Particle Swarm Optimization

This is Modeled by particles in Multidimensional Space that have a position and a Velocity . These particles are flying through hyperspace and remember the best position that they have seen. Members of a swarm communicate good positions to each other and adjust their own position and velocity based on these good positions. There are two main ways this communication is done:

a swarm best that is known to all

local bests are known in neighborhoods of particles

$x \leftarrow x + v$

$v \leftarrow wv + c_1 r_1 (\hat{x}-x) + c_2 r_2 (\hat{x}_g-x)$

--- $w$ is the inertial constant. Good values are usually slightly less than 1.

--- $c_1$ and $c_2$ are constants that say how much the particle is directed towards good positions. Good values are usually right around 1.

--- $r_1$ and $r_2$ are random values in the range $1$ .

--- $\hat{x}$ is the best the particle has seen.

--- $\hat{x}_g$ is the global best seen by the swarm. This can be replaced by $\hat{x}_l$ , the local best, if neighborhoods are being used.

ALGORITHM

Initialize $x$ and $v$ of each particle to a random value. The range of these values may be domain specific.

Initialize each $\hat{x}$ to the current position.

Initialize $\hat{x}_g$ to the position that has the best fitness in the swarm.

Loop while the fitness of $\hat{x}_g$ is below a threshold and the number of iterations is less than some predetermined maximum.

--- For each particle do the following:

-- Update $x$ according to the above equation.

-- Calculate fitness for new position.

-- If it is better than the fitness of $\hat{x}$ , replace $\hat{x}$ .

-- It it is better than the fitness of $\hat{x}_g$ , replace $\hat{x}_g$ .

-- Update $v$ according to the above equation.

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Particle Swarm Optimization