Particle Swarm Optimization (PSO) Explorer

PSO updates each particle's velocity as a weighted sum of three influences: its own momentum (inertia), attraction toward its personal best position (cognitive), and attraction toward the swarm's global best (social) — as introduced by Kennedy & Eberhart (1995), with the inertia weight refinement by Shi & Eberhart (1998). The animation cycles through the velocity decomposition for a single particle with fixed parameters; the right panel gives the update equations and a pseudocode sketch of the full algorithm. For a fully interactive simulation with adjustable parameters and live fitness landscapes, see Tab ② PSO Simulation.

Velocity decomposition Initializing…

Current position x Personal best p ✦Global best g →Current velocity v

➡ New Velocity

v_i(t+1) = ωv_i + φ_pr_p(p_i−x_i) + φ_gr_g(g−x_i) + η

Clamped to |v| ≤ v_max. Position updated: x_i(t+1) = x_i(t) + v_i(t+1). The η term is optional.

Velocity Update

① Inertia

ω · v_i(t)

Carries previous velocity forward. High ω → exploration; low ω → exploitation.

② Cognitive (Personal Best)

φ_p · r_p · (p_i − x_i(t))

Pulls toward the particle's own historical best. Stochastic weight r_p ∼ U(0,1) each step.

⊕ Velocity Noise — non-standard

η ∼ 𝒩(0, σ²I)

Additive Gaussian noise applied each step, independent of pbest and gbest. Not part of Kennedy & Eberhart's (1995) canonical PSO — inspired by the Vicsek (1995) flocking model and enables independent hill climbing by individuals.

Algorithm Sketch

initialize N particles randomly x_i ∼ U(lb, ub); v_i ∼ U(−v_max, v_max) p_i ← x_i; g ← argmax_i f(x_i) repeat for each particle i: r_p, r_g ∼ U(0,1) η_i ∼ 𝒩(0,σ²I) // non-standard v_i ← ωv_i + φ_pr_p(p_i−x_i) + φ_gr_g(g−x_i) + η_i // non-standard v_i ← clip(v_i, −v_max, v_max) x_i ← x_i + v_i if f(x_i) > f(p_i): p_i ← x_i if f(x_i) > f(g): g ← x_i until // mean|v| < ε or iter ≥ max

Convergence guarantee

Clerc & Kennedy (2002) analyzed convergence by writing the update as v(t+1) = χ [v(t) + φ_p r_p(p−x) + φ_g r_g(g−x)], where χ is the constriction factor. Convergence can be guaranteed under the conditions χ = 2 / |2−φ−√(φ²−4φ)| where φ ≜ φ_p + φ_g and φ_p + φ_g > 4 (which ensures χ is real and less than 1). The canonical choice φ_p = φ_g = 2.05 gives φ = 4.1 and χ ≈ 0.729 in this constriction-factor form. Distributing χ into the bracket recovers the inertia-weight form used here, with ω = χ ≈ 0.729 and effective coefficients χφ_p = χφ_g ≈ 1.494. The displayed φ_p, φ_g sliders on Tab ② control these distributed effective coefficients (which sum to > 4χ).

Watch the swarm navigate a 2-D fitness landscape in real time. Each particle shows its velocity arrow and a hollow ghost at its personal best. The global best is marked with a ✦. Adjust ω, φ_p, φ_g, swarm size (N), and maximum speed (v_max), then hit Run (or Resume after pausing).

Fitness landscape idle

Particle Personal best ✦Global best →Velocity

g-best / max p-best median p-best mean |v| (norm.) landscape max

Landscape

PSO Parameters

ω 0.72

φ_p 1.49

φ_g 1.49

N 20

v_max 1.00

Display

Show velocity arrows

Show personal bests

Velocity noise (non-standard)

σ 0.20

Controls

speed 1×

Iteration

—

g-best f

—

mean |v|

—

max |v|

—

Convergence The φ_p, φ_g sliders above control the distributed effective coefficients χφ_p, χφ_g, where ω = χ ≈ 0.729 is the constriction factor (Clerc & Kennedy, 2002). Convergence is guaranteed when the raw coefficients satisfy φ_p + φ_g > 4 in the constriction-factor form — equivalent to ω ≈ 0.729 with the default slider values of 1.49. See Tab ① for the full derivation.