An Extended Jump Functions Benchmark for the Analysis of Randomized Search Heuristics

Jump functions are the {most-studied} non-unimodal benchmark in theory of randomized search heuristics, particular, evolutionary algorithms (EAs). They have significantly improved our understanding how EAs escape from local optima. However, their particular structure -- to leave optimum one can only jump directly global raises question representative such results are. For this reason, we propose an extended class $\textsc{Jump}_{k,\delta}$ that contain a valley low fitness width $\delta$ starting at distance $k$ optimum. We prove several previous extend more general class: for all {$k \le \frac{n^{1/3}}{\ln{n}}$} and $\delta < k$, optimal mutation rate $(1+1)$~EA is $\frac{\delta}{n}$, fast runs faster than classical by factor super-exponential $\delta$. also observe some known do not generalize: algorithm with stagnation detection, which polynomial on $\textsc{Jump}_k$, slower $n$ instances. Computationally, new allows experiments wider valleys, especially when they lie further away