Plus Strategies are Exponentially Slower for Planted Optima of Random Height

We compare the $(1,\lambda)$-EA and the $(1 + \lambda)$-EA on the recently introduced benchmark DisOM, which is the OneMax function with randomly planted local optima. Previous work showed that if all local optima have the same relative height, then the plus strategy never loses more than a factor $O(n\log n)$ compared to the comma strategy. Here we show that even small random fluctuations in the heights of the local optima have a devastating effect for the plus strategy and lead to super-polynomial runtimes. On the other hand, due to their ability to escape local optima, comma strategies are unaffected by the height of the local optima and remain efficient. Our results hold for a broad class of possible distortions and show that the plus strategy, but not the comma strategy, is generally deceived by sparse unstructured fluctuations of a smooth landscape.

Hardest Monotone Functions for Evolutionary Algorithms

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.