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Artificial Immune Systems (AIS) employing hypermutations with linear staticmutation potential have recently been shown to be very effective at escapinglocal optima of combinatorial optimisation problems at the expense of beingslower during the exploitation phase compared to standard evolutionaryalgorithms. In this paper we prove that considerable speed-ups in theexploitation phase may be achieved with dynamic inversely proportional mutationpotentials (IPM) and argue that the potential should decrease inversely to thedistance to the optimum rather than to the difference in fitness. Afterwards wedefine a simple (1+1)~Opt-IA, that uses IPM hypermutations and ageing, forrealistic applications where optimal solutions are unknown. The aim of the AISis to approximate the ideal behaviour of the inversely proportionalhypermutations better and better as the search space is explored. We prove thatsuch desired behaviour, and related speed-ups, occur for a well-studied bimodalbenchmark function called \textsc{TwoMax}. Furthermore, we prove that the(1+1)~Opt-IA with IPM efficiently optimises a third bimodal function,\textsc{Cliff}, by escaping its local optima while Opt-IA with static potentialcannot, thus requires exponential expected runtime in the distance between thecliff and the optimum.

On Inversely Proportional Hypermutations with Mutation Potential