gbest The constriction factor balances the need for local and

– The constriction factor balances the need for local and global search depending how the social conditions are in place

z

The update velocity equation is

( )

_⎟^⎞

⎜⎛v_i,j t +

( ) ( ( ) )

( ( ) )

^⎟⎟⎟

⎟

⎜ ⎠

⎜⎜

⎜

⎝ −

+

−

= +

t x gbest

r c

t x pbest

r c t

v

j i j

j i j

i j

i

, 2

2

, ,

1 1

, 1

χ

2κ

[ ]

where ; ; and

2

(

4

)

2

−

−

= −

φ φ φ

χ κ

^{κ ∈} [ ] ^{0 , 1}

4

2 ;

1 + >

=

φ

φ

c₁ c ₂

φ

φ

z The parameter controls the convergence speed to the point of attraction.

z If is close to zero, will be close to zero, then the resulting

κ

κ χ

velocity will be small. Small velocity encourage local search, so the convergence speed is high

z If is close to one, high exploration behavior but slowest possible convergence speed

κ

possible convergence speed

z Experiment: Even without the velocity clipping criterion the constriction factor can prevent the particles from leaving the search space and ensure convergence

search space and ensure convergence

z

Gaussian Particle Swarm Model (GPSO)

–

Observation shows the expected values for

( b ) d ( b ) 0 729 d 0 85

(pbest-x) and (gbest- x) are 0.729 and 0.85

–

A probability distribution that generates random values with expected values of [0.729 0.85]

values with expected values of [0.729 0.85]

( ) ^{t randn} ( ^{pbest x} ( ) ^t ) ^randn ( ^{gbest x ( ) t} )

v

_i,j

+ 1 =

_i,j

−

_i,j

+

_j

−

_i,j

( ) ¹ ( ) ( ) ¹

–

|randn| and |randn| are positive random numbers generated according to abs[N(0,1)]

( ) ¹

_,

( )

_,

( ) ¹

,

t + = x t + v t +

x

_{i j i j i j}

g g [ ( , )]

R. A. Krohling, “Gaussian swarm: a novel particle swarm optimization algorithm,” Proceedings of the IEEE Conference on Cybernetics and Intelligent Systems, Singapore, pp. 372-376, 2004.

3. Neighborhood Topology

z

From the standard PSO equation, the movement of particles influence by both personal best (pbest) and global best (gbest)

global best (gbest)

z

Neighborhood topology- topology of a swarm (usually replace gbest)

( y p g )

z

Neighborhood topology- How the particles in the swarm are connected with each other in terms of sharing their knowledge

sharing their knowledge

z

The convergence rate can be estimated by calculating the average distance between two particles in the neighborhood topology

particles in the neighborhood topology

z

The shorter the average distance facilitates quick

convergence speed – This means lower degree of g p g

connectivity

z

Global Topology (STAR)

– Happens in every iteration

All th ti l t d i h th t th k l d i

– All the particles are connected in such that the knowledge is shared by all particles

– Best solution found by any particles in the swarm and will i fl ll th ti l t t it ti

influence all the particles at next iteration

g

e b

f a

g

d c

z

Wheel Topology

– When particle (example: b) finds the global best, particle a, will immediately drawn into it (at next iteration)

will immediately drawn into it (at next iteration)

– Only when particle a move to the location, that it influence the rest of the particles

O it ti i d b f ll th ti l i

– One or more iterations required before all the particles in the neighborhood are influence by the global best

b f

g

a f b

g a

The rest of particles e will

d

c a

e

d

c

will

drawn into particle a

z

Ring Topology (Circle or lbest)

– Each particle is connected with K immediate neighbors

When one particle (example: b) finds the global best only its

– When one particle (example: b) finds the global best, only its immediate neighbors (i.e., a & c), will be drawn to b

– Other particles are not influenced by b until their immediate neighbors have moved towards that location

neighbors have moved towards that location

– A few iterations may required before all the particles in the neighborhood are influence by the global best

b a f

g

b a f

g

e d

c f b

e d c

f b

z

Each neighborhood has strength and weakness

z

STAR – Fast information flow, converges fast, but with potential of premature convergence

with potential of premature convergence

z

Wheel – Moderate information flow, converges more quickly but sometimes to a suboptimal point in the space

z

Circle – Slow information flow, converge slower,

favor more exploration; might have more chances to find better solutions slowly

z

There are many more neighborhood topologies and

the choice of these topologies are depended by the

the choice of these topologies are depended by the

problems (i.e., problem dependent)

4. Mutation/Perturbation Operators

z

Problem of PSO – lack of diversity and easily trapped in a local optimum.

A t ti ti l b bl t l d th th

z

A mutating particle may be able to lead the other particles away from their current position if this particle becomes the global best.

p g

z

Hence, apply mutation operators to PSO – A

strategy to enhance the exploration of the particles and to escape the local minima

and to escape the local minima.

z

Where to apply the mutation operators in PSO?

– Apply to the updated particle’s position (decision variable)

A l t th d fi d i l it th h ld

– Apply to the user defined maximum velocity threshold

z

The mutation operators are the mutation approaches use for GA or MOEA. Example:

R d t ti N if t ti N l di t ib t d

– Random mutation; Non-uniform mutation; Normal distributed mutation

– Example of Gaussian mutation used for PSO*:

where σ is set to be 0.1 times the length of the search space in

( ) ^{x x} ( ^Gaussian ( ) ^σ )

mutation = × 1 −

g

one dimension; OR σ can be set at 1.0 and linearly decreases to 0 as the iteration counts reach maximum criteria

5. Multiple-Swarm Concept in PSO

z

Improve performance by promoting exploration and diversity

– Counter its tendency of premature convergence z

Three main groups

– Improve the performance of PSO by promoting diversityImprove the performance of PSO by promoting diversity

– Solve multimodal problems

– Locate and track the optima of a multimodal problem in a dynamic environment

dynamic environment

z

Kennedy proposed using a k-means clustering

algorithm to identify the centers of different clusters of particles in the population, and then use these of particles in the population, and then use these cluster centers to substitute the personal bests

– Require a pre-specified number of iterations to determine the cluster centers and pre-specified number of clustersp p

– Not suitable for multimodal problems since the cluster

centers don’t necessarily the best-fit particles in that cluster

*J. Kennedy, “Stereotyping: improving particle swarm performance with cluster analysis,” Proceedings of Congress on Evolutionary Computation, San Diego, CA. pp. 1507-1512, 2000

–

Number of clusters and number of iterations to identify the cluster centers must be

predetermined predetermined.

Cluster A’s center performs better than performs better than all members of

cluster A, whereas

l t B’ t

cluster B’s center performs better than some and worse

than others*

z

Chen and Yu’s TPSO

– First subswarm will optimize following the global best;

second subswarm will move in the opposite direction pp

– Particle’s pbest is updated based on its local best, their corresponding subswarm’s best, and the global best collected from two subswams

– If the global best has not improved for 15 successive

iterations, the worst particles of a subswarm are replaced by the best ones of the other subswarm. Then, the subswarms switch their flight directions

switch their flight directions

G. Chen and J. Yu, “Two sub-swarms particle swarm optimization algorithm,” Proceeding of International Conference on Natural Computation, Changsha, China, pp. 515-524, 2005

z

Multi-population cooperative optimization (MCPSO)

– Based on the concept of master-slave mode

S l ti ill h t d lti l

– Swarm population will have a master swarm and multiple slave swarms

– Slave swarms explore the search space independently to i t i di it f ti l

maintain diversity of particles

– Master swarm updates via the best particles collected from the slave swarms

B. Niu, Y. Zhu, and X. He, “Multi-population cooperative particle swarm optimization,” Proceeding of European Conference on Artificial Life, Canterbury, UK, pp. 874-883, 2005

z

Speciation-based PSO (SPSO) is proposed by Parrott and Li

– Notion of species: A group of individuals sharing commonNotion of species: A group of individuals sharing common attributes according to some similarity metric

– A radius, r_s, is measured in Euclidean distance from the center of a species to its boundary

– The center of a species, species seed, is always the fittest individual in the species

– All particles that fall within the distance from the species

d l ifi d h i

seed are classified as the same species

D. Parrott and X. Li, “Locating and tracking multiple synamic optima by a particle swarm model using speciation,” IEEE Transactions on Evolutionary Computations, Vol. 10, No. 4, pp. 440-458, 2006

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