Show & Tell 37.1 — ParticleSwarmOptimizer: PSO Velocity Update & MOPSO Archive

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Show & Tell: ParticleSwarmOptimizer (Phase 37.1)

PSO Velocity Update — Core Algorithm

The canonical PSO velocity update (Kennedy & Eberhart 1995, modified by Shi & Eberhart 1998):

v_i(t+1) = ω·v_i(t) + c₁·r₁·(pbest_i − x_i(t)) + c₂·r₂·(gbest − x_i(t))
x_i(t+1) = x_i(t) + v_i(t+1)

Where:

• ω = inertia weight (0.9 → 0.4 linear decay)
• c₁ = cognitive coefficient (typically 2.0)
• c₂ = social coefficient (typically 2.0)
• r₁, r₂ = uniform random in [0,1]

Constriction Factor (Clerc & Kennedy 2002)

When φ = c₁ + c₂ > 4:

χ = 2κ / |2 − φ − √(φ² − 4φ)|

With κ=1, φ=4.1 → χ ≈ 0.7298. Replaces inertia weight for guaranteed convergence.

MOPSO Pareto Archive (Coello Coello 2004)

Multi-objective extension:

1. Maintain external archive of non-dominated solutions
2. Divide objective space into hypercubes
3. Select leader from less crowded hypercube regions
4. Archive bounded by size limit with crowding-distance pruning

Implementation Skeleton

class ParticleSwarmOptimizer:
    def update_velocities(self, state: SwarmState, config: PSOConfig) -> SwarmState:
        new_particles = []
        for p in state.particles:
            r1 = np.random.uniform(0, 1, config.dimensions)
            r2 = np.random.uniform(0, 1, config.dimensions)

            cognitive = config.cognitive_coeff * r1 * (p.personal_best - p.position)
            social = config.social_coeff * r2 * (state.global_best_position - p.position)

            if config.constriction_factor is not None:
                new_vel = config.constriction_factor * (p.velocity + cognitive + social)
            else:
                new_vel = config.inertia_weight * p.velocity + cognitive + social

            if config.velocity_clamp is not None:
                new_vel = np.clip(new_vel, -config.velocity_clamp, config.velocity_clamp)

            new_particles.append(Particle(
                id=p.id, position=p.position, velocity=new_vel,
                personal_best=p.personal_best,
                personal_best_fitness=p.personal_best_fitness,
                current_fitness=p.current_fitness
            ))
        return SwarmState(
            particles=tuple(new_particles),
            global_best_position=state.global_best_position,
            global_best_fitness=state.global_best_fitness,
            iteration=state.iteration,
            convergence_history=state.convergence_history
        )

Benchmark Results (Expected)

Function	Dimensions	Known Optimum	PSO Target
Sphere	30	0.0	< 1e-6
Rosenbrock	30	0.0	< 1.0
Rastrigin	30	0.0	< 5.0
Ackley	30	0.0	< 0.01

Ref: Issue #755 | Wiki: Phase-37-Particle-Swarm-Optimizer