On Inversely Proportional Hypermutations with Mutation Potential

Artificial Immune Systems (AIS) employing hypermutations with linear static mutation potential have recently been shown to be very effective at escaping local optima of combinatorial optimisation problems at the expense of being slower during the exploitation phase compared to standard evolutionary algorithms. In this paper we prove that considerable speed-ups in the exploitation phase may be achieved with dynamic inversely proportional mutation potentials (IPM) and argue that the potential should decrease inversely to the distance to the optimum rather than to the difference in fitness. Afterwards we define a simple (1+1)~Opt-IA, that uses IPM hypermutations and ageing, for realistic applications where optimal solutions are unknown. The aim of the AIS is to approximate the ideal behaviour of the inversely proportional hypermutations better and better as the search space is explored. We prove that such desired behaviour, and related speed-ups, occur for a well-studied bimodal benchmark 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 potential cannot, thus requires exponential expected runtime in the distance between the cliff and the optimum.