8.3.3 Mutation

Mutation is the random change of (usually small) parts of the genotype of an individual. The first two mutation operators defined in this section are the most important ones for continuous variables. The Gaussian mutation operator is advantageous since it is very flexible and supports both fine tuning of solutions and searching the domain.

8.3.3.1 The Random Mutation Operator

Let $x\in[a,b]$ be a real variable. The random mutation operator $M_U$ changes $x$ to

$$M_U(x):=U([a,b]),$$

where $U([a,b])$ denotes the uniform distribution on the interval $[a,b]$. Thus the new value of $x$ is independent from its previous value. $M_U$ is applied with a probability of a few percent. However a much better and more versatile mutation operator is the Gaussian mutation operator described in the following.

8.3.3.2 The Gaussian Mutation Operator

Let $x\in[a,b]$ be a real variable. Then the Gaussian mutation operator $M_G$ changes $x$ to

$$M_G(x):=\min( \max( N(x,\sigma), a), b),$$

where $\sigma$ may depend on the length $\ell:=b-a$ of the interval and typically $\sigma/\ell=1/10$. $M_G$ is applied with probability $p_m$ to each variable, where $p_m$ generally has a value of a few percent. The value of $\sigma$ may also depend on time, i.e., the number of the current generation, and usually decreases with time. The reason for a decreasing $\sigma$ is that stronger mutation during the beginning of an optimization supports the sampling of the search space and smaller displacements towards the end aid in fine tuning extremum values.

Another advantageous setup is to use two mutation operators: one with a high and one with a low value of $\sigma$. This enables to achieve a similar effect as a decreasing $\sigma$.