The heavy-tailed mutation operator, proposed by Doerr, Le, Makhmara, and Nguyen (2017) for evolutionary algorithms, is based on the power-law assumption of mutation rate distribution. Here we generalize the power-law assumption using a regularly varying constraint on the distribution function of mutation rate. In this setting, we generalize the upper bounds on the expected optimization time of the $(1+(λ,λ))$ genetic algorithm obtained by Antipov, Buzdalov and Doerr (2022) for the OneMax function class parametrized by the problem dimension $n$. In particular, it is shown that, on this function class, the sufficient conditions of Antipov, Buzdalov and Doerr (2022) on the heavy-tailed mutation, ensuring the $O(n)$ optimization time in expectation, may be generalized as well. This optimization time is known to be asymptotically smaller than what can be achieved by the $(1+(λ,λ))$ genetic algorithm with any static mutation rate. A new version of the heavy-tailed mutation operator is proposed, satisfying the generalized conditions, and promising results of computational experiments are presented.

翻译：重尾变异算子由Doerr、Le、Makhmara与Nguyen（2017）提出，用于进化算法，其基础是变异率分布的幂律假设。本文基于变异率分布函数的正则变化约束，对该幂律假设进行广义化推广。在此设定下，我们推广了Antipov、Buzdalov与Doerr（2022）针对以问题维度$n$参数化的OneMax函数类，所获得的$(1+(λ,λ))$遗传算法期望优化时间的上界。特别地，研究表明：在该函数类上，Antipov、Buzdalov与Doerr（2022）提出的确保期望优化时间为$O(n)$的重尾变异充分条件同样可被推广。该优化时间已知在渐近意义上优于任何静态变异率下的$(1+(λ,λ))$遗传算法所能达到的结果。本文提出一种满足广义条件的新型重尾变异算子，并展示了具有前景的计算实验结果。