OneMax in Black-Box Models with Several Restrictions

As in classical runtime analysis the \textsc{OneMax} problem is the most prominent test problem also in black-box complexity theory. It is known that the unrestricted, the memory-restricted, and the ranking-based black-box complexities of this problem are all of order $n/\! \log n$, where $n$ denotes the length of the bit strings. The combined memory-restricted ranking-based black-box complexity of \textsc{OneMax}, however, was not known. We show in this work that it is $\Theta(n)$ for the smallest possible size bound, that is, for (1+1) black-box algorithms. We extend this result by showing that even if elitist selection is enforced, there exists a linear time algorithm optimizing \textsc{OneMax} with failure probability $o(1)$. This is quite surprising given that all previously regarded algorithms with $o(n\log n)$ runtime on \textsc{OneMax}, in particular the quite natural $(1+(\lambda,\lambda))$~GA, heavily exploit information encoded in search points of fitness much smaller than the current best-so-far solution. Also for other settings of $\mu$ and $\lambda$ we show that the $(\mu+\lambda)$ \emph{elitist memory-restricted ranking-based black-box complexity} of \textsc{OneMax} is as small as (an advanced version of) the information-theoretic lower bound. Our result enlivens the quest for natural evolutionary algorithms optimizing \textsc{OneMax} in $o(n \log n)$ iterations.