We study the $(1 + 1)$-EA in dynamic linear environments, where in every generation selection is performed with respect to a freshly sampled linear function with positive weights. We consider the Dynamic Binary Value problem, where each generation uses a uniformly random permutation of $1,2,4,\dots,2^{n-1}$, and a Uniform weight variant, where the weights are drawn independently from $\mathrm{Unif}(0,1)$. Both of them have recently been integrated into the IOHprofiler platform and empirically studied. For both models we prove a sharp threshold in the mutation parameter $χ$ for mutation rate $χ/n$. Below the threshold, the expected optimisation time is $\mathcal{O}(n\log n)$, whereas above it the runtime becomes $2^{Ω(n)}$. For the Dynamic Binary Value problem in the exponential regime, we also quantify at what distance from the optimum the optimisation process stagnates. We show that there is a second threshold: a distance that is efficiently reached, but reaching any smaller distance takes exponential time. This quantifies and proves previous empirical findings.

翻译：我们研究动态线性环境中的$(1 + 1)$-EA，其中每一代都基于新抽取的具有正权重的线性函数进行选择。我们考虑动态二进制值问题，其中每一代使用${1,2,4,\dots,2^{n-1}}$的均匀随机排列，以及均匀权重变体，其中权重独立地服从$\mathrm{Unif}(0,1)$分布。这两个问题最近已被集成到IOHprofiler平台并进行了实证研究。对于这两种模型，我们在突变参数$\chi$（对应突变率$\chi/n$）上证明了一个尖锐阈值。低于该阈值时，期望优化时间为$\mathcal{O}(n\log n)$，而高于阈值时，运行时间变为$2^{\Omega(n)}$。对于指数机制下的动态二进制值问题，我们还量化了优化过程停滞时与最优解的距离。我们证明存在第二个阈值：一个可高效达到的距离，但达到任何更小的距离都需要指数时间。这量化和验证了先前的实证发现。