Editing Mutation (evolutionary algorithm) (section)

=== Mutation with consideration of restrictions ===
One possible form of changing the value of a gene while taking its value range <math>[x_{\min}, x_{\max}]</math> into account is the mutation ''relative parameter change'' of the evolutionary algorithm GLEAM (General Learning Evolutionary Algorithm and Method),<ref>{{Citation |last1=Blume |first1=Christian |last2=Jakob |first2=Wilfried |title=GLEAM - An Evolutionary Algorithm for Planning and Control Based on Evolution Strategy |date=2002 |url=https://publikationen.bibliothek.kit.edu/170053025/3814288 |work=Conf. Proc. of Genetic and Evolutionary Computation Conference (GECCO 2002) |volume=Late Breaking Papers |pages=31–38 |access-date=2023-01-01 }}</ref> in which, as with the mutation presented earlier, small changes are more likely than large ones.
[[File:Probabilty distribution of the muatation relative parameter change.png|thumb|264x264px|Distribution of probabilities for k=10 sub-areas of the total change interval. The sub-areas each cover 1/k of the width of the total change interval.]]
First, an equally distributed decision is made as to whether the current value <math>x</math> should be increased or decreased and then the corresponding total change interval is determined. [[Without loss of generality]], an increase is assumed for the explanation and the total change interval is then <math>[x, x_\max]</math>. It is divided into <math>k</math> sub-areas of equal size with the width <math>\delta</math>, from which <math>k</math> sub-change intervals of different size are formed:
:<math>i</math>-th sub-change interval: <math>[x, x + \delta \cdot i]</math> with
:<math>\delta = \frac{(x_\text{max} - x)}{k}</math> and <math>i = 1, \dots, k</math>
Subsequently, one of the <math>k</math> sub-change intervals is selected in equal distribution and a random number, also equally distributed, is drawn from it as the new value <math>x'</math> of the gene. The resulting summed probabilities of the sub-change intervals result in the probability distribution of the <math>k</math> sub-areas shown in the adjacent figure for the exemplary case of <math>k=10</math>. This is not a normal distribution as before, but this distribution also clearly favours small changes over larger ones. 

This mutation for larger values of <math>k</math>, such as 10, is less well suited for tasks where the optimum lies on one of the value range boundaries. This can be remedied by significantly reducing <math>k</math> when a gene value approaches its limits very closely.