Hardest Monotone Functions for Evolutionary Algorithms

In this paper we revisit the question how hard it can be for the $(1+1)$ Evolutionary Algorithm to optimize monotone pseudo-Boolean functions. By introducing a more pessimistic stochastic process, the partially-ordered evolutionary algorithm (PO-EA) model, Jansen first proved a runtime bound of $O(n^{3/2})$. More recently, Lengler, Martinsson and Steger improved this upper bound to $O(n \log^2 n)$ by an entropy compression argument. In this work, we analyze monotone functions that may adversarially vary at each step of the optimization, so-called dynamic monotone functions. We introduce the function Switching Dynamic BinVal (SDBV) and prove, using a combinatorial argument, that for the $(1+1)$-EA with any mutation rate $p \in [0,1]$, SDBV is drift minimizing within the class of dynamic monotone functions. We further show that the $(1+1)$-EA optimizes SDBV in $\Theta(n^{3/2})$ generations. Therefore, our construction provides the first explicit example which realizes the pessimism of the \poea model. Our simulations demonstrate matching runtimes for both the static and the self-adjusting $(1,\lambda)$-EA and $(1+\lambda)$-EA. Moreover, devising an example for fixed dimension, we illustrate that drift minimization does not equal maximal runtime beyond asymptotic analysis.