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7/23/2019 Msdo 2015 Lecture 8 Pso
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IN THE NAME OF A
THE MOST BENEFTHE MOST MER
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[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]
TEL: +92-320-9595510
WEB:
www.ist.edu.pk/qasim-zeeshan LINKEDIN: pk.linkedin.com/pub/qasim-zeeshan/67/554/ba7
Dr Qasim Zeeshan
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MULTIDISCIPLI
SY
DOPTIMIZA
LECTURE # 8
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PARTICLE SWARM
OPTIMIZATION
Dr. Qasim Zeeshan
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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
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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
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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 ?
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In this LECTURE
PARTICLE SWARM OPTIMIZATION SWARM INTELLIGENCE
INTRO
HISTORY
ALGORITHM SWARM TOPOLOGY
PARAMETER SELECTION
VARIANTS, ATTRIBUTES AND EXAMPLES
APPLICATION IN MATLAB
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PARTICLE SWARM OPTIMIZA _______________________
SWARM INTELL
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SWARM INTELLIGENCE
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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
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SWARM INTELLIGENCE
Inspired by simulation social behavior.
To model human intelligence, we should model individuals in a social con
with one another.
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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.
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SWARM INTELLIGENCE - APPLICA
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.
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SWARM INTELLIGENCE - APPLICA
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
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PARTICLE SWARM OPTIMIZA _______________________
PSO - INTRODU
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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INTRODUCTION
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PARTICLE SWARM OPTIMIZA _______________________
HISTORICAL PERSP
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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.
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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.
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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.
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HISTORICAL PERSPECTIVE
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.
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HISTORICAL PERSPECTIVE
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
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PARTICLE SWARM OPTIMIZA _______________________
THE ALGO
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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PSO – ALGORITHM (FLOW CHD
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
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PSO – ALGORITHM (FLOW CHD
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
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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
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PARTICLE SWARM OPTIMIZA _______________________
PARTICLE MOVE STR
PARTICLE MOVE STRATEGY
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PARTICLE MOVE STRATEGY
Key issue: For each individual, how does iwhich direction to move along?
PARTICLE MOVE STRATEGY
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PARTICLE MOVE STRATEGY
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
PARTICLE MOVE STRATEGY
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PARTICLE MOVE STRATEGY
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
?
?
PARTICLE MOVE STRATEGY
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PARTICLE MOVE STRATEGY
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.
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PARTICLE SWARM OPTIMIZA _______________________
SWARM TOP
SWARM TOPOLOGY
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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
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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
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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
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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
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PARTICLE SWARM OPTIMIZA _______________________
PARAMETER SEL
PARAMETER SELECTION
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PARAMETER SELECTION
PARAMETER SELECTION
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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
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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
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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
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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
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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
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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
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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
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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
AN ILLUSTRATIVE EXAMPLE
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AN ILLUSTRATIVE EXAMPLE
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 )
AN ILLUSTRATIVE EXAMPLE
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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
pbest 1 f (pbest 1) 3
9
x 3
f (x 3)
2
4
pbest 1 f (pbest 1) 2
4
x 4 f (x 4) 6 36
pbest 1 f (pbest 1) 6
36
gbest f (gbest ) 2
4
AN ILLUSTRATIVE EXAMPLE
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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
AN ILLUSTRATIVE EXAMPLE
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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
AN ILLUSTRATIVE EXAMPLE
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The solutions corresponding to individuals at itera
iteration
0
1
2
x 1 f (x 1) 8
64
-1.6 2.56
pbest 1 f (pbest 1) 8
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
pbest 1 f (pbest 1) 2
4
x 4 f (x 4) 6
36
-0.4 0.16
pbest 1 f (pbest 1) 6
36
gbest f (gbest ) 2
4
AN ILLUSTRATIVE EXAMPLE
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The best solutions for individuals and group are u
iteration
0
1
2
x 1 f (x 1) 8
64
-1.6 2.56
pbest 1 f (pbest 1) 8
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
pbest 1 f (pbest 1) 2
4
2 4
x 4 f (x 4) 6
36
-0.4 0.16
pbest 1 f (pbest 1) 6
36
-0.4 0.16
gbest f (gbest ) 2
4
-0.4 0.16
AN ILLUSTRATIVE EXAMPLE
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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 + a 1·r 1·(pbest1-x 11) + a 2·r 2·(gbest-x 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
AN ILLUSTRATIVE EXAMPLE
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v 32= Ф·v 3
1 + a 1·r 1·(pbest3-x 31) + a 2·r 2·(gbest-x 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 + a 1·r 1·(pbest4-x 41) + a 2·r 2·(gbest-x 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
AN ILLUSTRATIVE EXAMPLE
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The best solutions for individuals and group are u
iteration
0
1
2
x 1 f (x 1) 8
64
-1.6 2.56 -8.08 65.29
pbest 1 f (pbest 1) 8
64
-1.6 2.56 -1.6 2.56
x 2 f (x 2) 3 9 1.4 1.96 -1.68 2.82
pbest 1 f (pbest 1) 3
9
1.4 1.96 1.4 1.96
x 3
f (x 3)
2
4
2 4 -0.4 0.16
pbest 1 f (pbest 1) 2
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
AN ILLUSTRATIVE EXAMPLE
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v 33= Ф·v 3
2 + a 1·r 1·(pbest3-x 32) + a 2·r 2·(gbest-x 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 + a 1·r 1·(pbest4-x 42) + a 2·r 2·(gbest-x 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
AN ILLUSTRATIVE EXAMPLE
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The best solutions for individuals and group are u
iteration
0
1
2
x 1 f (x 1) 8
64
-1.6 2.56 -8.08 65.29
pbest 1 f (pbest 1) 8
64
-1.6 2.56 -1.6 2.56
x 2 f (x 2) 3
9
1.4 1.96 -1.68 2.82
pbest 1 f (pbest 1) 3
9
1.4 1.96 1.4 1.96
x 3
f (x 3)
2
4
2 4 -0.4 0.16
pbest 1 f (pbest 1) 2
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
AN ILLUSTRATIVE EXAMPLE
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Iteration process
0
5
10
15
20
25
30
1 2 3 4 5
Average
Best
PSO – SIMULATION
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x
y
PSO – SIMULATION
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x
y
PSO – SIMULATION
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x
y
PSO – SIMULATION
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x
y
PSO – SIMULATION
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x
y
PSO – SIMULATION
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x
y
PSO – SIMULATION
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x
y
PSO – SIMULATION
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x
y
SCHWEFEL'S FUNCTION
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420.9=
4=)(
minimumglobal
500
where
)()(1
i
i
n
i
i
x
n x f
x
x x f
EVOLUTION INITIALIZATION
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EVOLUTION 5 ITERATION
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EVOLUTION 10 ITERATION
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EVOLUTION 15 ITERATION
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EVOLUTION 20 ITERATION
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EVOLUTION 25 ITERATION
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EVOLUTION 100 ITERATION
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EVOLUTION 500 ITERATION
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PARTICLE SWARM OPTIMIZA _______________________
VARIANTS OF PSO – DISCRE
VARIANTS OF PSO
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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
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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
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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
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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
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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
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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
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PARTICLE SWARM OPTIMIZA
_______________________VARIANTS OF PSO – PS
CONSTRICTION FACTO
PSO WITH CFA
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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
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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
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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
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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
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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
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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
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PARTICLE SWARM OPTIMIZA _______________________SOME ATTRIBUTES O
ATTRIBUTES OF PSO Like GA PSO is population based stochastic optimization
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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.
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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.
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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.
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PARTICLE SWARM OPTIMIZA _______________________PITFALLS O
PITFALLS OF PSO
Tendency to cluster very quickly
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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
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PARTICLE SWARM OPTIMIZA _______________________OPTIMIZATION IN DYNAMIC ENVIRO
OPTIMIZATION IN DYNAMIC ENVIRO
Many real-world optimization problems are dynamic and require
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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??????????????????
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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
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PARTICLE SWARM OPTIMIZA _______________________EXA
EXAMPLE 1: COMBINATORIAL OPTIMIZ
Combinatorial optimization problem
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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
EXAMPLE 1: COMBINATORIAL OPTIMIZ
Choose the length of binary string
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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
EXAMPLE 1: COMBINATORIAL OPTIMIZ
Suppose the population size is four.
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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)
EXAMPLE 1: COMBINATORIAL OPTIMIZ
Iteration: iter=1
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Binary Integer objective function
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
EXAMPLE 1: COMBINATORIAL OPTIMIZ
Iteration: iter=2
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Binary Integer objective function
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
EXAMPLE 1: COMBINATORIAL OPTIMIZ
Iteration: iter=10
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Binary Integer objective function
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
EXAMPLE 1: COMBINATORIAL OPTIMIZ
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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
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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.
EXAMPLE 2: WEAPON TARGET ALLOCA
Mathematical Model
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
v s = Value of asset s
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
The probability that weapons deployed in d th layer w
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
The probability that a single attacking weapon of type
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
The objective from the defending side is to maximi
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
Constraints
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
This is an Integer Nonlinear Programming prob
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
By constructing a penalty function, transform i
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
Consider that two types of weapons are available to defend three a
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
The values of the first, second and third assets are 40
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
Effectiveness of defending weapons and damage probabilitie
weapons
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
Problem : Determine an optimal defense plan against the k
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
Binary coding
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
For instance
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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
EXAMPLE 2: WEAPON TARGET ALLOCA
800
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-600
-400
-200
0
200
400
600
800
0 10 20 30 40 50 60 70 80 90
Iter
f (x )
EXAMPLE 2: WEAPON TARGET ALLOCA
Optimal solution
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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
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THE FUTURE?
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REFERENCE BOOKS
Swarm Intelligence
/
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James Kennedy, Russell C Eberhart, Academic Press /
Morgan Kaufmann, 2001
Practical Optimization: Algorithms and Engineering
Applications
Andreas Antoniou and Wu-Sheng Lu 2007
PARTICLE SWARM OPTIMIZA
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PARTICLE SWARM OPTIMIZA
_______________________PSO IN M
1)Sphere Function
STANDARD BENCHMARK FUNCT
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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
STANDARD BENCHMARK FUNCT
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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
STANDARD BENCHMARK FUNCT
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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
STANDARD BENCHMARK FUNCT
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ii nn 11
The Ackley problem has several local
minima but only one global minimum.
It is a widely used multi-modal test function.
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THANK YOU FOR YOUR INT