# Real-Coded Genetic Algorithm #

## Name #

Real-Coded Genetic Algorithm (RCGA)

## Taxonomy #

The Real-Coded Genetic Algorithm is a variant of the Genetic Algorithm (GA) that belongs to the field of Evolutionary Computation, a subfield of Computational Intelligence. It is closely related to other GA variants such as the Binary-Coded Genetic Algorithm and the Steady-State Genetic Algorithm.

- Computational Intelligence
- Evolutionary Computation
- Evolutionary Algorithms
- Genetic Algorithms
- Real-Coded Genetic Algorithm

- Genetic Algorithms

- Evolutionary Algorithms

- Evolutionary Computation

## Strategy #

The Real-Coded Genetic Algorithm operates on a population of candidate solutions, where each solution is represented as a vector of real numbers. This representation allows for a more natural mapping between the problem space and the search space, especially for optimization problems with continuous variables.

### Initialization #

The algorithm begins by initializing a population of randomly generated real-valued vectors. Each vector represents a potential solution to the optimization problem at hand. The size of the population is a user-defined parameter and can impact the algorithm’s exploration and exploitation capabilities.

### Fitness Evaluation #

Each candidate solution in the population is evaluated using a fitness function that measures its quality or performance with respect to the optimization objective. The fitness function is problem-specific and must be designed to accurately reflect the desired characteristics of the optimal solution.

### Selection #

The selection process determines which individuals from the current population will be chosen as parents to create the next generation. Common selection methods include tournament selection, where a fixed number of individuals compete based on their fitness, and fitness-proportionate selection, where individuals are chosen with a probability proportional to their relative fitness.

### Crossover #

Crossover is a key genetic operator that combines genetic information from two parent solutions to create one or more offspring solutions. In the Real-Coded Genetic Algorithm, crossover operators are designed to work with real-valued vectors. Examples include blend crossover (BLX-α), simulated binary crossover (SBX), and arithmetic crossover.

### Mutation #

Mutation introduces random changes to the genetic material of the offspring solutions, helping to maintain diversity in the population and explore new regions of the search space. Mutation operators for real-coded GAs include uniform mutation, where each gene is modified with a certain probability, and Gaussian mutation, where a random value drawn from a Gaussian distribution is added to each gene.

### Replacement #

After the offspring solutions have been generated through crossover and mutation, they are evaluated using the fitness function. The replacement strategy determines which individuals from the current population and the offspring population will form the next generation. Common replacement strategies include generational replacement, where the entire population is replaced by the offspring, and steady-state replacement, where only a subset of the population is replaced.

### Termination #

The algorithm iterates through the selection, crossover, mutation, and replacement steps until a termination criterion is met. Common termination criteria include reaching a maximum number of generations, achieving a satisfactory fitness level, or observing convergence in the population.

## Procedure #

### Data Structures #

- Individual: A real-valued vector representing a candidate solution
- Population: An array of individuals
- Fitness: An array of fitness values corresponding to each individual in the population

### Parameters #

- PopulationSize: The number of individuals in the population
- CrossoverProbability: The probability of applying crossover to a pair of parent solutions
- MutationProbability: The probability of applying mutation to an offspring solution
- MaxGenerations: The maximum number of generations before termination

### Pseudocode #

- Initialize:
- Generate PopulationSize random individuals
- Evaluate the fitness of each individual in the population

- While termination criteria are not met:
- Selection:
- Select parent solutions from the population based on their fitness

- Crossover:
- With probability CrossoverProbability, apply crossover to the selected parents to create offspring solutions

- Mutation:
- With probability MutationProbability, apply mutation to the offspring solutions

- Evaluation:
- Evaluate the fitness of each offspring solution

- Replacement:
- Replace individuals in the population with the offspring solutions based on the replacement strategy

- Selection:
- Return the best solution found

## Considerations #

### Advantages #

- Real-coded representation allows for a more natural mapping between the problem space and the search space
- Suitable for optimization problems with continuous variables
- Can efficiently explore and exploit the search space through the use of specialized crossover and mutation operators

### Disadvantages #

- The choice of crossover and mutation operators can significantly impact the algorithm’s performance
- May require tuning of the population size, crossover probability, and mutation probability parameters for optimal performance
- Can be computationally expensive, especially for large population sizes and complex fitness evaluations

## Heuristics #

### Population Size #

- A larger population size can improve exploration but may slow down convergence
- A smaller population size can lead to faster convergence but may result in premature convergence to suboptimal solutions
- As a general rule, the population size should be proportional to the problem complexity and the desired solution quality

### Crossover Probability #

- A higher crossover probability encourages exploration by creating more diverse offspring solutions
- A lower crossover probability can help exploit good solutions by preserving their genetic material
- Typical values range from 0.6 to 0.9

### Mutation Probability #

- A higher mutation probability can help maintain diversity and escape local optima
- A lower mutation probability can help fine-tune solutions and converge to the optimal solution
- Typical values range from 0.01 to 0.1

### Termination Criteria #

- The maximum number of generations should be set based on the problem complexity and available computational resources
- The fitness threshold for termination should be set based on the desired solution quality and the known optimal fitness (if available)
- Convergence can be detected by monitoring the diversity of the population or the improvement in the best fitness over a number of generations