The (1+1)-EA on Noisy Linear Functions with Random Positive Weights

We study the well-known black-box optimisation algorithm (1 + 1)-EA on a novel type of noise model. In our noise model, the fitness function is linear with positive weights, but the absolute values of the weights may fluctuate in each round. Thus in every state, the fitness function indicates that one-bits are preferred over zero-bits. In particular, hillelimbing heuristics should be able to find the optimum fast. We show that the (1+1)-EA indeed finds the optimum in time O(n log n) if the mutation parameter is c/n for a constant c < c(0):= 1.59... However, we also show that for c > c(0) the ( 1 + 1)-EA needs superpolynomial time to find the optimum. Thus the choice of mutation parameter is critical even for optimisation tasks in which there is a clear path to the the optimum. A similar threshold phenomenon has recently been shown for noise-free monotone fitness functions.