  The OneMax problem, alternatively known as the Hamming distance problem, is
often referred to as the "drosophila of evolutionary computation (EC)", because
of its high relevance in theoretical and empirical analyses of EC approaches.
It is therefore surprising that even for the simplest of all mutation-based
algorithms, Randomized Local Search and the (1+1) EA, the optimal mutation
rates were determined only very recently, in a GECCO 2019 poster.
  In this work, we extend the analysis of optimal mutation rates to two
variants of the $(1+\lambda)$ EA and to the $(1+\lambda)$ RLS. To do this, we
use dynamic programming and, for the $(1+\lambda)$ EA, numeric optimization,
both requiring $\Theta(n^3)$ time for problem dimension $n$. With this in hand,
we compute for all population sizes $\lambda \in \{2^i \mid 0 \le i \le 18\}$
and for problem dimension $n \in \{1000, 2000, 5000\}$ which mutation rates
minimize the expected running time and which ones maximize the expected
progress.
  Our results do not only provide a lower bound against which we can measure
common evolutionary approaches, but we also obtain insight into the structure
of these optimal parameter choices. For example, we show that, for large
population sizes, the best number of bits to flip is not monotone in the
distance to the optimum. We also observe that the expected remaining running
time are not necessarily unimodal for the $(1+\lambda)$ EA$_{0 \rightarrow 1}$
with shifted mutation.
