Rigorous Hitting Times for Binary Mutations

In the binary evolutionary optimization framework, two mutation operators are theoretically investigated. For both the standard mutation, in which all bits are ipped independently with the same probability, and the 1-bit-ip mutation, which ips exactly one bit per bitstring, the statistical distribution of the rst hitting times of the target are thoroughly computed (expectation and variance) up to terms of order l (the size of the bitstrings) in two distinct situations: without any selection, or with the deterministic (1+1)-ES selection on the OneMax problem. In both cases, the 1-bit-ip mutation convergence time is smaller by a constant (in terms of l) multi-plicative factor. These results extend to the case of multiple independent optimizers.