# (μ+1)-Evolution Strategy #

## Name #

(μ+1)-Evolution Strategy, (μ+1)-ES

## Taxonomy #

The (μ+1)-Evolution Strategy is a stochastic optimization algorithm belonging to the field of Evolutionary Computation, a subfield of Computational Intelligence. It is closely related to other Evolution Strategies, such as (1+1)-ES and (μ+λ)-ES.

- Computational Intelligence
- Biologically Inspired Computation
- Evolutionary Computation
- Evolutionary Algorithms
- Evolution Strategies
- (μ+1)-Evolution Strategy

- Evolution Strategies

- Evolutionary Algorithms

- Evolutionary Computation

- Biologically Inspired Computation

## Strategy #

The (μ+1)-Evolution Strategy is a population-based optimization technique that maintains a population of μ individuals, each representing a candidate solution to the problem at hand. The algorithm iteratively generates new offspring by applying mutation to a single parent, selected uniformly at random from the population. The mutation operator perturbs the parent’s decision variables, typically by adding a normally distributed random value to each variable. The resulting offspring is then evaluated using a fitness function, which measures its quality as a solution to the problem.

After the offspring is generated and evaluated, the algorithm performs a selection step to determine which individuals will survive to the next generation. In the (μ+1)-ES, the population for the next generation is selected from the union of the current population and the newly generated offspring. The best μ individuals, as determined by their fitness values, are retained, while the worst individual is discarded. This selection mechanism ensures that the population size remains constant at μ throughout the optimization process.

The (μ+1)-ES continues to iterate, generating new offspring and performing selection, until a termination criterion is met. Common termination criteria include reaching a maximum number of generations, finding a solution of satisfactory quality, or observing convergence in the population.

## Procedure #

### Data Structures #

- Individual: A structure representing a candidate solution, which consists of:
- Object Variables: An array of real-valued object variables.
- Strategy Parameters: An array of real-valued strategy parameters (mutation step sizes).
- Fitness: The fitness value of the individual.

- Population: An array of individuals.
- Offspring: A single individual generated by mutating a parent.
- Best Individual: The individual with the best fitness value found so far.

### Parameters #

- μ (mu): The number of parent individuals in the population.
- Initial Step Size: The initial value for the strategy parameters (mutation step sizes).
- Termination Criterion: The condition for terminating the algorithm, such as reaching a maximum number of iterations or achieving a satisfactory fitness value.

### Steps #

- Initialize the population
- For each individual in the population (μ individuals):
- Randomly initialize the object variables.
- Set the strategy parameters to the initial step size.
- Evaluate the fitness of the individual.

- Update the best individual if necessary.

- For each individual in the population (μ individuals):
- While the termination criterion is not met:
- Select a parent individual
- Randomly select one individual from the population.

- Create an offspring by mutating the parent
- Copy the object variables and strategy parameters from the parent to the offspring.
- Mutate the strategy parameters of the offspring.
- For each strategy parameter:
- Multiply the strategy parameter by a random factor drawn from a lognormal distribution.

- For each strategy parameter:
- Mutate the object variables of the offspring.
- For each object variable:
- Add a random value drawn from a normal distribution with mean 0 and standard deviation equal to the corresponding mutated strategy parameter.

- For each object variable:
- Evaluate the fitness of the offspring.

- Select the survival individual
- Compare the fitness of the offspring with the fitness of the parent.
- If the offspring has a better (or equal) fitness than the parent:
- Replace the parent with the offspring in the population.

- Otherwise:
- Discard the offspring.

- Update the best individual if necessary.

- Select a parent individual
- Return the best individual found.

## Considerations #

### Advantages #

- Simplicity: The (μ+1)-ES is easy to understand and implement, making it a good choice for practitioners new to Evolutionary Computation.
- Robustness: The algorithm can handle noisy and multimodal fitness landscapes, as well as problems with non-differentiable or discontinuous objective functions.
- Parallelization: The fitness evaluations of the population can be easily parallelized, allowing for efficient use of computational resources.

### Disadvantages #

- Limited Exploration: The (μ+1)-ES relies on a single offspring per generation, which may limit its ability to explore the search space effectively, especially in high-dimensional problems.
- Slow Convergence: The algorithm may converge slowly, particularly in the later stages of the optimization process, as it relies on small, random perturbations to improve the solution quality.
- Parameter Sensitivity: The performance of the (μ+1)-ES can be sensitive to the choice of the population size (μ) and the mutation operator, requiring careful tuning for each problem.

## Heuristics #

### Population Size #

- Start with a moderate population size (e.g., μ = 10) and adjust based on the problem complexity and available computational resources.
- Larger population sizes can improve exploration but may slow down convergence.
- Smaller population sizes can lead to faster convergence but may increase the risk of premature convergence to suboptimal solutions.

### Mutation Operator #

- Use a normally distributed random perturbation with zero mean and a problem-dependent standard deviation.
- Adapt the mutation step size during the optimization process, e.g., by using a self-adaptive strategy or a deterministic schedule.
- Consider using correlated mutations or other advanced mutation schemes for high-dimensional problems or problems with strong variable interactions.

### Termination Criterion #

- Set a maximum number of generations based on the available computational budget and the problem complexity.
- Monitor the progress of the best fitness value and stop the algorithm if no significant improvement is observed over a predefined number of generations.
- Use a convergence criterion based on the diversity of the population, e.g., stop when the variance of the fitness values falls below a threshold.

### Hybridization #

- Consider combining the (μ+1)-ES with local search techniques, such as hill climbing or pattern search, to improve the exploitation of promising regions in the search space.
- Incorporate domain-specific knowledge or heuristics into the mutation operator or the fitness function to guide the search process more effectively.