Run-time analysis of the (1+1) evolutionary algorithm optimizing linear functions over a finite alphabet

We analyze the run-time of the (1 + 1) Evolutionary Algorithm optimizing an arbitrary linear function <i>f</i> : {0,1,...,r}ⁿ -> R. If the mutation probability of the algorithm is <i>p</i> = <i>c/n</i>, then (1 + o(1))(e^c/c))<i>rn</i> log <i>n</i> + <i>O</i>(<i>r</i>³<i>n</i> log log <i>n</i>) is an upper bound for the expected time needed to find the optimum. We also give a lower bound of (1 + <i>o</i>(1))(1/<i>c</i>)<i>rn</i> log <i>n</i>. Hence for constant <i>c</i> and all <i>r</i> slightly smaller than (log <i>n</i>)^1/3, our bounds deviate by only a constant factor, which is <i>e</i>(1 + <i>o</i>(1)) for the standard mutation probability of 1/<i>n</i>. The proof of the upper bound uses multiplicative adaptive drift analysis as developed in a series of recent papers. We cannot close the gap for larger values of <i>r</i>, but find indications that multiplicative drift is not the optimal analysis tool for this case.