While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the LeadingOnes and Jump benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $\sigma$ into another one $\tau$, but also the precise cycle structure of $\sigma \tau^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $\Theta(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{\Theta(m)}$ on jump functions with jump size $m$. A short empirical analysis confirms these findings, but also reveals that small implementation details like the rate of void mutations can make an important difference.
翻译:虽然对演化算法(EAs)的理论分析在过去25年中在假Boolean优化问题上取得了重大进展,但对于EAs如何解决基于变异性的问题,只有零星的理论结果存在。为了克服缺乏基于变异的基准问题,我们提出了一个将经典假Boolean基准转换成一套变异基准的一般方法。我们随后对Scharnow、Tinnefeld和Wegener(2004年)提议的基于变异性计算$(1+1)的EA进行了严格的运行时间分析。我们发现,在“领导一”和跳跃基准的类比喻中,“领导一”和“跳跃”的比喻中,对于EAts解决基于变异性问题的方法,只有零星的理论结果。为了克服缺乏调异性基准,我们提出了一种一般的方法,将典型的假数值转换成另一套“美元”的基准。我们发现,对于调异性变异性调的变异性变变异性变异性变异性变异性比较小。我们发现,它不仅导致更简单的证据,而且还会降低跳动的大小。</s>