We present a time complexity analysis of the Opt-IA artificial immune system (AIS). We first highlight the power and limitations of its distinguishing operators (i.e., hypermutations with mutation potential and ageing) by analysing them in isolation. Recent work has shown that ageing combined with local mutations can help escape local optima on a dynamic optimisation benchmark function. We generalise this result by rigorously proving that, compared to evolutionary algorithms (EAs), ageing leads to impressive speed-ups on the standard

benchmark function both when using local and global mutations. Unless the stop at first constructive mutation (FCM) mechanism is applied, we show that hypermutations require exponential expected runtime to optimise any function with a polynomial number of optima. If instead FCM is used, the expected runtime is at most a linear factor larger than the upper bound achieved for any random local search algorithm using the artificial fitness levels method. Nevertheless, we prove that algorithms using hypermutations can be considerably faster than EAs at escaping local optima. An analysis of the complete Opt-IA reveals that it is efficient on the previously considered functions and highlights problems where the use of the full algorithm is crucial. We complete the picture by presenting a class of functions for which Opt-IA fails with overwhelming probability while standard EAs are efficient.

中文翻译：

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当超变异和衰老使人工免疫系统的性能优于进化算法时

我们介绍了Opt-IA人工免疫系统（AIS）的时间复杂度分析。我们首先通过孤立地分析算子来突出其区分运算符的功能和局限性（即具有突变潜力和老化的超变异）。最近的工作表明，老化与局部突变相结合可以帮助摆脱动态优化基准函数上的局部最优。我们通过严格证明与进化算法（EA）相比，老化导致标准上令人印象深刻的加速，从而概括了这一结果

使用局部和全局突变时的基准功能。除非在第一个建设性突变时停止（FCM）机制被应用，我们证明超变需要指数期望的运行时间来优化多项式为最优值的任何函数。如果改为使用FCM，则预期运行时间最多是一个线性因子，该线性因子大于使用人工适应水平方法的任何随机局部搜索算法所达到的上限。尽管如此，我们证明了在避免局部最优的情况下，使用超变的算法可以比EA更快。对完整的Opt-IA的分析显示，它对先前考虑的功能有效，并突出了使用完整算法至关重要的问题。通过展示一类功能，Opt-IA会以压倒性的优势失败，而标准EA则是有效的，从而使图片更加完整。