A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems — general results which translate an expected movement away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) negative drift in populations method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof of this result, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a (1-ω(n-1/2)) factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than Θ(1/n), which includes the heavytailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally use our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on ONEMAX for arbitrary population size.

中文翻译：

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通过负乘法漂移的非精英进化算法的下限

到目前为止，已经显示了大量基于非精英群体的进化算法的下界。由于（难以避免）使用负漂移定理，它们中的大多数在技术上要求很高——一般结果将预期的远离目标的运动转化为高击球时间。我们为乘法漂移场景提出了一个简单的负漂移定理，并表明它可以简化现有的分析。我们更详细地讨论了 Lehre (PPSN 2010) 种群中的负漂移方法，这是证明离散搜索空间的基于非精英突变的进化算法运行时间下界的最通用工具之一。与其他论点一起，我们获得了该结果的另一种更简单的证明，这也加强和简化了这种方法。特别是，现在只需要验证先前结果的五个技术条件中的三个。我们获得的下界是明确的，而不仅仅是渐近的。这允许计算具体算法的具体下限，但也使我们能够证明当再现率仅低于阈值 (1-ω(n-1/2)) 因子时，超多项式运行时间已经出现。对于使用具有随机突变率的标准位突变的算法的特殊情况（在超启发式语言中称为均匀混合），我们证明了 Dang 和 Lehre (PPSN 2016) 所述的结果并将其扩展到 Θ 以外的突变率(1/n)，其中包括 Doerr、Le、Makhmara 和 Nguyen 提出的重尾变异算子（GECCO 2017）。