particle swarm optimization: method and applications · • particle description: each particle has...

1© Rania Hassan 3/2004

Engineering Systems Division - Massachusetts Institute of Technology

Rania HassanRania HassanPostPost--doctoral Associatedoctoral Associate

Engineering Systems DivisionEngineering Systems Division

Particle Swarm Optimization:Method and Applications

Particle Swarm Optimization:Particle Swarm Optimization:Method and ApplicationsMethod and Applications



Particle Swarm OptimizationParticle Swarm OptimizationA pseudo-optimization method (heuristic) inspired by the collective intelligence of swarms of biological populations.

Flocks of Birds Colonies of Insects



Particle Swarm OptimizationParticle Swarm OptimizationA pseudo-optimization method (heuristic) inspired by the collective intelligence of swarms of biological populations.

Schools of Fish Herds of Animals



InventorsInventors

James KennedySocial Psychologist

US Department of Labor

Russell EberhartDean of Engineering Research

Indiana Univ. Purdue Univ. Indianapolis

Introduced in 1995: Kennedy, J. and Eberhart, R., “Particle Swarm Optimization,” Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia 1995, pp. 1942-1945.



Swarming in System DesignSwarming in System Design

• Swarming theory has been used in system design. Examples of aerospace systems utilizing swarming theory include formation flying of aircraft and spacecraft.

Flocks of birds fly in V-shaped formations to reduce drag and save energy on long migrations.



Swarming in System DesignSwarming in System Design

A study of great white pelicans has found that birds flying in formation use up to a fifth less energy than

those flying solo (Weimerskirch et al.).

Weimerskirch, H. et al. "Energy saving in flight formation." Nature 413, (18 October 2001): 697 - 698.



Swarming in System DesignSwarming in System Design• A space system in formation flying combines data from several

spacecraft rather than flying all the instruments on one costly satellite.

• It also allows for the collection of data unavailable from a single satellite, such as stereo views or simultaneously collecting data of the same ground scene at different angles.

Image taken from NASA's website. http://www.nasa.gov.



Lecture OverviewLecture Overview

• Introduction• Motivation• PSO Conceptual Development• PSO Algorithm

– Basic Algorithm– Constraint Handling– Discretization

• Applications– PSO Demo– Benchmark test problems

• Unconstrained• Constrained

– System Problem: Spacecraft Design

• Final Comments on PSO• References



““Personal” MotivationPersonal” MotivationPSO

– is a zero-order, non-calculus-based method (no gradients are needed).– can solve discontinuous, mutlimodal, non-convex problems.– includes some probabilistic features in the motion of particles.– is a population-based search method, i.e. it moves from a set of points

(particles’ positions) to another set of points with likely improvement in one iteration (move). [is that good or bad ??]

Does it remind you of another heuristic?

The Genetic Algorithm (GA)– The GA is inherently discrete (in terms of handling design variables)– PSO is inherently continuous (in terms of handling design variables)

– Some researchers report that PSO requires less function evaluations than the GA (most problems studied are continuous).

– In Compindex: there are 18150 hits for the GA from 1990 to 2004,whereas there are only 105 hits for PSO - many version of PSO are likely to appear.



PSO Conceptual DevelopmentPSO Conceptual Development

• Social Behavior Simulation Optimizer

• The social model was intended to answer the following questions:

– How do large numbers of birds (or other populations exhibiting swarming behavior) produce seamless, graceful flocking choreography, while often, but suddenly changing direction, scattering and regrouping?

– Are there any advantages to the swarming behavior for an individual in a swarm?

– Do humans exhibit social interaction similar to the swarming behavior in other species?

• Keep these questions in mind while watching clips from the French documentary “Winged Migration” by Jacques Perrin.




• How do large numbers of birds produce seamless, graceful flocking choreography, while often, but suddenly changing direction, scattering and regrouping?

– “Decentralized” local processes.– Manipulation of inter-individual distances (keep pace and avoid collision).

• Are there any advantages to the swarming behavior for an individual in a swarm?

– Can profit from the discoveries and previous experience of other swarm members in search for food, avoiding predators, adjusting to the environment, i.e. information sharing yields evolutionary advantage.

• Do humans exhibit social interaction similar to the swarming behavior in other species?

– Absolutely, humans learn to imitate physical motion early on; as they grow older, they imitate their peers on a more abstract level by adjusting their beliefs and attitudes to conform with societal standards.




• Phase I:

– Randomly generated particle positions in 2-d space.

– Randomly generated velocity vectors for each particle in 2-d space.

– For each swarm movement (iteration), each particle (agent) matches the velocity of its nearest neighbor to provide synchrony.

– Random changes in velocities (craziness) are added in each iteration to provide variation in motion and “life-like” appearance – artificial.




• Phase II:– Heppner’s simulation used a roost (cornfield), that the birds

flocked around before landing there – eliminating the need for “craziness “.

– The mathematical formulation for the roosting model included:

• each particle evaluates its current fitness by comparing its position to the roost position (100, 100).

• each particle remembers its best ever fitness value, , and theposition associated with it, , and adjusts its velocity accordingly.

( ) ( )−+−= ik

ik

ik yxfkth time unit

ith particle

ifiii yxp =

randcVxVxxx

randcVxVxxxik

ik

iik

ik

ik

iik

+=∴<

−=∴>

+

+ Uniformly distributed random number

update the velocity in the y-direction in the same manner



PSO Conceptual DevelopmentPSO Conceptual Development• Phase II:

• each particle knows the position of the best particle in the current swarm, . This is the particle with the lowest fitness value, (closest to the roost).

• in the simulation, when and are set high, the flock is sucked violently into the cornfield; whereas if they are set low, the flock swirls around the cornfield and slowly approach it.

– The simulation with the known cornfield position appeared real and raised the question of how do birds find “optimal” food sources in reality without knowing the “cornfield” location?

• Example 1: scientists proved that parks in affluent neighborhoods attract more birds than parks in poor neighborhoods.

• Example 2: put a bird feeder in your balcony and see what happens in few hours.

global best

gkf

gk

gk

gk yxp =

randcVxVxxx

randcVxVxxx

ik

ik

gk

ik

ik

ik

gk

ik

+=∴<

−=∴>

+

+

c c



PSO AlgorithmPSO Algorithm



Basic PSO AlgorithmBasic PSO Algorithm

• Phase III

– The swarming behavior of the birds could be the reason for finding optimal food resources.

– The model developed in Phase II could be used (with minor modifications) to find optimal solutions for N-dimensional, non-convex, multi-modal, nonlinear functions.

– In this current basic version of PSO, craziness and velocity matching are eliminated.

Algorithm Description

• Particle Description: each particle has three features

– Position (this is the ith particle at time k, notice vector notation) – Velocity (similar to search direction, used to update the position)– Fitness or objective (determines which particle has the best value in

the swarm and also determines the best position of each particle over time.

ikxikv

( )ikf x




• Initial Swarm– No well established guidelines for swarm size, normally 15 to 30.– particles are randomly distributed across the design space.

where and are vectors of lower and upper limit values respectively.

– Evaluate the fitness of each particle and store:• particle best ever position (particle memory here is same as )• Best position in current swarm (influence of swarm )

– Initial velocity is randomly generated.

( )xxxx −+= randi

x x

( ) =−+=t

randi xxxv

ixipgp




• Velocity Update– provides search directions– Includes deterministic and probabilistic parameters.– Combines effect of current motion, particle own memory,

and swarm influence.

( )t

randct

randcw

ik

gk

ik

iik

ik ∆

−+

∆

−+=+

xpxpvv

New velocity

current motion particle memory influence

swarm influence

inertia factor

0.4 to 1.4

self confidence

1.5 to 2

self confidence

2 to 2.5




• Position Update– Position is updated by velocity vector.

• Stopping Criteria– Maximum change in best fitness smaller than specified

tolerance for a specified number of moves (S).

tik

ik

ik ∆+= ++ vxx

ik

x ik

v

gk

pip

ik

x +

( ) ( ) ,..S,qff gqk

gk =≤− − εpp



Constraint HandlingConstraint Handling

• Side Constraints– Velocity vectors can drive particles to “explosion”.– Upper and lower variable limits can be treated as regular constraints.– Particles with violated side constraints could be reset to the nearest

limit.

• Functional Constraints– Exterior penalty methods (linear, step linear, or quadratic).

– If a particle is infeasible, last search direction (velocity) was not feasible. Set current velocity to zero.

( ) ( ) [ ]( )+==

conN

iii ,grf xx φ

fitness function

objective function penalty function

penalty multipliers

( )t

randct

randc

ik

gk

ik

iik ∆

−+

∆

−=+

xpxpv



DiscretizationDiscretization

• System problems typically include continuous, integer, and discrete design variables.

• Basic PSO works with continuous variables.

• There are several methods that allows PSO to handle discrete variables.

• The literature reports that the simple method of rounding particle position coordinates to the nearest integers provide the best computational performance.



ApplicationsApplications



PSO DemoPSO Demo

• Demo provided by Gerhard Venter of Vanderplatts R&D.• Smooth function with three local minima and a known global

minimum of [0,0] and an optimal function value of 0.

Courtesy of Gerhard Venter. Used with permission.



PSO DemoPSO Demo

• Demo provided by Gerhard Venter of Vanderplatts R&D.• Noisy function with many local minima and a known global

minimum of [0,0] and an optimal function value of 1000.

Courtesy of Gerhard Venter. Used with permission.



Unconstrained Benchmark ProblemsUnconstrained Benchmark ProblemsEggcrate FunctionEggcrate Function

w

• Known global minimum of [0,0] and an optimal function value of 0.

• PSO parameters: – Swarm Size = 30– Inertia, = 0.5 (static)– Self Confidence, = 1.5– Swarm Confidence, = 1.5– Stopping Tolerance, = 0.001

( ) ( )xxxxf +++=xMinimize

ccε



Unconstrained Benchmark ProblemsUnconstrained Benchmark ProblemsEggcrate FunctionEggcrate Function

-6 -4 -2 0 2 4 6-8

-6

-4

-2

0

2

4

6Results for move0

x1

x2

-8 -6 -4 -2 0 2 4 6 8 10-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Results for move5

x1

x2

-5 -4 -3 -2 -1 0 1 2 3 4-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


x1

x2

-6 -5 -4 -3 -2 -1 0 1 2-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Results for move28

x1

x2

Statistics

Min = 15.1

Mean = 49.9

Max = 92.6

Statistics

Min = 0.3E-5

Mean = 4.69

Max = 41.9

Statistics

Min = 0.25

Mean = 26.5

Max = 86.6

Statistics

Min = 0.02

Mean = 13.6

Max = 66.5



Unconstrained Benchmark ProblemsUnconstrained Benchmark ProblemsBanana (Rosenbrock) FunctionBanana (Rosenbrock) Function

w

• Known global minimum of [1,1] and an optimal function value of 0.

• PSO parameters: – Swarm Size = 30– Inertia, = 0.5 (static)– Self Confidence, = 1.5– Swarm Confidence, = 1.5– Stopping Tolerance, = 0.001

Minimize

ccε

( ) ( ) ( )xxxf −+−=x



Unconstrained Benchmark ProblemsUnconstrained Benchmark ProblemsBanana (Rosenbrock) FunctionBanana (Rosenbrock) Function

-5 0 5-5

-4

-3

-2

-1

0

1

2

3

4

5Results for move0

x1

x2

-4 -2 0 2 4 6-2

-1

0

1

2

3

4Results for move5

x1

x2

-4 -2 0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

0.8

1


x1

x2

-5 -4 -3 -2 -1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4


x1

x2

Statistics

Min = 1.03

Mean = 1.4E4

Max = 5.3E4

Statistics

Min = 0.6E-3

Mean = 1.3E3

Max = 2.7E4

Statistics

Min = 0.26

Mean = 5.8E3

Max = 6.9E4

Statistics

Min = 0.36

Mean = 2.6E4

Max = 7.2E5



Constrained Benchmark ProblemsConstrained Benchmark ProblemsGolinski Speed ReducerGolinski Speed Reducer

w

• Full problem description can be found at http://mdob.larc.nasa.gov/mdo.test/class2prob4.html

• This problem represents the design of a simple gear box such as might be used in a light airplane between the engine and propeller to allow each to rotate at its most efficient speed.

• The objective is to minimize the speed reducer weight while satisfying a number of constraints (11) imposed by gear and shaft design practices.

• Seven design variables are available to the optimizer, and each has an upper and lower limit imposed.

• PSO parameters: – Swarm Size = 60– Inertia, = 0.5 (static)– Self Confidence, = 1.5– Swarm Confidence, = 1.5– Stopping Tolerance, = 5 kg

ccε

Image taken from NASA's website. http://www.nasa.gov.




0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

7 PSO Statistics

Move

Min

, M

ean,

and

Max

Fitn

ess

( )xf= [3.50 0.7 17 7.3 7.30 3.35 5.29]= 2985 kg

x• Known solution

• PSO solution

( )xf= [3.53 0.7 17 8.1 7.74 3.35 5.29]= 3019 kg

x




0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

7 Results for Move 0, # of Feasible Particles (out of 60) = 0

Particle number

Fitn

ess

Val

ue

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10


Particle number

Fitn

ess

Val

ue

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3x 10


Particle number

Fitn

ess

Val

ue

0 10 20 30 40 50 600

2

4

6

8

10

12

14x 10


Particle number

Fitn

ess

Val

ue



System ProblemSystem ProblemCommunication Satellite DesignCommunication Satellite Design

MissionRequirements

Acquisition Cost

Reliability

Payload / Launch Mass

Value




Combinatorial Problem discreteintegercontinuous

OptimizationAlgorithm

Performance Estimation

Tools

Multidisciplinary Framework many design variablescomputationally-expensive disciplinary design tools

Use non-calculus based search methods like the Genetic Algorithm (GA)

Focus on the conceptual design stage (less variables, simpler models)

Challenges

Solutions




Launch vehicle choice from 8 options14

E/W STK thruster technology (bi-propellant or hydrazine)13

N/S STK thruster technology (xenon plasma, arcjets, bi-propellant, and hydrazine monopropellant)

12

N/S thermal coupling (no coupling or coupling)11

Battery cell type (NiCd or NiH2)10

Solar array cell type (GaAs single junction, GaAs multi-junction, Si thin, Si normal, or hybrid Si with GaAs multi-junction)

9

HPA type (TWTA or SSPA)1 to 8

Description and discrete valuesDesign

parameter

Design VariablesThere are 27 design variables that can be combined into 2.2 trillion different designs.




Design Variables continued …

Solar array area redundancy level (0, 1.02, 1.04 or 1.06 × Area redundancy level)

27

TCR subsystem redundancy level (0 or full redundancy)26

ADCS subsystem redundancy level (0 or full redundancy)25

Propulsion subsystem redundancy level (0 or full redundancy)24

Redundancy levels for last C-band transponder that has 2 operating HPAs (0 or 1 redundant HPA)

23

Redundancy levels for C-band transponders, each has 12 operating HPAs (0, 1, 2, or 4 redundant HPAs)

16 to 22

Redundancy level for Ku-band transponder that has 12 operating HPAs (0, 1, 2, or 4 redundant HPAs)

15

Description and discrete valuesDesign

parameter

Stowed Sidewall Antenna

Stowed SolarArray

FairingDiameter

Stowed Satellite Configuration in Launch Vehicle Fairing




w

• PSO parameters: – Swarm Size = 60– Inertia, = 0.5 (static)– Self Confidence, = 1.5– Swarm Confidence, = 1.5– Stopping Tolerance, = 5 kg

ccε

( )xf= [1 0 1 1 1 1 1 1 4 2 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 3]= 3443 kg

x• Known solution

• PSO solution

( )xf= [1 1 1 0 1 1 1 1 4 2 1 1 0 1 0 2 1 1 0 1 1 0 1 0 1 1 2]= 3461 kg

x




0 10 20 30 40 50 600

2

4

6

8

10

12

14

16x 10


Particle number

Fitn

ess

Val

ue

0 10 20 30 40 50 600

1

2

3

4

5

6

7

8

9x 10


Particle number

Fitn

ess

Val

ue

0 10 20 30 40 50 600

2

4

6

8

10

12

14x 10


Particle number

Fitn

ess

Val

ue

0 10 20 30 40 50 600.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10


Particle number

Fitn

ess

Val

ue



Final Comments on PSOFinal Comments on PSO

• This is a method “in the making” - many versions are likely to appear.

• Poor repeatability in terms of:– finding optimal solution– computational cost

• More robust constraint (side and functional) handling approaches are needed.

• Guidelines for selection of swarm size, inertia and confidence parameters are needed.



ReferencesReferences

• Kennedy, J. and Eberhart, R., “Particle Swarm Optimization,” Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia 1995, pp. 1942-1945.

• Venter, G. and Sobieski, J., “Particle Swarm Optimization,” AIAA 2002-1235, 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, CO., April 2002.

• Kennedy, J. and Eberhart, R., Swarm Intelligence, Academic Press, 1st ed., San Diego, CA, 2001.

particle swarm optimization: method and applications · • particle description: each particle has...

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