of real numbers and produces a real number as output which indicates the objective function value of the given candidate solution. The gradient of f is not known. The goal is to find a solution a for which f(a) ≤ f(b) for all b in the search-space, which would mean a is the global minimum. Let S be the number of particles in the swarm, each having a position xi ∈ ℝn in the search-space and a velocity vi ∈ ℝn. Let pi be the best known position of particle i and let g be the best known position of the entire swarm. A basic PSO algorithm to minimize the cost function is then: for each particle i = 1, ..., S do Initialize the particle's position with a uniformly distributed random vector: xi ~ U(blo, bup) Initialize the particle's best known position to its initial position: pi ← xi if f(pi) < f(g) then update the swarm's best known position: g ← pi Initialize the particle's velocity: vi ~ U(-|bup-blo|, |bup-blo|) while a termination criterion is not met do: for each particle i = 1, ..., S do for each dimension d = 1, ..., n do Pick random numbers: rp, rg ~ U(0,1) Update the particle's velocity: vi,d ← w vi,d + φp rp (pi,d-xi,d) + φg rg (gd-xi,d) Update the particle's position: xi ← xi + vi if f(xi) < f(pi) then Update the particle's best known position: pi ← xi if f(pi) < f(g) then Update the swarm's best known position: g ← pi The values blo and bup represent the lower and upper boundaries of the search-space respectively. The w parameter is the inertia weight. The parameters φp and φg are often called cognitive coefficient and social coefficient. The termination criterion can be the number of iterations performed, or a solution where the adequate objective function value is found. The parameters w, φp, and φg are selected by the practitioner and control the behaviour and efficacy of the PSO method (below). Parameter selection The choice of PSO parameters can have a large impact on optimization performance. Selecting PSO parameters that yield good performance has therefore been the subject of much research. To prevent divergence ("explosion") the inertia weight must be smaller than 1. The two other parameters can be then derived thanks to the constriction approach, or freely selected, but the analyses suggest convergence domains