Hardest Monotone Functions for Evolutionary Algorithms

Marc Kaufmann, Maxime Larcher, Johannes Lengler, Oliver Sieberling

Abstract: The hardness of optimizing monotone functions using the \((1+1)\)-EA has been an open problem for a long time. By introducing a more pessimistic stochastic process, the partially-ordered evolutionary algorithm (PO-EA) model, Jansen proved a runtime bound of \(O(n^{3/2})\). In 2019, Lengler, Martinsson and Steger improved this upper bound to \(O(n \log ^2 n)\) leveraging an entropy compression argument. We continue this line of research by analyzing monotone functions that may vary at each step, 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, SDBV is drift minimizing within the class of dynamic monotone functions. We further show that the \((1+1)\)-EA optimizes SDBV in \(\varTheta (n^{3/2})\) generations. Therefore, our construction provides the first explicit example which realizes the pessimism of the PO-EA model. Our simulations demonstrate matching runtimes for both static and self-adjusting \((1,\lambda )\) and \((1+\lambda )\)-EA. We additionally demonstrate, devising an example of fixed dimension, that drift minimization does not equal maximal runtime.