Msdo 2015 Lecture 8 Pso

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IN THE NAME OF A

THE MOST BENEFTHE MOST MER



[email protected] ; 0321-9

_________________PhD, FLIGHT VEHICLE DESIGNBEIJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS, BUAA, P.R.CHINA, 2009

MS, FLIGHT VEHICLE DESIGNBEIJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS, BUAA, P.R.CHINA, 2006

BE, MECHANICAL ENGINEERINGNATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY, NUST, PAKISTAN, 2000

EMAIL: [email protected]

[email protected]

TEL: +92-320-9595510

WEB:

www.ist.edu.pk/qasim-zeeshan LINKEDIN: pk.linkedin.com/pub/qasim-zeeshan/67/554/ba7

Dr Qasim Zeeshan

http://pk.linkedin.com/pub/qasim-zeeshan/67/554/ba7

mailto:[email protected]


http://www.ist.edu.pk/qasim-zeeshan














MULTIDISCIPLI

SY

DOPTIMIZA

LECTURE # 8



PARTICLE SWARM

OPTIMIZATION

Dr. Qasim Zeeshan



STATUS

PHASE-I

Introduction to Multidisciplinary System Design Optimizatio

Terminology and Problem Statement

Introduction to Optimization

Classification of Optimization Problems

Numerical/ Classical Optimization

MSDO Architectures

Practical Applications: Structure, Aero etc



STATUS

PHASE-II WEEK 8: Genetic Algorithm

WEEK 9: Particle Swarm Optimization

WEEK 10: Simulated Annealing

WEEK 11: MID TERM

WEEK 12:

Ant Colony Optimization, Tabu Search, Pattern Search

WEEK 13:

LAB, Practical Applications [PLATFORMS]

00

20

40

60

-0.5

0

0.5

1



STATUS

PHASE-III WEEK 14: Design of Experiments, Meta-modeling, and Ro

WEEK 15: Multi-objective Optimization

Hybrid Optimization & Hyper Heuristic Optimiz

WEEK 16: Post Optimality Analysis/ Revision & Discussion

WEEK 17: END TERM/ Paper Presentations ?



In this LECTURE

PARTICLE SWARM OPTIMIZATION SWARM INTELLIGENCE

INTRO

HISTORY

ALGORITHM SWARM TOPOLOGY

PARAMETER SELECTION

VARIANTS, ATTRIBUTES AND EXAMPLES

APPLICATION IN MATLAB



PARTICLE SWARM OPTIMIZA _______________________

SWARM INTELL



SWARM INTELLIGENCE



SWARM INTELLIGENCE Swarm intelligence (SI) is an artificial intelligence technique bas

study of collective behavior in decentralized, self-organized systems

SI systems are typically made up of a population of simple age

locally with one another and with their environment.

Although there is normally no centralized control structure dictating

agents should behave, local interactions between such agents oftemergence of global behavior.

Examples of systems like this can be found in nature, including an

flocking, animal herding, bacteria molding and fish schooling (from W



SWARM INTELLIGENCE

Inspired by simulation social behavior.

To model human intelligence, we should model individuals in a social con

with one another.



SWARM INTELLIGENCE - APPLICA

Swarm-bots, an EU project led by Marco Dorigo, aimed to study ne

to the design and implementation of self-organizing and self-assem

(http://www.swarm-bots.org/).

A 1992 paper by M. Anthony Lewis and George A. Bekey discusseof using swarm intelligence to control nanobots within the body for

killing cancer tumors.

http://www.swarm-bots.org/













Artists are using swarm technology as ameans of creating complex interactive

environments.

Disney's The Lion King was the first movie to

make use of swarm technology (the stampede

of the bisons scene).

The movie "Lord of the Rings” has also made

use of similar technology during battle scenes.




U.S. Military is applying SI techniques to control of unmanned vehicl

NASA is applying SI techniques for planetary mapping.

Medical Research is trying SI based controls for nanobots to fight ca

SI techniques are applied to load balancing in telecommunication ne

Entertainment industry is applying SI techniques for battle and crowd




PSO - INTRODU



INTRODUCTION Inspired by simulation social behavior.

Related to bird flocking, fish schooling and swarming theory Steer toward the center

Match neighbors’ velocity

Avoid collisions

Suppose

A group of birds are randomly searching food in an area. There is only one piece of food in the area being searched.

All the birds do not know where the food is. But they know how far the food is in

So what's the best strategy to find the food? The effective one is to follow the bito the food.



INTRODUCTION

In PSO, each single solution is a "bird" in the search space. Call it "particle".

All of particles have fitness values

Which are evaluated by the fitness function to be optimized, and

have velocities Which direct the flying of the particles.

The particles fly through the problem space by followingoptimum particles.



INTRODUCTION

In computer science, particle swarm optimization (PSO)is “a computational method that optimizes a problem by

iteratively trying to improve a candidate solution with

regard to a given measure of quality” .

PSO optimizes a problem by having a population ofcandidate solutions, here dubbed particles, and movingthese particles around in the search-space according tosimple mathematical formulae over the particle'sposition and velocity.



INTRODUCTION

Each particle's movement is influenced by its local bestknown position and it's also guided toward the best

known positions in the search-space, which are updated

as better positions are found by other particles.

This is expected to move the swarm toward the best

solutions.

PSO is a metaheuristic as it makes few or no

assumptions about the problem being optimized and

can search very large spaces of candidate solutions.



INTRODUCTION

Each particle keeps track of its coordinates in the solution spaassociated with the best solution (fitness) that has achieved sparticle. This value is called personal best, pbest.

Another best value that is tracked by the PSO is the best valu

far by any particle in the neighborhood of that particle. This vglobal best, gbest.

The basic concept of PSO lies in accelerating each particle towand the gbest locations, with a random weighted accelerationstep.



INTRODUCTION

PSO does not use the gradient of the problem beingoptimized, which means PSO does not require that the

optimization problem be differentiable as is required

by classic optimization methods such as gradient descent

and quasi-newton methods.

PSO can therefore also be used on optimization

problems that are partially irregular, noisy, change over

time, etc.



INTRODUCTION

Particle swarm optimization (PSO) is a global optimization dealing with problems in which a best solution can be rep

point or surface in an n-dimensional space.

Hypotheses are plotted in this space and seeded with an init

well as a communication channel between the particles.

Particles then move through the solution space, and a

according to some fitness criterion after each time step.



INTRODUCTION

Over time, particles are accelerated towards thoseparticles within their communication grouping which have

better fitness values.

The main advantage of such an approach over other

global minimization strategies such as simulated

annealing is that the large number of members that

make up the particle swarm make the technique

impressively resilient to the problem of local minima.



INTRODUCTION

PSO was aimed to treat nonlinear optimization problems wvariables originally.

PSO has been expanded to handle combinatorial optimizaand both discrete and continuous variables as well.

One of PSO’s advantages:

Efficient treatment of mixed-integer nonlinear optimization problems (Ma small program.

http://1.bp.blogspot.com/_SHzEVRv5TV4/SqvgvZoex-I/AAAAAAAAH9Q/V0Cy6JygSN8/s1600-h/swarm_escape-708935.jpg



INTRODUCTION

“Swarm intelligence”

refers to the more general set of algorithms. Application of swarm principles to robots is called “swarm robotics”.

“Swarm prediction” has been used in the context of forecasting problem

The typical swarm intelligence system has the following properties:

It is composed of many individuals;

The individuals are relatively homogeneous (i.e., they are either all identical or thtypologies);

The interactions among individuals are based on simple behavioral rules that exploit othat the individuals exchange directly or via the environment (stigmergy);

The overall behavior of the system results from the interactions of individuals with each

environment, that is, the group behavior self-organizes.

http://1.bp.blogspot.com/_SHzEVRv5TV4/SqvgvZoex-I/AAAAAAAAH9Q/V0Cy6JygSN8/s1600-h/swarm_escape-708935.jpg



INTRODUCTION




HISTORICAL PERSP



PSO PRECURSORS

Reynolds (1987)’s simulation Boids – a simple flocking modthree simple local rules:

Collision avoidance: pull away before they crash into one anothe

Velocity matching: try to go about the same speed as their n

flock;

Flock centering: try to move toward the center of the flock as the

Heppner (1990) interests in rules that enabled large numbe

flock synchronously.



PSO’S LINK TO EVOLUTIONARY COMPU

Both PSO and EC are population based.

PSO also uses the fitness concept, but, less-fit particles do not die. Nthe fittest”.

No evolutionary operators such as crossover and mutation.

Each particle (candidate solution) is varied according to its past erelationship with other particles in the population.

Having said the above, there are hybrid PSOs, where some ECadopted, such as selection, mutation, etc.



HISTORICAL PERSPECTIVE

James Kennedy (Nov 5, 1950) is an Americansocial psychologist, best known as an originator

and researcher of particle swarm optimization.

Russell C. Eberhart , an American electricalengineer, best known as the co-developer of

particle swarm optimization concept.




The particle swarm paradigm draws on social-psychological simuin which Kennedy had participated at the University of Nintegrated with evolutionary computation methods that Eberh

working with in the 1990s.

The result was a problem-solving or optimization algorithm

principles of human social interaction. Individuals begin the program with random guesses at the problem solutio

As the program runs, the "particles" share their successes with their topoleach particle is both teacher and learner.

Over time, the population converges reliably on optimal vectors.




The first papers on the topic, by Kennedy and Russell were presented in 1995; since then more than a thous

have been published on particle swarms.

The Academic Press / Morgan Kaufmann book, Swarmby Kennedy and Eberhart with Yuhui Shi, was published




THE ALGO



PSO – ALGORITHM

Particle Swarm Optimization is a relatively recent evolutionarpopulation-based computer algorithm for problem solving.

Mechanics of PSO took inspiration from the swarming or collabo

of biological populations (flock of birds, schools of fish, and herds

Social-psychological principles form the basis of PSO, and it provi

social behavior, as well as contributing to engineering applications



PSO – ALGORITHM

PSO was originally aimed at treating nonlinear optimization continuous variables.

Moreover, PSO has been expanded to handle combinatori

problems and both discrete and continuous variables.

Unlike other optimization techniques, PSO can be realized wit

program and it requires only primitive mathematical ope

computationally inexpensive in terms of both memory requirements



PSO – ALGORITHM

Working of PSO algorithm is summarized as under: Define the problem to search and develop solution criteri

Initialize population via random initial positions and

velocities.

Determine global best position. Determine personal best position.

Update velocity and position equations.



PSO – ALGORITHM

Searching procedure by PSO can be described as follo A flock of agents optimizes an objective function.

Each agent knows its personal best value, while the bes

group, global best is also known.

New position and velocity of each agent are calculacurrent position and best values as below:

)]([)]([ 2211)1(ik

g k i

ik

ii

ik

ik

x pa x pavv



PSO – ALGORITHM

Where,

i is the particle index,

k is discrete time index,

v is the velocity of the ith particle,

x is the position of the ith particle,

pi is the best position found by the ith particle (personal best),

γ 1,2 are random numbers on the interval applied to the ith particle,

Ф is the inertia function

a1,2 are acceleration constants.

)]([)]([ 2211)1(

i

k

g

k i

i

k

i

i

i

k

i

k x pa x pavv

No



PSO – ALGORITHM

The right side of equation consists of three terms (vectors)

The first term is the previous velocity of the agent.

The second and third terms are utilized to change the velocity o

Without the second and third terms, the agent will keep on "

same direction until it hits the boundary.

Try to explore new areas and, therefore, the first term cor

diversification in the search procedure.

)]([)]([ 2211)1(

i

k

g

k i

i

k

i

i

i

k

i

k x pa x pavv



PSO – ALGORITHM

Without the first term, the velocity of the "flying" agent is only

using its current position and its best positions in history.

Try to converge to the their pbests and/or gbest and, there

correspond with intensification in the search procedure. For example, set Фmax =0.9 and Фmin =0.4.

At the beginning of the search procedure, diversification is heavily

intensification is heavily weighted at the end of the search procedu

)]([)]([ 2211)1(

i

k

g

k i

i

k

i

i

i

k

i

k x pa x pavv



PSO – ALGORITHM

Concept of modification of a searching point by PSO

X k : current searching point, X k+ 1 : modified searching point

v k : current velocity, v k +1: modified velocity

v pbest: velocity based on pbest, v gbest: velocity based on gbest



PSO – ALGORITHM

Position update is the last step in each iteration. The poparticle is updated using its velocity vector.

ik p

i pk

ik p

v x x)1()1(

Swarm

Particl

Influen

Current Motion Inf

p

i

k v1

i

k x

g k

p



PSO – ALGORITHM

In PSO, a set of randomly generated solutions (initial swarmin the design space towards the optimal solution over

iterations (moves) based on a large amount of informat

design space that is assimilated and shared by all members

Particles evolve in the search space motivated by three facto

Inertia,

Memory,

Cooperation.



PSO – ALGORITHM

Inertia implies a particle keeps moving in the direction it had previ

Memory factor influences the particle to remember the best positio

space it has ever visited.

Cooperation factor induces the particles to move closer to the bes

found by all particles.

Each particle is a candidate solution to the optimization problem

own position and velocity.



PSO – ALGORITHM (FLOW CHD

PSO O

Pop

Gene

Evalu

Modifica

No

Generation of initial condition of each agent

Initial searching points (X i0) and velocities (vi

0) of each

agent are usually generated randomly within the

allowable range.

The current searching point is set to pbest for each agent.

The best-evaluated value of pbest is set to gbest.

The agent number with the best value is stored.




PSO O

Pop

Gene

Evalu

Modifica

No

Evaluation of search point.

The objective function value is calculated for each agent.

If the value is better than the current pbest of the agent,

the pbest value is replaced by the current value.

If the best value of pbest is better than the current gbest ,

gbest is replaced by the best value and the agent number

with the best value is stored.

PSO ALGORITHM (FLOW CH




PSO O

Pop

Gene

Evalu

Modifica

No

Modification of each search point

The current searching point of each agent is changed using

PSO equations.

Stopping Criteria

The current iteration number reaches the predeterminedmaximum iteration number, then exits.

Otherwise, the process proceeds to step 2.

PSO ALGORITHM (PSEUDOC



PSO – ALGORITHM (PSEUDOCFor each particle

Initialize particle

END

DoFor each particle

Calculate fitness valueIf the fitness value is better than its peronal bestset current value as the new pBest

End

Choose the particle with the best fitness value of all as gBFor each particle

Calculate particle velocity according equation (a)Update particle position according equation (b)

End

While maximum iterations or minimum error criteria is not at




PARTICLE MOVE STR

PARTICLE MOVE STRATEGY




Key issue: For each individual, how does iwhich direction to move along?





Because the group has a common goal, so the following

reasonable:

The position and the corresponding objective function of each indiv

completely open to the public.

A particle can get two messages: The current known best position found by the entire group

The best position a particle has reached so far





Based on the “benchmark” psychology and “ego” psy

each particle has two options:

Move closer to the global best position (gbest)

Maintain its own best position (pbest)

i

gbest Global best position s

pbest Personal best positi

？

？





How to determine the weights of the two options?

Determine by God (according to probability)

Generate two random numbers as weights

r1 ∈ [0, 1], r

2 ∈ [0, 1]

r1 indicates the extent to which a particle expects to maintain its own b

r2 indicates the extent to which a particle expects to move closer to

position.




SWARM TOP

SWARM TOPOLOGY



SWARM TOPOLOGY

There have been two basic topologies used in the literatu

Ring Topology Star Topology

I4

I0

I1

I2I3

I4

I0

II3

SWARM TOPOLOGY



SWARM TOPOLOGY RING TOPOLOGY

Also known as circle topology. In the Circle topology, parts of the population that are

distant from one another are also independent of oneanother, but neighbors are closely connected.

Thus one segment of the population might converge on alocal optimum, while another segment converges on a

different optimum or keeps searching. Influence spreads from neighbor to neighbor in this

topology, until, if an optimum really is the best found byany part of the population, it will eventually pull all theparticles in.

Circles were defined with k = 2.

I4

I3

SWARM TOPOLOGY



SWARM TOPOLOGY

STAR TOPOLOGY

Also known as wheel topology. This topology effectively isolates individuals from one

another, as all information has to be communicated

through the focal individual.

This focal individual compares performances of all

individuals in the population and adjusts its trajectorytoward the very best of them.

If adjustments result in improvement in the focal

individual’s performance, then that performance is

eventually communicated to the rest of the population.

I4

I3

SWARM TOPOLOGY



SWARM TOPOLOGY

STAR TOPOLOGY

Thus the focal individual serves as a kind of buffer orfilter, slowing the speed of transmission of good solutions

through the population.

It should be noted that the highly centralized STAR is acommon configuration for many business and government

organizations.

The buffering effect of the focal particle should prevent

overly rapid convergence on local optima.

It is a way to preserve diversity of potential problem

solutions, though it was expected that it might entirely

destroy the ability of the population to collaborate.

I4

I3




PARAMETER SEL

PARAMETER SELECTION



PARAMETER SELECTION

PARAMETER SELECTION



PARAMETER SELECTION

The choice of PSO parameters though simple but can have a

on optimization performance.

Selecting PSO parameters that yield good performance been the subject of much research.

The PSO parameters can also be tuned by using anothoptimizer, a concept known as meta-optimization.

Parameters have also been tuned for various optimization sc

PARAMETER SELECTION



PARAMETER SELECTION

The parameter selection for PSO is relatively simple (so

involving two categories: Inertia coefficient

Accelerating coefficients

Eberhart et al. tried to examine the parameter selecAccording to their examination, the following parameters arand the values do not depend on problems:

a1 = a2= 2.0, Фmax = 0.9, Фmin = 0.4

PARAMETER SELECTION



PARAMETER SELECTION

Parameter selection: different opinion

Inertia coefficient can be set to decrease linearly to 0 in the ite

Itermax is the predefined maximum iteration number,

η 1 is a positive real number (the value of η depends on spec

iter is the current iteration number.

max

max.iter

iter iter

PARAMETER SELECTION



PARAMETER SELECTION

The control parameter “acceleration constant,” turns out to be ve

determining the type of trajectory the particle travels.

If a = 0.0, it is obvious that v = v + 0, and as x = x + v it s

linearly.

If a is set to a very small value, the trajectory of x rises and falls s

The accelerating coefficients a1 and a2 satisfy

a1 + a2 = 4.0

a1 = a2= 2.0

PARAMETER SELECTION



PARAMETER SELECTION

INERTIA WEIGHT:

Large inertia weight facilitates global exploration, while smal

facilitates local exploration.

Inertia weight must be selected carefully and/or decreased ov

By linearly decreasing the inertia weight from a relatively la

small value through the course of the PSO run gives the best PScompared with fixed inertia weight settings.

Inertia weight seems to have attributes of temperature in simula

PARAMETER SELECTION



PARAMETER SELECTION

MAX VELOCITY:

An important parameter in PSO; typically the only one ad

Clamps particles velocities on each dimension.

Determines “fineness” with which regions are searched

If too high, can fly past optimal solutions.

If too low, can get stuck in local minima.

PARAMETER SELECTION (LITERAT



PARAMETER SELECTION (LITERAT

Eberhart R., Kennedy J. A New Optimizer Using Particle Swa

Proceedings of the Sixth International Symposium on MicroHuman Science[C], 1995, 10: 39-43

Eberhart R. , Simpson P., Dobbins R. Computational Intellig[M]. Boston, MA: Academic. 1996: 212-226

Van Den Bergh F., Engelbrecht A.P. A Cooperative ApproaSwarm Optimization [J]. IEEE Transactions on Evolutionary2004, 8(3): 225 – 239

AN ILLUSTRATIVE EXAMPLE




Problem: min f (x )=x 2

Assume there are four individuals in the group: x 1, x 2,

Set

a 1 = a 2 = 2

iter max = 10

Ф

= (iter max - iter )/ iter max = (10- iter ) / 10

v 10 = v 2

0 = v 30 = v 4

0 = 0





Iteration 0: generate four initial agents (solutions)

iteration

0

1

2

x 1 f (x 1) 8

64

pbest 1 f (pbest 1)

x 2 f (x 2) 3 9

pbest 2 f (pbest 2)

x 3

f (x 3)

2

4

pbest 3 f (pbest 3)

x 4 f (x 4) 6 36

pbest 4 f (pbest 4)

gbest f (gbest )




The best solutions for individuals and group can b

iteration

0

1

2

x 1 f (x 1) 8

64

pbest 1 f (pbest 1) 8

64

x 2 f (x 2) 3 9


9

x 3

f (x 3)

2

4


4

x 4 f (x 4) 6 36


36

gbest f (gbest ) 2

4




Iteration 1: assume r 1 = 0.2, r 2 = 0.8

Inertia coefficient Ф = (10-1)/10 = 0.9

v 11 = Ф·v 1

0+a 1·r 1·(pbest1-x 10)+a 2·r 2·(gbest-x 1

0)

= 0.9*0 + 2*0.2*(8-8) + 2*0.8*(2-8)

= -9.6

x 11= x 1

0+v 11 = 8-9.6 = -1.6

v 21 =

Ф·v 20+a 1·r 1·(pbest2-x 20)+a 2·r 2·(gbest-x 20)

= 0.9*0 + 2*0.2*(3-3) + 2*0.8*(2-3)

= -1.6

x 21= x 2

0+v 21 = 3-1.6= 1.4




v 31 = Ф·v 3

0 + a 1·r 1·(pbest3-x 30) + a 2·r 2·(gbest-x 3

0)

= 0.9*0 + 2*0.2*(2-2) + 2*0.8*(2-2)= 0

x 31 = x 3

0+v 31 = 2+0 = 2

v 41= Ф·v 4

0+ a 1·r 1·(pbest4-x 40) + a 2·r 2·(gbest-x 4

0)

=0.9*0 + 2*0.2*(6-6) + 2*0.8*(2-6)

= -6.4

x 41= x 4

0+v 41 = 6 - 6.4 = -0.4




The solutions corresponding to individuals at itera

iteration

0

1

2

x 1 f (x 1) 8

64

-1.6 2.56


64

x 2 f (x 2) 3

9

1.4 1.96

pbest 1 f (pbest 1) 3 9

x 3

f (x 3)

2

4

2 4


4

x 4 f (x 4) 6

36

-0.4 0.16


36

gbest f (gbest ) 2

4




The best solutions for individuals and group are u

iteration

0

1

2

x 1 f (x 1) 8

64

-1.6 2.56


64

-1.6 2.56

x 2 f (x 2) 3

9

1.4 1.96

pbest 1 f (pbest 1) 3 9 1.4 1.96

x 3 f

(x

3)

2

4

2 4


4

2 4

x 4 f (x 4) 6

36

-0.4 0.16


36

-0.4 0.16

gbest f (gbest ) 2

4

-0.4 0.16




Iteration 2: assume r 1= 0.3, r 2 = 0.5

Inertia coefficient Ф = (10-2) / 10 = 0.8

v 12= Ф·v 1


1)

= 0.8*(-9.6) + 2*0.3*(-1.6-(-1.6)) + 2*0.5*(-0.4-(-1.6))

= -6.48

x 12= x 1

1+v 12 = -1.6 - 6.48 = -8.08

v 22=Ф·v 21+a 1·r 1·(pbest2-x 21)+a 2·r 2·(gbest-x 21)

=0.8*(-1.6) + 2*0.3*(1.4-1.4) + 2*0.5*(-0.4-1.4)

= -3.08

x 22= x 2

1+v 22 = 1.4 - 3.08 = -1.68




v 32= Ф·v 3


1)

= 0.8*0 + 2*0.3*(2-2) + 2*0.5*(-0.4-2)

= -2.4

x 32= x 3

1+v 32 = 2 - 2.4 = -0.4

v 42= Ф·v 4


1)

= 0.8*(-6.4) + 2*0.3*(-0.4-(-0.4)) + 2*0.5*(-0.4-(-0.4))

= -5.12

x 42= x 4

1+v 42 = -0.4 - 5.12 = -5.52





iteration

0

1

2

x 1 f (x 1) 8

64

-1.6 2.56 -8.08 65.29


64

-1.6 2.56 -1.6 2.56

x 2 f (x 2) 3 9 1.4 1.96 -1.68 2.82


9

1.4 1.96 1.4 1.96

x 3

f (x 3)

2

4

2 4 -0.4 0.16


4

2 4 -0.4 0.16

x 4 f (x 4) 6 36 -0.4 0.16 -5.52 30.47


36

-0.4 0.16 -0.4 0.16

gbest f (gbest ) 2

4

-0.4 0.16 -0.4 0.16




v 33= Ф·v 3


2)

= 0.7*(-2.4) + 2*0.5*(-0.4-(-0.4)) + 2*0.6*(-0.4-(-0.4))

= -1.68

x 33= x 3

2+v 33 = -0.4 - 1.68 = -2.08

v 43= Ф·v 4


2)

= 0.7*(-5.12) + 2*0.5*(-0.4-(-5.52)) + 2*0.6*(-0.4-(-5.52

= 7.68

x 43= x 4

2+v 43 = -5.52 + 7.68 = 2.16





iteration

0

1

2

x 1 f (x 1) 8

64

-1.6 2.56 -8.08 65.29


64

-1.6 2.56 -1.6 2.56

x 2 f (x 2) 3

9

1.4 1.96 -1.68 2.82


9

1.4 1.96 1.4 1.96

x 3

f (x 3)

2

4

2 4 -0.4 0.16


4

2 4 -0.4 0.16

x 4 f (x 4) 6

36

-0.4 0.16 -5.52 30.47

pbest 1 f (pbest 1) 6 36 -0.4 0.16 -0.4 0.16

gbest f (gbest ) 2

4

-0.4 0.16 -0.4 0.16




Iteration process

0

5

10

15

20

25

30

1 2 3 4 5

Average

Best

PSO – SIMULATION



x

y

PSO – SIMULATION



x

y

SCHWEFEL'S FUNCTION



420.9=

4=)(

minimumglobal

500

where

)()(1

i

i

n

i

i

x

n x f

x

x x f

EVOLUTION INITIALIZATION



EVOLUTION 5 ITERATION




VARIANTS OF PSO – DISCRE

VARIANTS OF PSO



Discrete PSO ……………… can handle discrete

binary variables

MINLP PSO………… can handle both discrete

binary and continuous variables.

Hybrid PSO…………. Utilizes basic mechanism of

PSO and the natural selection mechanism, which is

usually utilized by EC methods such as GAs.

DISCRETE PSO



The original PSO treats nonlinear optimization pro

continuous variables.

Practical management and engineering problems

formulated as combinatorial optimization problems.

Kennedy and Eberhart developed a discrete version o

combinatorial optimization problems.

DISCRETE PSO



Kennedy and Eberhart proposed a model wherein the probability of an a

yes or no, true or false, or making some other decision is a function of

social factors as follows:

The parameter v, an agent’s tendency to make one or the other choice, wil

probability threshold.

If v is higher, the agent is more likely to choose 1, and lower values favor 0

1( 1) ( , , ,k k k

i i i i p x f x v pbest gbe

DISCRETE PSO



Such a threshold requires staying in the range [0, 1]. One of

accomplishing this feature is the sigmoid function.

The agent’s tendency should be adjusted for success of the agent a

In order to accomplish this, a formula for each vik that will be so

the difference between the agent’s current position and the best p

so far by itself and by the group should be developed.

1( )

1 exp( )

k

i k

i

sig vv

DISCRETE PSO



Like the basic continuous version, the formula for the binary versio

be described as follows:

r1 and r2 are positive random numbers with a uniform distribution,

vector of random number of [0, 1].

1

1 1 2 2( ) ( )k k k k

i i i i iv v c r pbest x c r gbest x

1 1 1

1

if ( ) then 1

else 0

k k k

i i i

k

i

sig v x

x

DISCRETE PSO



These formulas are iterated repeatedly over each dimension of eac

vik can be limited so that sig(vi

k ) does not approach too closely to 0

This ensures that there is always some chance of a bit flipping.

A constant parameterV

max (limited value ofv

i

k

) can be set at the stIn practice, V max is often set in [-4.0, +4.0].

The entire algorithm of the binary version of PSO is almost the sa

the basic continuous version except for the state equations.

PARTICLE SWARM OPTIMIZA




_______________________VARIANTS OF PSO – PS

CONSTRICTION FACTO

PSO WITH CFA



When using PSO, it is possible for the magnitude of the velocities t

large.

Performance can suffer if V max is inappropriately set.

Two methods were developed for controlling the growth of velocitie

A dynamically adjusted inertia factor, and

A constriction coefficient.

PSO WITH CFA



The basic system equation of PSO

can be considered as a kind of difference equation.

The system dynamics, that is, the search procedure, can be a

eigenvalues of the difference equation.

1

1 1 2 2( ) (

k k k k

i i i i iv wv c r pbest X c r gbest X

1 1k k k

i i i X X v

max minmax

max

w ww w k

iter

PSO WITH CFA



Using a simplified state equation of PSO, Clerc and Kennedy devPSO by eigenvalues.

The velocity of the constriction factor approach (simplest constexpressed as

and K are coefficients. If =4.1, then K=0.729.

As increases above 4.0, K gets smaller. For example, If =5.0, then K=0

)]()([ 2211

1 k

i

k

ii

k

i

k

i X gbest r c X pbest r cv K v

4 , where,42

221

2

cc K

PSO WITH CFA



The convergence characteristics of the system can be c

.

Clerc et al. found that the system behavior can be contr

the system behavior has the following features:

The system does not diverge in a real-valued region and finally

The system can search different regions efficiently by avoi

convergence.

PSO WITH CFA



The whole PSO algorithms by WEIGHTS and CFA are the

that CFA utilizes a different equation for calculation of veloc

PSO with CFA ensures the convergence of the search proc

on mathematical theory.

PSO with CFA can generate higher-quality solutions for so

than PSO with WEIGHTS.

SOME OTHER VARIANTS



Tribes (Clerc, 2006) – aims to adapt population size, so that it does not have to be

ARPSO (Riget and Vesterstorm, 2002) – uses a diversity measure to alternate betw

Dissipative PSO (Xie, et al., 2002) – increasing randomness;

PSO with self-organized criticality (Lovbjerg and Krink, 2002) – aims to improve

Self-organizing Hierachicl PSO (Ratnaweera, et al. 2004);

FDR-PSO (Veeramachaneni, et al., 2003) – using nearest neighbour interactions;

PSO with mutation (Higashi and Iba, 2003; Stacey, et al., 2004)

Cooperative PSO (van den Bergh and Engelbrecht, 2005) – a cooperative approach

DEPSO (Zhang and Xie, 2003) – aims to combine DE with PSO;

CLPSO (Liang, et al., 2006) – incorporate learning from more previous best partic



PARTICLE SWARM OPTIMIZA _______________________SOME ATTRIBUTES O

ATTRIBUTES OF PSO Like GA PSO is population based stochastic optimization



Like GA, PSO is population based stochastic optimization.

Algorithms starts with a group of a randomly generated population

PSO has fitness values to evaluate the population.

Update the population and search for the optimium with random tec

PSO does not have genetic operators like crossover and mutation. Pthemselves with the internal velocity.

They also have memory, which is important to the algorithm.

ATTRIBUTES OF PSO Particles do not die.



The information sharing mechanism in PSO is significantly different Info from best to others.

PSO has a memory NOT “what” that best solution was, but “where” that best solution was

Quality: Population responds to quality factors pbest and gbest .

Diverse Response: Responses allocated between pbest and gbest .

Stability: Population changes state only when gbest changes.

ATTRIBUTES OF PSO Adaptability: Population does change state when gbest changes.



p y p g g g

There is no selection in PSO All particles survive for the length of the run. PSO is the only EA that does not remove candidate population members.

In PSO, topology is constant; a neighbor is a neighbor.

Simple in concept

Easy to implement

Computationally efficient.



PARTICLE SWARM OPTIMIZA _______________________PITFALLS O

PITFALLS OF PSO

Tendency to cluster very quickly



Tendency to cluster very quickly

Reinitialization

Use multiple velocity update strategies

Particles may move into infeasible region

Disregard the particles

Modify or repair the particle to move it back into feasible region

Problem specific



PARTICLE SWARM OPTIMIZA _______________________OPTIMIZATION IN DYNAMIC ENVIRO

OPTIMIZATION IN DYNAMIC ENVIRO

Many real-world optimization problems are dynamic and require



y p p y qalgorithms capable of adapting to the changing optima over time.

E.g., Traffic conditions in a city change dynamically

and continuously. What might be regarded as an

optimal route at one time might not be optimal in

the next minute.

In contrast to optimization towards a static optimum, in a dynamic egoal is to track as closely as possible the dynamically changing optim

OPTIMIZATION IN DYNAMIC ENVIRO

WHY PSO??????????????????



With a population of candidate solutions, a PSO algorithm can

information about characteristics of the environment. PSO, as characterized by its fast convergence behavior, has an in-built

to a changing environment.

Some early works on PSO have shown that PSO is effective for locatoptima in both static and dynamic environments.

Two major issues must be resolved when dealing with dynamic prob

How to detect that a change in the environment has actually occurred?

How to respond appropriately to the change so that the optima can still



PARTICLE SWARM OPTIMIZA _______________________EXA

EXAMPLE 1: COMBINATORIAL OPTIMIZ

Combinatorial optimization problem



Combinatorial optimization problem

Issue: How to ensure x take integers in the iteration process?

Use the binary version of PSO, represent x as a binary string.

2 21 2

1

2

1 2

min ( ) ( 1) ( 2)

. . 0 6

0 10

, are integers

f x x x

s t x

x

x x


Choose the length of binary string



g y g

For x 1

For x 2

The binary length for (x 1, x 2) is 3+4=7

For Instance1 0 0 0 1 0 1

x 1=4 x 2=5

1 12 2 2

6 0log log log 6

accuracy required 1

b alength

2 22 2 2

10 0log log log 1

accuracy required 1

b alength


Suppose the population size is four.



Initialization

Binary Integer objective function

1 0 0 1 0 1 1 (4, 11) f =90

0 1 1 1 1 1 0 (3,12) f =104

0 0 0 0 1 1 1 (0,7) f =26

1 1 1 0 0 1 1 (7,3) f =37 Set v 1

0= v 20=v 3

0=v 40=(0 0 0 0 0 0 0)

gbest=(0 0 0 0 1 1 1)


Iteration: iter=1




Agent1 0 1 0 0 0 0 0 (2,0) f =5Agent2 1 0 0 1 0 0 1 (4,11) f =90

Agent3 1 0 1 0 1 0 1 (5,5) f =25

Agent4 0 0 0 0 1 0 1 (0,7) f =26

pbest1 0 1 0 0 0 0 0 (2,0) f =5

pbest2 1 0 0 1 0 0 1 (4,11)f =90

Pbest3 1 0 1 0 1 0 1 (5,5) f =25

pbest4 0 0 0 0 1 0 1 (0,7) f =26

gbest 0 1 0 0 0 0 0 (2,0) f =5


Iteration: iter=2




Agent1 0 1 0 0 0 0 0 (2,0) f =5Agent2 1 1 0 0 1 0 1 (6,7) f =50

Agent3 0 0 0 0 0 0 0 (0,0) f =5

Agent4 0 0 0 0 1 0 1 (0,5) f =10

pbest1 0 1 0 0 0 0 0 (2,0) f =5

pbest2 1 1 0 0 1 0 1 (6,7) f =50

pbest3 0 0 0 0 0 0 0 (0,0) f =5

pbest4 0 0 0 0 1 0 1 (0,5) f =10

gbest 0 0 0 0 0 0 0 (0,0) f =5


Iteration: iter=10




Agent1 0 0 1 0 0 1 0 (1,2) f =0Agent2 0 0 0 0 0 1 0 (0,2) f =1

Agent3 0 0 1 0 0 1 0 (1,2) f =0

Agent4 0 0 1 0 1 1 0 (1,4) f =4

pbest1 0 0 1 0 0 1 0 (1,2) f =0

pbest2 0 0 0 0 0 1 0 (0,2) f =1

pbest3 0 0 1 0 0 1 0 (1,2) f =0

pbest4 0 0 1 0 1 1 0 (1,4) f =4

gbest 0 0 1 0 0 1 0 (1,2) f =0




0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9

iter

f (x )

EXAMPLE 2: WEAPON TARGET ALLOCA

Weapon Target Allocation is a reactive assignment of



p g g

engage incoming missile threats.

Decision making on deployment of various types of defending

Multilayer is complex due to various modern and sophistica

weapons, different types of defending weapons required to

attacking weapons so as to increase the survivability of the assets

spent for procurement, deployment and operation of such defend

Manning of these weapons, area availability at the assvalues/preference/priorities set for each asset by decision maker

asset more than the other etc.


Mathematical Model



Let

D = Types of defending weapons availableS = Number of assets

A = Types of attacking weapons

k dsa = Probability of successful interception by one defentype d deployed to defend an asset s against an attactype a

xdsa = Number of defending weapons of type d that are deploattacking weapon of type a to defend asset s (defense p

n sa = Number of attacking weapons of type a aimed at asset

g sa = The probability that a single attacking weapons of typasset s when it is able to penetrate the defending weap


v s = Value of asset s



cd = Cost of operating one defending weapon of type d

md = Manpower required per defending weapon of type d

Bd = Number of defending weapons of type d

Ra = Number of attacking weapons of type a

G s = Ground area available at asset s

t d = Ground area required by a defending weapon of type dC max = Maximum operating cost of weapons deployed

M maxd = Maximum available manpower to operate defending w


The probability that weapons deployed in d th layer w



p y p p y y

able to intercept a single attacking weapon of type a ogiven by

The probability that a single attacking weapon of typ

intercepted by any layer on asset s is given by

(1 )dsa

sa

x

n

dsak

1

(1 )dsa

sa

x D

n

dsad

k


The probability that a single attacking weapon of type



p y g g p yp

the asset s is given by

The survival probability of asset s by multiple layer deattacked by all types of attacking weapons is given by

1

(1 )dsa

sa

x D

n

dsa sad

k g

1 1

( ) 1 (1 )

sadsa

sa

n x

A Dn

dsa saa d

H s k g


The objective from the defending side is to maximi



j g

expected value of the surviving assets which is given by

From the attacking side, the problem is to minimizobjective function.

1

( )S

s s

v H s


Constraints



Weapon availability

Area availability

Cost

Manpower

1 1

, 1, , D A

d dsa sd a

t x G s S

max1 1 1

D S A

d dsad s a c x C

max1 1

, 1, ,S A

d dsa d s a

m x M d D

1 1

, 1, ,S A

dsa d s a

x B d D


This is an Integer Nonlinear Programming prob



g g g p

constraints.

S.t.1 1

, 1, , D A

d dsa sd a

t x G s S

max1 1 1

D S A

d dsad s a

c x C

max1 1

, 1, ,S A

d dsa d s a

m x M d D

1 1

, 1, ,S A

dsa d s a

x B d D

1 1 1

max 1 (1 )

sadsa

sa

n x

A DS n s dsa sa

s a d

v k g


By constructing a penalty function, transform i



y g p y

optimization problem without constraint

1 1 1

2

1 1 1

2

1 1 1

1 1 1

max ( ) 1 (1 )

max[0, ]

max[0, ]

max[0,

sadsa

sa

n x

A DS n

s dsa sa s a d

D S A

d dsa d d s a

S D A

s d dsa s s d a

S A

d dsad s a

f x v k g

x B

t x G

c x

2

max

2

max1 1 1

]

max[0, ]

D

D S A

d d dsa d d s a

C

m x M


Consider that two types of weapons are available to defend three a



two types of attacking weapons.

The maximum number of available defending weapons of the first

and that of the second type is 50.

B1=100, B2=50

The number of attacking weapons of the first and the second type a

respectively.

R1=50, R2=29


The values of the first, second and third assets are 40



200 respectively

v1=400, v2=300, v3=200

Assume the attack plan is known, the allocation o

weapons is as follows:

n11=5, n12=9, n21=25, n22=7, n31=20, n32=13

n11+n21+n31=5+25+20=50, n12+n22+n32=9+7+13=29


Effectiveness of defending weapons and damage probabilitie

weapons



pDefending weapon type (d ) Asset (s ) Attacking weapon type (a ) k dsa

1

1

1

0.20

2 1 1 0.60

1 1 2 0.35

2 1 2 0.50

1 2 1 0.25

2 2 1 0.50

1 2 2 0.20

2 2 2 0.45

1 3 1 0.35

2 3 1 0.45

1 3 2 0.25

2 3 2 0.65


Problem : Determine an optimal defense plan against the k



plan

Consider the constraint of weapon availability alone. C

penalty function

Set 1= 2=100

2 23 2 3

1 1 11 1

max ( ) 1 (1 ) max[0,

sadsa

sa

n x

n

s dsa sa d s d s a d

f x v k g


Binary coding



Since B1=100, so 0 x111, x112, x121, x122, x131, x132 100

Since B2=50, so 0 x211, x212, x221, x222, x231, x232 50

A solution x=( x111, x112, x121, x122, x131, x132 , x211, x212, x221, x222, x231, x2

encoded as a 78-bit binary string (7 bits6+6 bits 6= 78)

2 2log log 100 6.64 (take 7)accuracy required

b alength

2 2log log 50 5.64 (take 6)

accuracy required

b alength


For instance



1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1

indicates

x=( x111, x112, x121, x122, x131, x132 , x211, x212, x221, x222, x231, x232 )

=(92, 106, 62, 121, 102, 16, 47, 56, 63, 36, 14, 19)

Set Number of agents=10, iter max=100


800



-600

-400

-200

0

200

400

600

800

0 10 20 30 40 50 60 70 80 90

Iter

f (x )


Optimal solution



x 111= 2 x 112=40 x 121= 8 x 122= 10 x 131=34 x 132= 6 ( =100)

x 211=3 x 212=8 x 221= 25 x 222= 0 x 231=2 x 232= 12 ( =50)

Optimal objective function: 544.8

PSO



THE FUTURE?



REFERENCE BOOKS

Swarm Intelligence

/



James Kennedy, Russell C Eberhart, Academic Press /

Morgan Kaufmann, 2001

Practical Optimization: Algorithms and Engineering

Applications

Andreas Antoniou and Wu-Sheng Lu 2007





_______________________PSO IN M

1）Sphere Function

STANDARD BENCHMARK FUNCT



n

n

i

i x x x f 5,5,1

2

It is continuous, convex, uni-modal and

one of the simplest benchmark tests.

Also known as DeJong’s function.

1

222

1 10,10,1100n

n

iii x x x x x f

2）Rosenbrock Function




1i

Rosenbrock function is a classic

optimization problem, also known as banana

function or the second function of de Jong.

The global minimum is present inside a long,

narrow, parabolic-shaped flat valley.

To locate the valley is trivial, however,convergence to the global optimum is difficult

and hence it has been frequently used to assess

the performance of optimization algorithms.

D

2

3）Rastrigin Function




i

ii x x x f

1

2 102cos10

Rastrigin's function is often used to evaluate

global optimizers.

This function is comparatively a difficult

problem because of its large search space and

the large number of local minima.

The function is highly multi-modal, and the

locations of the minima are regularly

distributed.

4）Ackley Function

nn

i

n

x xn

xn

e x f 32,32,2cos1

exp1

2.0exp2020 2




ii nn 11

The Ackley problem has several local

minima but only one global minimum.

It is a widely used multi-modal test function.



THANK YOU FOR YOUR INT

Msdo 2015 Lecture 8 Pso

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