# Extremal Optimization #

## Name #

Extremal Optimization (EO)

## Taxonomy #

Extremal Optimization is a nature-inspired optimization technique that falls under the broad category of Computational Intelligence and Stochastic Optimization. It is closely related to other techniques such as Simulated Annealing and Genetic Algorithms.

- Computational Intelligence
- Stochastic Optimization
- Nature-Inspired Algorithms
- Extremal Optimization

- Nature-Inspired Algorithms

- Stochastic Optimization

## Strategy #

Extremal Optimization is inspired by the concept of self-organized criticality in complex systems, where systems naturally evolve towards a critical state characterized by power-law distributions. In EO, the optimization process is driven by the iterative improvement of the worst components of a solution, rather than focusing on the overall solution quality.

### Solution Representation #

In EO, a solution is typically represented as a set of components, each contributing to the overall fitness or cost of the solution. The specific representation depends on the problem being solved but could include binary strings, permutations, or real-valued vectors.

### Fitness Evaluation #

The fitness of each component in a solution is evaluated independently, allowing the identification of the worst-performing components. This local fitness evaluation is a key distinguishing feature of EO compared to other optimization techniques that evaluate the global fitness of an entire solution.

### Iterative Improvement #

At each iteration, EO selects the worst component (or a subset of the worst components) and replaces them with new, randomly generated values. This process of replacing the least fit components drives the solution towards a more optimal state.

### Acceptance Criteria #

The newly generated solution is accepted if it improves upon the current solution’s fitness. Some variations of EO may include an acceptance probability function, similar to Simulated Annealing, to allow the occasional acceptance of worse solutions to escape local optima.

## Procedure #

- Initialize:
- Generate an initial solution randomly or using problem-specific heuristics.
- Evaluate the fitness of each component in the solution.

- Repeat until termination criteria are met:
- Identify the worst component(s) based on their individual fitness values.
- Replace the worst component(s) with new, randomly generated values.
- Evaluate the fitness of the modified solution.
- If the new solution is better than the current solution, accept it as the current solution.
- Update the fitness values of the affected components.

- Return the best solution found.

### Data Structures #

- Solution: Represents a candidate solution to the problem, typically an array or vector of components.
- Component Fitness: An array or vector storing the individual fitness values of each component in the solution.

### Parameters #

- Population Size: The number of candidate solutions maintained during the optimization process (optional, as EO typically operates on a single solution).
- Replacement Rate: The number or percentage of worst components replaced at each iteration.
- Termination Criteria: Conditions for stopping the optimization process, such as a maximum number of iterations, a target fitness value, or a time limit.

## Considerations #

### Advantages #

- Simplicity: EO is conceptually simple and easy to implement compared to many other optimization algorithms.
- Flexibility: EO can be applied to a wide range of optimization problems, as long as a suitable solution representation and fitness evaluation can be defined.
- Local Search: The focus on improving the worst components allows EO to perform a targeted local search, which can be effective in escaping local optima.

### Disadvantages #

- Parameter Sensitivity: The performance of EO can be sensitive to the choice of parameters, such as the replacement rate, and may require problem-specific tuning.
- Lack of Diversity: As EO operates on a single solution, it may lack the population diversity that other algorithms, like Genetic Algorithms, can provide.
- Convergence Speed: EO may have slower convergence compared to some other optimization algorithms, particularly on problems with complex fitness landscapes.

## Heuristics #

### Replacement Rate #

- Start with a low replacement rate (e.g., replacing only the single worst component) and gradually increase it over the course of the optimization to balance exploration and exploitation.
- For problems with a large number of components, consider replacing a fixed percentage of the worst components rather than a fixed number.

### Initialization #

- Use problem-specific knowledge, when available, to generate initial solutions that are closer to the optimal solution, rather than purely random initialization.
- If the problem has known constraints, ensure that the initial solution satisfies them to avoid wasting computational resources on infeasible solutions.

### Fitness Evaluation #

- If the fitness evaluation is computationally expensive, consider caching the fitness values of components and only recalculating them when necessary (i.e., after a component has been modified).
- For problems with a large number of components, consider using approximate or surrogate fitness models to reduce the computational cost of fitness evaluations.

### Termination Criteria #

- Use a combination of termination criteria, such as a maximum number of iterations and a target fitness value, to ensure that the optimization process stops when a satisfactory solution is found or when further improvements are unlikely.
- Monitor the rate of improvement in the solution fitness and consider terminating the optimization if the rate falls below a certain threshold, indicating convergence.

### Variation Operators #

- Consider incorporating problem-specific variation operators, such as local search heuristics or specialized mutation operators, to improve the efficiency and effectiveness of the optimization process.
- Experiment with different replacement strategies, such as replacing the worst components with values drawn from a specific probability distribution rather than purely random values.