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**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:**

Initialize the swarm from the solution space Evaluate fitness of each particle Update individual and global bests Update velocity and position of each particle Go to step 2, and repeat until termination condition

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**PSO Algorithm (cont.) Original velocity update equation: with**

: acceleration constant Inertia Cognitive Component Social Component

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**PSO Algorithm (cont.) Original velocity update equation:**

with : acceleration constant Position Update: Inertia Cognitive Component Social 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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**Initialize swarm and evaluate fitness (t=0)**

Simple 1D Example Initialize swarm and evaluate fitness (t=0) gbest

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**Update velocity and position (t=1)**

Simple 1D Example Update velocity and position (t=1) gbest

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**Update personal and global best (t=2)**

Simple 1D Example Evaluate fitness Update personal and global best (t=2) gbest

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**Update personal and global best (t=2)**

Simple 1D Example Evaluate fitness Update personal and global best (t=2) gbest

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**Update velocity and position (t=2)**

Simple 1D Example Update velocity and position (t=2) gbest Inertia Personal Social Total

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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 Ring Wheel Von Neumann gbest 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 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. 1 Vid

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**PSO Variants Hybrid PSO Adaptive PSO PSO in complex environments**

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: The particle movement is defined as: 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**

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