Adaptive Drift Analysis

We show that, for any c >0, the (1+1) evolutionary algorithm using an arbitrary mutation rate p n = c / n finds the optimum of a linear objective function over bit strings of length n in expected time Θ( n log n ). Previously, this was only known for c ≤1. Since previous work also shows that universal drift functions cannot exist for c larger than a certain constant, we instead define drift functions which depend crucially on the relevant objective functions (and also on c itself). Using these carefully-constructed drift functions, we prove that the expected optimisation time is Θ( n log n ). By giving an alternative proof of the multiplicative drift theorem, we also show that our optimisation-time bound holds with high probability.