Particle Swarm Optimization(PSO)

Mansour Nejati

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Introduction : Swarm Intelligence

Study of collective behavior in decentralized, self-organized systems.

Originated from the study of colonies, or swarms of social organisms.

Collective intelligence arises from interactions.

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Introduction

Particle Swarm Optimization: Introduced by Kennedy & Eberhart 1995 Inspired by social behavior of birds and shoals of fish Swarm intelligence-based optimization Nondeterministic Population-based optimization Performance comparable to Genetic algorithms

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Particle Swarm Optimization

Swarm : a set of particles (S) Particle: a potential solution

Position, Velocity ,

Each particle maintains Individual best position:

Swarm maintains its global best:

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PSO Algorithm

Basic algorithm of PSO:1. Initialize the swarm from the solution space2. Evaluate fitness of each particle3. Update individual and global bests4. Update velocity and position of each particle5. Go to step 2, and repeat until termination condition

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PSO Algorithm (cont.)

Original velocity update equation:

with : acceleration constant

InertiaCognitive ComponentSocial Component

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PSO Algorithm (cont.)

Original velocity update equation:

with : acceleration constant

Position Update:

InertiaCognitive ComponentSocial Component

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PSO Algorithm - Parameters

Acceleration constant Small values limit the movement of the particles Large values : tendency to explode toward infinity In general

Maximum velocity Velocity is a stochastic variable => uncontrolled trajectory

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Simple 1D Example

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Initialize swarm and evaluate fitness (t=0)

gbest

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Simple 1D Example

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Update velocity and position (t=1)

gbest

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Simple 1D Example

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Evaluate fitnessUpdate personal and global best (t=2)

gbest

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Simple 1D Example

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Evaluate fitnessUpdate personal and global best (t=2)

gbest

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Simple 1D Example

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3gbest

Update velocity and position (t=2)

InertiaPersonalSocialTotal

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Rate of Convergence Improvement

Inertia weight:

Scaling the previous velocity Control search behavior

High values exploration Low values exploitation

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PSO with Inertia weight

can be decreased over time: Linear [0.9 to 0.4] Nonlinear

main disadvantage: once the inertia weight is decreased, the swarm loses

its ability to search new areas (can not recover its exploration mode).

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Rate of Convergence Improvement

Constriction Factor: Canonical PSO

Typically , Can converge without using Vmax (velocity clamping) Improve the convergence by damping the oscillations

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Swarm Topologies

Two general types of neighborhoods: Global best (gbest) : fully connected network Local best (lbest) : according to a topology

gbestRing Wheel Von Neumann

lbest

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Lbest vs. Gbest

Gbest converges fast but may be trapped in a local optima. Lbest is slower in convergence but has more chances to find

an optimal solution.

Most efficient neighborhood structure, in general, depends on the type of problem.

Fully Informed PSO (FIPS): Each individual is influenced by successes of all its neighbors.

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Diversity Improvement

Based on lbest model. Usually slow down the convergence rate.

Spatial Neighborhoods: Partition particles based on spatial location. Calculate the largest distance between any two particles. Select neighboring particles according to ratio: Selection threshold can be varied over time.

Start with small ratio (lbest) and gradually increase the ratio.

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Diversity Improvement

Neighborhood Topologies: In lbest model, all particles can exchange information

indirectly.

Average path length depends on the topology. Topology significantly affects the performance (experimentally). Randomly change some connections can change average path

length.

i i+1 i+2

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Diversity Improvement

Subpopulations: Previously used in GA. Original swarm is partitioned to subpopulations. PSO is applied to each subpopulation. An interaction scheme is used for information sharing

between subpopulations. Each subpopulation can search the smaller region of

search space.

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Discrete PSO

Binary PSO: Introduces by kennedy and Eberhart. Each individual (particle) has to take a binary decision.

Predisposition is derived based on individual and group performance:

Previous state predisposition

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Binary PSO (cont.)

determines a threshold in the probability function and therefore should be bounded in the range of [0.0, 1.0].

state of the dth position in the string at time t:

Where is a random number with a uniform distribution.

Vid

1

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PSO Variants

Hybrid PSO Incorporate the capabilities of other evolutionary

computation techniques. Adaptive PSO

Adaptation of PSO parameters for a better performance. PSO in complex environments

Multiobjective or constrained optimization problems or tracking dynamic systems.

Other variants variations to the original formulation to improve its performance.

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Hybrid PSO

GA-PSO: combines the advantages of swarm intelligence and a

natural selection mechanism.

jump from one area to another by the selection mechanism accelerating the convergence speed.

capability of “breeding”.

replacing agent positions with low fitness values, with those with high fitness, according to a selection rate

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Hybrid PSO

EPSO: Evolutionary PSO Incorporates a selection procedure Self-adapting of parameters

The particle movement is defined as:

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Hybrid PSO : EPSO

Mutation of weights and global best:

Learning parameters can be either fixed or dynamically changing as strategic parameters.

Survival Selection: Stochastic tournament.

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Hybrid PSO : EPSO

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Hybrid PSO : DEPSO

Hybrid of Differential Evolution and PSO. A DE operator applied to the particle’s best position to

eliminate the particles falling into local minima. Alternation:

Original PSO algorithm at the odd iterations. DE operator at the even iterations.

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Hybrid PSO : DEPSO

DE mutation on particle’s best positions:

where k is a random integer value within [1,n] which ensures the mutation in at least one dimension.

Trial point:

For each dth dimention:

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Hybrid PSO : DEPSO

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Dynamic Tracking in PSO

The classical PSO is very effective in solving static optimization problems but is not as efficient when applied to a dynamic system in which the optimal value may change repeatedly.

An adaptive approach has been introduced for this problem: Detection of environmental changes:

changed-gbest-value fixed-gbest-values

rerandomizing a certain number of particles

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Applications

Convenience of realization, properties of low constraint on the continuity of objective function and joint of search space, and ability of adapting to dynamic environment, make PSO be applied in more and more fields.

Some PSO applications: Electronics and electromagnetic Signal, Image and video processing Neural networks Communication networks …

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Thanks for your attention