We study the $(1,\lambda)$-EA with mutation rate $c/n$ for $c\le 1$, where the population size is adaptively controlled with the $(1:s+1)$-success rule. Recently, Hevia Fajardo and Sudholt have shown that this setup with $c=1$ is efficient on \onemax for $s<1$, but inefficient if $s \ge 18$. Surprisingly, the hardest part is not close to the optimum, but rather at linear distance. We show that this behavior is not specific to \onemax. If $s$ is small, then the algorithm is efficient on all monotone functions, and if $s$ is large, then it needs superpolynomial time on all monotone functions. In the former case, for $c<1$ we show a $O(n)$ upper bound for the number of generations and $O(n\log n)$ for the number of function evaluations, and for $c=1$ we show $O(n\log n)$ generations and $O(n^2\log\log n)$ evaluations. We also show formally that optimization is always fast, regardless of $s$, if the algorithm starts in proximity of the optimum. All results also hold in a dynamic environment where the fitness function changes in each generation.

翻译：我们研究突变率为$c/n$（$c\le 1$）的$(1,\lambda)$-EA，其中种群规模通过$(1:s+1)$-成功规则进行自适应控制。最近，Hevia Fajardo与Sudholt证明，当$c=1$时该配置在\onemax问题上对$s<1$有效，但对$s\ge 18$无效。令人惊讶的是，最困难的部分并非接近最优解，而是位于线性距离处。我们证明这种行为并非\onemax所特有：若$s$较小，则算法对所有单调整函数高效；若$s$较大，则算法在所有单调整函数上均需超多项式时间。在前一种情况下，对于$c<1$，我们给出$O(n)$代次和$O(n\log n)$函数评估次数的上界；对于$c=1$，我们给出$O(n\log n)$代次和$O(n^2\log\log n)$评估次数的上界。我们还严格证明，若算法从最优解附近启动，则无论$s$取值如何，优化过程始终快速。所有结论在适应度函数每代变化的动态环境中同样成立。