# Local Best Particle Swarm Optimization #

## Name #

Local Best Particle Swarm Optimization (LBPSO), also known as:

- Lbest PSO
- Local Best PSO
- Local Neighborhood PSO

## Taxonomy #

Local Best Particle Swarm Optimization is a variant of the Particle Swarm Optimization algorithm, which falls under the broader category of Swarm Intelligence, a subfield of Computational Intelligence.

- Computational Intelligence
- Swarm Intelligence
- Particle Swarm Optimization (PSO)
- Local Best Particle Swarm Optimization (LBPSO)

- Particle Swarm Optimization (PSO)

- Swarm Intelligence

## Strategy #

The Local Best Particle Swarm Optimization algorithm is an extension of the standard PSO that incorporates a local neighborhood topology to guide the search process. In LBPSO, each particle is connected to a subset of the swarm, forming a local neighborhood, rather than being influenced by the global best position as in the original PSO.

### Local Neighborhood Topology #

The local neighborhood topology defines the connections between particles in the swarm. Common topologies include the ring topology, where each particle is connected to its k nearest neighbors, and the von Neumann topology, which arranges particles in a grid and connects each particle to its immediate neighbors.

The choice of topology affects the flow of information within the swarm and the balance between exploration and exploitation. The local neighborhood allows particles to explore their local regions more thoroughly while reducing the influence of the global best position, potentially leading to a more diverse search and improved performance on multimodal problems.

### Position and Velocity Update #

Similar to the standard PSO, LBPSO updates the position and velocity of each particle iteratively. However, instead of using the global best position to guide the search, each particle is influenced by the best position found within its local neighborhood (lbest).

The velocity update equation incorporates the particle’s personal best position (pbest) and the local best position (lbest) of its neighborhood. The updated velocity is then used to compute the new position of the particle in the search space.

### Convergence and Termination #

The LBPSO algorithm continues iterating until a specified termination criterion is met, such as reaching a maximum number of iterations or finding a solution of sufficient quality. The local neighborhood topology can slow down the convergence of the swarm compared to the global best PSO, as information propagates more gradually through the local connections. However, this slower convergence can be beneficial in maintaining population diversity and avoiding premature convergence to suboptimal solutions.

## Procedure #

### Data Structures #

- Particle: Represents a candidate solution in the search space, consisting of:
- Position: The current position of the particle in the search space.
- Velocity: The velocity of the particle, determining its direction and step size.
- Personal Best (pbest): The best position found by the particle so far.

- Swarm: A collection of particles, each connected to a local neighborhood based on the chosen topology.

### Parameters #

- Population Size: The number of particles in the swarm.
- Dimension: The dimensionality of the search space.
- Inertia Weight (w): Controls the influence of the previous velocity on the current velocity.
- Cognitive Coefficient (c1): Determines the attraction towards the particle’s personal best position.
- Social Coefficient (c2): Determines the attraction towards the local best position in the particle’s neighborhood.
- Maximum Velocity (vmax): Limits the maximum velocity of particles to prevent excessive oscillations.
- Neighborhood Size (k): The number of neighbors connected to each particle in the local topology.

### Pseudocode #

- Initialize the swarm
- For each particle in the swarm:
- Randomly initialize the particle’s position and velocity
- Evaluate the fitness of the particle’s position
- Set the particle’s personal best (pbest) to its current position

- Determine the local best (lbest) position for each particle based on the chosen neighborhood topology

- For each particle in the swarm:
- While termination criteria are not met:
- For each particle in the swarm:
- Update the particle’s velocity using the velocity update equation:
- v = w * v + c1 * rand() * (pbest - position) + c2 * rand() * (lbest - position)

- Update the particle’s position using the position update equation:
- position = position + v

- Evaluate the fitness of the particle’s new position
- If the new position is better than the particle’s personal best (pbest):
- Update the particle’s personal best (pbest)

- Update the particle’s velocity using the velocity update equation:
- Update the local best (lbest) position for each particle based on the new positions and fitness values

- For each particle in the swarm:
- Return the best solution found by the swarm

## Considerations #

### Advantages #

- Improved exploration: The local neighborhood topology allows particles to explore their local regions more thoroughly, potentially leading to better coverage of the search space and reduced risk of premature convergence.
- Increased diversity: By limiting the influence of the global best position, LBPSO maintains a higher level of diversity in the swarm, which can be beneficial for multimodal problems and avoiding local optima.
- Robustness: LBPSO is less sensitive to the choice of initial conditions and can be more robust to parameter settings compared to the global best PSO.

### Disadvantages #

- Slower convergence: The local neighborhood topology can slow down the convergence of the swarm, as information propagates more gradually through the local connections. This may require more iterations to reach a satisfactory solution.
- Increased computational complexity: Maintaining and updating the local neighborhood topology adds computational overhead compared to the global best PSO, especially for large swarm sizes and complex topologies.
- Parameter sensitivity: The performance of LBPSO can be sensitive to the choice of neighborhood topology and size, requiring careful tuning for different problem instances.

## Heuristics #

### Neighborhood Topology Selection #

- Use the ring topology for problems with a small number of local optima, as it allows for a balanced exploration-exploitation trade-off.
- Consider the von Neumann topology for problems with a large number of local optima or when a more thorough local search is desired.
- Experiment with different neighborhood sizes (k) to find the optimal balance between local and global information sharing. Smaller neighborhoods promote local exploration, while larger neighborhoods increase the influence of the global best position.

### Parameter Tuning #

- Set the inertia weight (w) to a value between 0.5 and 0.9 to balance the influence of the previous velocity and the current attraction forces. Higher values favor exploration, while lower values promote exploitation.
- Adjust the cognitive coefficient (c1) and social coefficient (c2) based on the problem characteristics. A higher cognitive coefficient encourages particles to explore their personal best regions, while a higher social coefficient promotes movement towards the local best position.
- Limit the maximum velocity (vmax) to a fraction of the search space range (e.g., 10-20%) to prevent particles from overshooting the optimal solution and oscillating excessively.

### Swarm Size and Iteration Count #

- Start with a swarm size of 20-50 particles and adjust based on the problem complexity and dimensionality. Larger swarms can explore the search space more thoroughly but may require more iterations to converge.
- Set the maximum number of iterations based on the available computational resources and the desired solution quality. Monitor the convergence behavior and adjust the iteration count accordingly.

### Hybrid Approaches #

- Consider combining LBPSO with other techniques, such as local search methods or evolutionary operators, to further improve its performance. For example, applying a local search procedure to the best solutions found by LBPSO can help refine the results and escape local optima.
- Experiment with adaptive or self-adaptive strategies for adjusting the LBPSO parameters during the optimization process based on the search progress and population diversity.