The study of hardest and easiest fitness landscapes is an active area of research. Recently, Kaufmann, Larcher, Lengler and Zou conjectured that for the self-adjusting $(1,\lambda)$-EA, Adversarial Dynamic BinVal (ADBV) is the hardest dynamic monotone function to optimize. We introduce the function Switching Dynamic BinVal (SDBV) which coincides with ADBV whenever the number of remaining zeros in the search point is strictly less than $n/2$, where $n$ denotes the dimension of the search space. We show, using a combinatorial argument, that for the $(1+1)$-EA with any mutation rate $p \in [0,1]$, SDBV is drift-minimizing among the class of dynamic monotone functions. Our construction provides the first explicit example of an instance of the partially-ordered evolutionary algorithm (PO-EA) model with parameterized pessimism introduced by Colin, Doerr and F\'erey, building on work of Jansen. We further show that the $(1+1)$-EA optimizes SDBV in $\Theta(n^{3/2})$ generations. Our simulations demonstrate matching runtimes for both static and self-adjusting $(1,\lambda)$ and $(1+\lambda)$-EA. We further show, using an example of fixed dimension, that drift-minimization does not equal maximal runtime.

翻译：[translated abstract in Chinese] 关于最难与最易适应度景观的研究是当前活跃的研究领域。近期，Kaufmann、Larcher、Lengler 和 Zou 猜想，对于自适应 $(1,\lambda)$-EA，对抗性动态BinVal（ADBV）是最难优化的动态单调函数。我们引入了切换动态BinVal（SDBV）函数，该函数在搜索点中剩余零的个数严格小于 $n/2$ 时（其中 $n$ 表示搜索空间维度）与 ADBV 保持一致。通过组合论论证，我们证明对于采用任意变异率 $p \in [0,1]$ 的 $(1+1)$-EA，SDBV 在动态单调函数类中具有最小漂移量。该构造首次为 Colin、Doerr 和 F\'erey 在 Jansen 工作基础上提出的参数化悲观偏序进化算法（PO-EA）模型提供了显式实例。进一步研究表明，$(1+1)$-EA 可在 $\Theta(n^{3/2})$ 代内优化 SDBV。仿真实验表明，静态与自适应 $(1,\lambda)$ 和 $(1+\lambda)$-EA 均展现出匹配的运行时间。此外，通过固定维度的示例，我们证明最小漂移量并不等同于最长运行时间。