  Understanding how the time-complexity of evolutionary algorithms (EAs) depend
on their parameter settings and characteristics of fitness landscapes is a
fundamental problem in evolutionary computation. Most rigorous results were
derived using a handful of key analytic techniques, including drift analysis.
However, since few of these techniques apply effortlessly to population-based
EAs, most time-complexity results concern simplified EAs, such as the (1+1) EA.
  This paper describes the level-based theorem, a new technique tailored to
population-based processes. It applies to any non-elitist process where
offspring are sampled independently from a distribution depending only on the
current population. Given conditions on this distribution, our technique
provides upper bounds on the expected time until the process reaches a target
state.
  We demonstrate the technique on several pseudo-Boolean functions, the sorting
problem, and approximation of optimal solutions in combinatorial optimisation.
The conditions of the theorem are often straightforward to verify, even for
Genetic Algorithms and Estimation of Distribution Algorithms which were
considered highly non-trivial to analyse. Finally, we prove that the theorem is
nearly optimal for the processes considered. Given the information the theorem
requires about the process, a much tighter bound cannot be proved.
