Selection

Basic selection (s0)

1. Select a simplex $S$ (i.e. $D+1$ individuals) at random. Be sure that the current one is in this list. If not, replace the last of the list by the current one.
2. Select the three first ones.
3. Sort them, by increasing order of value (fitness). They are $x_{best}$, $x_{worst2}$, and $w_{worst}$.

If $x_{best}$ is better than the best ever found (i.e. $Best$), set $Best=x_{best}$.

Adaptive probability

The rationale is the following:
the more the simplex volume increases, the less the algorithm is successful, and the more one needs randomness to increase the diversity.

Compute the volume $V(1)$ of the simplex. If the previous volume is zero (in practice, too small) the probability $p$ is set to 0.5. If not, the formula is

$$p=\frac{1}{1+e^{-\frac{V(1)-V(0)}{V(0)}}}$$

After that, we simply set $V(0)=V(1)$, to save the simplex volume as the "previous" one.