Variation Operators in Real-Valued Genetic Algorithms

For real-valued genetic algorithms, we need crossover and mutation operators that directly operate on real-valued chromosomes.

Crossover

Methods:

• Arithmetic Crossover
• Blend Crossover
• Simulated Binary Crossover

Mutation

Mutation acts on a single parent at a time. Given a parent $\mathbf{x}$, mutation samples:

$$x^{\prime }=x+ϵ$$

where $ϵ$ is a random perturbation.

Gaussian Mutation

A Gaussian mutation is a standard operator for real-valued chromosomes. For each gene:

$$x_i^{\prime }=x_i+ϵ,ϵ\sim \mathcal{N}(0,{\sigma }^2)$$

This is centered around the parent value $x_i$, with mostly small mutations and occasional larger steps. The $\sigma$ parameter sets the mutation strength.

Adaptive Gaussian Mutation

Mutation strength can be changed over generations, while the mutation distribution remains Gaussian.

At generation $g$:

$$x_i^{\prime }=x_i+{ϵ}_i(g),{ϵ}_i(g)\sim \mathcal{N}(0,\sigma (g{)}^2)$$

We use large $\sigma (g)$ early for strong exploration, then small $\sigma (g)$ later for finer exploration.

Boundary Handling

Real-valued mutation can produce infeasible values outside of our constraints:

$$x_i<x_i^{\text{min}}\text{or}{x}_i>x_i^{\text{max}}$$

This is common when mutation strength $\sigma$ is large (early generations). We can use the same strategies as covered in Constraints in Real-Valued GA: clipping, reflection, resampling.

Mutation Strength vs. Convergence

If we have $\sigma$ too large, we will have random walk behavior with poor convergence. If it is too small, we will converge prematurely due to loss of diversity.

Note that some varialbes require finer perturbations than others!