Particle swarm optimization

ParticleSwarmArrowsAnimation

PSO_Meta-Fitness_Landscape_(12_benchmark_problems)

Optimization algorithm

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 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