Particle Swarm Optimization

Algorithm 6 (PSO)

Inputs: population size \(n\), dimension \(d\), inertia weight \(w\), acceleration coefficients \(c_1, c_2\)
Output: best individual \(\mathbf{g}\)

1. Initialize \(X\)

2. while stopping criterion not met do

   1. for \(i = 1, 2, \ldots, n\) do

      1. \(V_i \leftarrow wV_i + c_1\mathbf{R}^\intercal (\mathbf{p}_i - X_i) + c_2\mathbf{R}^\intercal (\mathbf{g} - X_i)\)

      2. \(X_i \leftarrow X_i + V_i\)

      3. if \(f(X_i) < f(\mathbf{p}_i)\) then

         1. \(\mathbf{p}_i \leftarrow X_i\)

         2. if \(f(X_i) < f(\mathbf{g})\) then

            1. \(\mathbf{g} \leftarrow X_i\)

Notation

• \(n\): population size

• \(d\): dimension of particles

• \(X\): population of particles, \(X \in \mathbb{R}^{N \times D}\)

• \(X_i\): \(i\)-th particle

• \(x_{i, j}\): \(j\)-th dimension of \(X_i\)

• \(V\): velocity of particles

• \(V_i\): velocity of \(i\)-th particle

• \(v_{i, j}\): \(j\)-th dimension of \(V_i\)

• \(w\): inertia weight

• \(c_1, c_2\): acceleration coefficients

• \(\mathbf{R}\): random vector in \([0, 1]^D\)

• \(\mathbf{p}_i\): personal best of particle \(i\)

• \(\mathbf{g}\): global best of particles

Update Equations

\[ V_i \leftarrow wV_i + c_1\mathbf{R}^\intercal (\mathbf{p}_i - X_i) + c_2\mathbf{R}^\intercal (\mathbf{g} - X_i) \]

\[ X_i \leftarrow X_i + V_i \]