Lernen Particle Swarm Optimization | Swarm-Based Algorithms

Definition

Particle Swarm Optimization (PSO) is a population-based, bio-inspired algorithm that draws its inspiration from the collective behavior of bird flocks and fish schools. In PSO, you work with a swarm of simple agents called particles. Each particle represents a potential solution to the optimization problem and moves through the search space by updating its position and velocity.

Core Concepts

The movement of each particle is influenced by two main factors:

Personal Best: The best-known position that the particle itself has discovered so far;
Global Best: The best-known position found by the entire swarm at any point during the search.

This dynamic allows the swarm to explore the solution space efficiently, balancing exploration (searching new areas) and exploitation (refining known good areas) as particles communicate and learn from each other.

Key Concepts in PSO:

Particles: agents that represent possible solutions;
Position: the current location of a particle in the search space;
Velocity: the direction and speed of particle movement;
Personal best position: the best position found by each particle so far;
Global best position: the best position found by any particle in the swarm.

By iteratively updating positions and velocities based on these principles, PSO enables you to solve complex optimization problems in a flexible and efficient manner.

Example: Basic PSO Implementation


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import numpy as np

# Objective function: simple 1D quadratic (minimum at x=3)
def objective(x):
    return (x - 3) ** 2 + 2

# PSO parameters
num_particles = 20
num_iterations = 50
w = 0.5  # Inertia coefficient
c1 = 1.5  # Cognitive (personal) coefficient
c2 = 1.5  # Social (global) coefficient

# Initialize particle positions and velocities
positions = np.random.uniform(-10, 10, num_particles)
velocities = np.zeros(num_particles)
personal_best_positions = positions.copy()
personal_best_scores = objective(positions)
global_best_idx = np.argmin(personal_best_scores)
global_best_position = personal_best_positions[global_best_idx]

# PSO loop
for iteration in range(num_iterations):
    # Evaluate current positions
    scores = objective(positions)
    # Update personal bests
    better_mask = scores < personal_best_scores
    personal_best_scores[better_mask] = scores[better_mask]
    personal_best_positions[better_mask] = positions[better_mask]
    # Update global best
    global_best_idx = np.argmin(personal_best_scores)
    global_best_position = personal_best_positions[global_best_idx]
    # Update velocities and positions
    r1 = np.random.rand(num_particles)
    r2 = np.random.rand(num_particles)
    velocities = (
        w * velocities
        + c1 * r1 * (personal_best_positions - positions)
        + c2 * r2 * (global_best_position - positions)
    )
    positions += velocities

print(f"Best position found: {global_best_position:.3f}")
print(f"Objective value: {objective(global_best_position):.3f}")

Step-by-Step: Position and Velocity Updates in PSO

Evaluate Each Particle: For every particle, calculate the objective function at its current position;
Update Personal Best: If the current position yields a better score than any previous position, store it as the new personal best;
Update Global Best: Identify the best score among all personal bests and update the global best position accordingly;
Update Velocity: For each particle, calculate the new velocity using:
- The inertia term (w * velocity), which encourages the particle to keep moving in its current direction;
- The cognitive term (c1 * r1 * (personal_best - position)), which pulls the particle towards its own best experience;
- The social term (c2 * r2 * (global_best - position)), which pulls the particle towards the best found by the swarm;
Update Position: Add the new velocity to the current position to obtain the particle's new location in the search space.

Particles repeat these steps for a set number of iterations or until convergence criteria are met.

Example: Visualizing Particle Trajectories


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import numpy as np
import matplotlib.pyplot as plt

# Objective function: simple 1D quadratic (minimum at x=3)
def objective(x):
    return (x - 3) ** 2 + 2

# PSO parameters
num_particles = 20
num_iterations = 50
w = 0.5  # Inertia coefficient
c1 = 1.5  # Cognitive (personal) coefficient
c2 = 1.5  # Social (global) coefficient

# Re-run the PSO loop, recording particle trajectories
positions = np.random.uniform(-10, 10, num_particles)
velocities = np.zeros(num_particles)
personal_best_positions = positions.copy()
personal_best_scores = objective(positions)
global_best_idx = np.argmin(personal_best_scores)
global_best_position = personal_best_positions[global_best_idx]

trajectories = [positions.copy()]

for iteration in range(num_iterations):
    scores = objective(positions)
    better_mask = scores < personal_best_scores
    personal_best_scores[better_mask] = scores[better_mask]
    personal_best_positions[better_mask] = positions[better_mask]
    global_best_idx = np.argmin(personal_best_scores)
    global_best_position = personal_best_positions[global_best_idx]
    r1 = np.random.rand(num_particles)
    r2 = np.random.rand(num_particles)
    velocities = (
        w * velocities
        + c1 * r1 * (personal_best_positions - positions)
        + c2 * r2 * (global_best_position - positions)
    )
    positions += velocities
    trajectories.append(positions.copy())

trajectories = np.array(trajectories)

# Plot trajectories
plt.figure(figsize=(10, 6))
for i in range(num_particles):
    plt.plot(trajectories[:, i], label=f"Particle {i+1}", alpha=0.5)
plt.axhline(3, color='red', linestyle='--', label='Optimum (x=3)')
plt.title('Particle Trajectories in PSO')
plt.xlabel('Iteration')
plt.ylabel('Position')
plt.legend(bbox_to_anchor=(1.05, 1), loc='upper left', fontsize='small')
plt.tight_layout()
plt.show()

Parameter Analysis

The choice of PSO parameters has a significant impact on the algorithm's behavior and performance:

Inertia Coefficient (w): Controls how much of the previous velocity is retained;
- Higher values encourage exploration and help particles escape local minima;
- Lower values favor exploitation and accelerate convergence.
Cognitive Coefficient (c1): Determines how strongly particles are pulled toward their own best positions;
- Raising this value encourages independent exploration and helps maintain swarm diversity.
Social Coefficient (c2): Governs the attraction to the global best position;
- Increasing this value accelerates convergence by promoting collective learning;
- Excessively high values can cause premature stagnation.

Careful tuning of these coefficients is essential for achieving robust optimization performance in different problem domains. Adjust the parameters to balance exploration and exploitation, ensuring that particles search effectively without getting stuck or converging too quickly.

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