All Mutation Rates $c/n$ for the $(1+1)$ Evolutionary Algorithm

TL;DR

The set of all $c \geq 1$ for which the mutation rate $c/n$ is optimal for the $(1+1)$ EA is dense in the interval $[1, \infty)$.

Abstract

For every real number $c \geq 1$ and for all $\varepsilon > 0$, there is a fitness function $f : \{0,1\}^n \to \mathbb{R}$ for which the optimal mutation rate for the $(1+1)$ evolutionary algorithm on $f$, denoted $p_n$, satisfies $p_n \approx c/n$ in that $|np_n - c| < \varepsilon$. In other words, the set of all $c \geq 1$ for which the mutation rate $c/n$ is optimal for the $(1+1)$ EA is dense in the interval $[1, \infty)$. To show this, a fitness function is introduced which is called HillPathJump.

Theorems & Definitions (21)

• Lemma 1
• proof
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof