  The analysis of randomized search heuristics on classes of functions is
fundamental for the understanding of the underlying stochastic process and the
development of suitable proof techniques. Recently, remarkable progress has
been made in bounding the expected optimization time of the simple (1+1) EA on
the class of linear functions. We improve the best known bound in this setting
from $(1.39+o(1))en\ln n$ to $en\ln n+O(n)$ in expectation and with high
probability, which is tight up to lower-order terms. Moreover, upper and lower
bounds for arbitrary mutations probabilities $p$ are derived, which imply
expected polynomial optimization time as long as $p=O((\ln n)/n)$ and which are
tight if $p=c/n$ for a constant $c$. As a consequence, the standard mutation
probability $p=1/n$ is optimal for all linear functions, and the (1+1) EA is
found to be an optimal mutation-based algorithm. The proofs are based on
adaptive drift functions and the recent multiplicative drift theorem.
