# Constriction Factor Particle Swarm Optimization #

## Name #

Constriction Factor Particle Swarm Optimization (CFPSO), Constricted PSO, Constriction PSO, CF-PSO

## Taxonomy #

Constriction Factor Particle Swarm Optimization is a variant of the Particle Swarm Optimization algorithm, which belongs to the field of Swarm Intelligence, a subcategory of Computational Intelligence and Biologically Inspired Computation.

- Computational Intelligence
- Biologically Inspired Computation
- Swarm Intelligence
- Particle Swarm Optimization (PSO)
- Constriction Factor Particle Swarm Optimization (CFPSO)

- Particle Swarm Optimization (PSO)

- Swarm Intelligence

- Biologically Inspired Computation

## Strategy #

### Swarm Initialization #

CFPSO begins by initializing a swarm of particles in the search space. Each particle represents a potential solution to the optimization problem and has a position and velocity. The particles’ positions are randomly initialized within the bounds of the search space, while their velocities are typically initialized to zero or small random values.

### Particle Evaluation #

After initialization, the fitness of each particle is evaluated using the objective function. The objective function measures the quality of the solution represented by the particle’s position. Each particle keeps track of its personal best position (pbest), which is the best solution it has encountered so far. The swarm also maintains a global best position (gbest), which is the best solution found by any particle in the swarm.

### Velocity Update #

In each iteration, the particles’ velocities are updated using the constriction factor formula. The constriction factor is a parameter that controls the convergence behavior of the swarm. It balances the influence of the particle’s previous velocity, its personal best position, and the global best position. The velocity update equation incorporates the constriction factor, cognitive component (attraction towards pbest), and social component (attraction towards gbest).

### Position Update #

After updating the velocities, the particles’ positions are updated by adding the new velocities to their current positions. This moves the particles through the search space, exploring new solutions. The updated positions are then evaluated using the objective function, and the pbest and gbest are updated if better solutions are found.

### Termination #

The CFPSO algorithm iterates until a termination criterion is met. Common termination criteria include reaching a maximum number of iterations, finding a solution with a satisfactory fitness value, or observing convergence of the swarm (i.e., particles converging to similar positions). Once the termination criterion is satisfied, the algorithm returns the best solution found (gbest).

## Procedure #

### Data Structures #

- Particle: Represents a potential solution, consisting of position and velocity vectors.
- Swarm: Collection of particles.
- pbest: Personal best position of each particle.
- gbest: Global best position found by the swarm.

### Parameters #

- num_particles: Number of particles in the swarm.
- num_iterations: Maximum number of iterations.
- phi1: Cognitive component weight.
- phi2: Social component weight.
- chi: Constriction factor.

### Algorithm Steps #

- Initialize swarm:
- For each particle in the swarm:
- Initialize particle position randomly within search space bounds.
- Initialize particle velocity to zero or small random values.
- Evaluate particle fitness using the objective function.
- Set particle’s pbest to its initial position.

- Set gbest to the position of the best particle in the swarm.

- For each particle in the swarm:
- While termination criterion is not met:
- For each particle in the swarm:
- Update particle velocity using constriction factor formula:
- v_new = chi * (v_old + phi1 * rand() * (pbest - x_old) + phi2 * rand() * (gbest - x_old))

- Update particle position:
- x_new = x_old + v_new

- Evaluate particle fitness using the objective function.
- If particle’s new fitness is better than its pbest fitness:
- Update particle’s pbest to its new position.

- If particle’s new fitness is better than the swarm’s gbest fitness:
- Update gbest to the particle’s new position.

- Update particle velocity using constriction factor formula:

- For each particle in the swarm:
- Return gbest as the best solution found.

## Considerations #

### Advantages #

- Improved convergence: The constriction factor helps balance exploration and exploitation, leading to faster and more stable convergence compared to the original PSO.
- Reduced parameter sensitivity: CFPSO is less sensitive to parameter tuning than the original PSO, making it easier to apply to various optimization problems.
- Adaptability: CFPSO can be applied to a wide range of optimization problems, including continuous, discrete, and mixed-variable problems.

### Disadvantages #

- Local optima: Like many metaheuristic algorithms, CFPSO may sometimes get trapped in local optima, especially in high-dimensional or multimodal problems.
- Premature convergence: If the constriction factor is not well-tuned, the swarm may converge prematurely, leading to suboptimal solutions.
- Computational cost: CFPSO requires evaluating the objective function for each particle in every iteration, which can be computationally expensive for problems with costly fitness evaluations.

## Heuristics #

### Parameter Settings #

- Set the constriction factor (chi) to a value between 0.6 and 0.8. A common choice is chi = 0.729.
- Set the cognitive and social component weights (phi1 and phi2) such that phi1 + phi2 > 4. A common choice is phi1 = phi2 = 2.05.
- Adjust the swarm size (num_particles) based on the problem complexity. Typically, 20 to 50 particles are used, but larger swarms may be necessary for high-dimensional problems.

### Initialization #

- Initialize particle positions uniformly within the search space bounds to ensure a diverse initial exploration.
- If problem-specific knowledge is available, use it to guide the initialization of particle positions.

### Termination Criteria #

- Set a maximum number of iterations (num_iterations) based on the problem complexity and available computational resources.
- Use a fitness threshold as a termination criterion if the optimal solution’s fitness is known or can be estimated.
- Monitor the swarm’s convergence by measuring the diversity of particle positions or the rate of improvement in the global best fitness. Terminate if the swarm has converged or if the improvement rate falls below a threshold.

### Boundary Handling #

- Apply boundary constraints to ensure particles remain within the valid search space.
- Use techniques like reflection, random reinitialization, or velocity clamping when particles violate boundary constraints.

### Problem-Specific Adaptations #

- Incorporate problem-specific heuristics or local search techniques to improve the quality of solutions found by CFPSO.
- Adapt the velocity update equation or introduce additional terms to balance exploration and exploitation based on the problem characteristics.
- Hybridize CFPSO with other optimization algorithms or techniques to leverage their strengths and overcome limitations.