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[[File:ParticleSwarmArrowsAnimation.gif|thumb|ParticleSwarmArrowsAnimation]] [[file:PSO_Meta-Fitness_Landscape_(12_benchmark_problems).JPG|right|thumb|PSO_Meta-Fitness_Landscape_(12_benchmark_problems)]] {{Short description|Optimization algorithm}}
{{Infobox algorithm
| name = Particle Swarm Optimization
| image =
| caption =
| class = [[Optimization algorithm]]
| creator = [[James Kennedy (social psychologist)|James Kennedy]] and [[Russell Eberhart]]
| year = 1995
| genre = [[Swarm intelligence]]
| license =
}}

'''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 of candidate solutions, here dubbed particles, and moving these particles around in the search-space according to simple mathematical formulae over the particle's position and velocity. Each particle's movement is influenced by its local best known position, but is 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.

== History ==
PSO was introduced by [[James Kennedy (social psychologist)|James Kennedy]] and [[Russell Eberhart]] in 1995. The algorithm was inspired by the social behavior of birds flocking or fish schooling.

== Algorithm ==
The PSO algorithm initializes a group of particles (solutions) and then iteratively moves these particles around the search-space. Each particle adjusts its position based on its own experience and the experience of neighboring particles, making use of the best position encountered by itself and its neighbors.

=== Initialization ===
Each particle is initialized with a random position and velocity. The particles are then evaluated using a fitness function that measures the quality of the solution.

=== Update Rules ===
The velocity and position of each particle are updated using the following equations:
* Velocity update:
\[ v_{i}(t+1) = w \cdot v_{i}(t) + c_1 \cdot r_1 \cdot (p_{i} - x_{i}(t)) + c_2 \cdot r_2 \cdot (g - x_{i}(t)) \]
* Position update:
\[ x_{i}(t+1) = x_{i}(t) + v_{i}(t+1) \]

Where:
* \( v_{i}(t) \) is the velocity of particle \( i \) at time \( t \)
* \( x_{i}(t) \) is the position of particle \( i \) at time \( t \)
* \( p_{i} \) is the best known position of particle \( i \)
* \( g \) is the global best known position
* \( w \) is the inertia weight
* \( c_1 \) and \( c_2 \) are cognitive and social coefficients
* \( r_1 \) and \( r_2 \) are random numbers between 0 and 1

=== Termination ===
The algorithm terminates when a stopping criterion is met, such as a maximum number of iterations or a satisfactory fitness level.

== Applications ==
PSO has been applied to a wide range of optimization problems, including:
* [[Function optimization]]
* [[Neural network]] training
* [[Fuzzy system]] control
* [[Robotics]]
* [[Signal processing]]

== Advantages and Disadvantages ==
=== Advantages ===
* Simple to implement
* Few parameters to adjust
* Effective for a wide range of problems

=== Disadvantages ===
* May converge prematurely
* Performance can be sensitive to parameter settings

== Variants ==
Several variants of PSO have been developed to improve its performance, including:
* [[Discrete Particle Swarm Optimization]]
* [[Multi-objective Particle Swarm Optimization]]
* [[Hybrid Particle Swarm Optimization]]

== See also ==
* [[Genetic algorithm]]
* [[Ant colony optimization]]
* [[Simulated annealing]]
* [[Differential evolution]]

== Related Pages ==
* [[Swarm intelligence]]
* [[Optimization algorithm]]
* [[James Kennedy (social psychologist)]]
* [[Russell Eberhart]]
* [[Function optimization]]
* [[Neural network]]
* [[Fuzzy system]]
* [[Robotics]]
* [[Signal processing]]

[[Category:Optimization algorithms]]
[[Category:Swarm intelligence]]
[[Category:Evolutionary algorithms]]
[[Category:1995 introductions]]

{{Swarm-intelligence-stub}}

Particle swarm optimization - Revision history

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