for all initial individuals.

The main reason, however, for not using the weaker condition (\[eq:average\]) is that this requires a deeper understanding of the probability distribution of $x^{(t)}$.

Let us stress that finding a potential function satisfying the stronger point-wise drift condition is usually very tricky. For example, one may ask why not take ${\textsc{OneMax}\xspace}(x)$ as potential function to bound the expected optimization time of the (1+1) EA for minimizing linear functions.

However, an easy observation reveals that there is an objective function $f$ and a search point $x$ such that $g$ yields to small a drift with respect to $f$. To see this, let $x=(x_1,\dots,x_n):=(0,\dots,0,1)$ and let $f:={\textsc{BinVal}\xspace}$ be the function to be minimized. Then the point-wise drift (\[eq:pointwise\]) with respect to ${\textsc{OneMax}\xspace}$ is only $1/n^2$. This example shows that finding a potential function yielding point-wise drift for all $x$ and all $f$ may be difficult. This observation is not to be confused with that in the discussion following (\[eq:binval\]). There, we determined the drift using the function [<span style="font-variant:small-caps;">BinVal</span>]{}itself as potential. Here, we use [<span style="font-variant:small-caps;">OneMax</span>]{}, that is, the 1-norm as potential function.

J[ä]{}gersk[ü]{}pper [@Jagerskupper08] was the first to overcome the difficulties of point-wise drift. While he still 