style="font-variant:small-caps;">OneMax</span>]{}and [<span style="font-variant:small-caps;">BinVal</span>]{}. When minimizing [<span style="font-variant:small-caps;">OneMax</span>]{}, the (1+1) EA accepts a new bit string in the selection step (Step 4) if the number of one-bits did not increase. In contrast, when minimizing [<span style="font-variant:small-caps;">BinVal</span>]{}, the inequality $2^k > \sum_{i=1}^{k-1}{2^{i-1}}$ implies that the (1+1) EA accepts a new bit string if and only if the highest-index bit that is touched in the mutation step (Step 3) is flipped from one to zero.

In spite of this difference in behavior, Droste, Jansen and Wegener showed in their seminal paper [@DrosteJW02] that for all linear functions the expected optimization time of the (1+1) EA is $\Theta(n \log n)$.

\[thm:ealinear\] For all positive integers $n\in\mathbb{N}$, the expected running time of the (1+1) EA on the class of linear functions with non-zero weights is $\Theta(n \log n)$.

The proof of Droste, Jansen and Wegener applies a level based argument to the potential function (called *artificial fitness function*) $g\colon \{0,1\}^n \to \mathbb{R}$ such that for all $x\in\{0,1\}^n$ $$\label{eq:droste}
g(x)=\sum_{i=1}^{\lfloor \frac{n}{2}\rfloor}x_i+\sum_{i=\lfloor \frac{n}{2}\rfloor +1}^n 2x_i.$$

A much easier proof avoiding partitioning arguments and instead working completely in the framework of drift analysis, was given by He and Yao in [@HeY04]. There, additive drift analysis is applied to 