One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed by the algorithm in increasing order, possibly skipping levels. An easy, but often strong upper bound for the run time can then be derived by adding the reciprocals of the probabilities to leave the levels (or upper bounds for these). Unfortunately, a similarly effective method for proving lower bounds has not yet been established. The strongest such method, proposed by Sudholt (2013), requires a careful choice of the viscosity parameters $\gamma_{i,j}$, $0 \le i < j \le n$. In this paper we present two new variants of the method, one for upper and one for lower bounds. Besides the level leaving probabilities, they only rely on the probabilities that levels are visited at all. We show that these can be computed or estimated without greater difficulties and apply our method to reprove the following known results in an easy and natural way. (i) The precise run time of the \oea on \leadingones. (ii) A lower bound for the run time of the \oea on \onemax, tight apart from an $O(n)$ term. (iii) A lower bound for the run time of the \oea on long $k$-paths.

翻译：最先且最容易使用的方法之一, 用来证明进化算法的运行时间限制。 不幸的是, 一个类似的证明较低限值的有效方法尚未建立。 Sudholt (2013年)提出的最强的这种方法需要谨慎地选择粘度参数 $\ gamma ⁇ i, j}$, $\le i < j\\le n$。 在本文中,我们提出两种新的方法变量, 一种是上限, 一种是下限 。 除了水平的概率, 它们只能依赖水平所访问的概率。 我们显示, 这些数据可以在没有更大困难的情况下计算或估计, 并且用我们的方法在已知的硬度参数上选择 $ $ $, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元 i < j\ le le n$。 在本文中, 我们提出两种新的方法, 一个是上限值, 一个是下限值, 一个是下限, 仅取决于所有水平所访问的概率。 我们表明, 可以在没有更大难度的情况下计算或估计这些方法, 以更低的 美元 美元 递增 rooe 。 a a rodeal a rodeal a a a a rodeal a rout a a ro ro rout a la la la la la la la la a rout a ro a la a la a la a la rout rout la la la routut ro a ro a rout ro ro a ro a rout ro a ro a ro a la a ro a ro a ro ro a ro ro a rout ro a rout ro ro a la ro ro a ro a ro a ro a ro ro ro rout rout rout a rout a rout a ro a ro ro a ro a ro a ro a ro a ro a ro a ro a ro a ro a ro a la ro a ro a la la ro a ro ro ro a