  The non-dominated sorting genetic algorithm II (NSGA-II) is the most
intensively used multi-objective evolutionary algorithm (MOEA) in real-world
applications. However, in contrast to several simple MOEAs analyzed also via
mathematical means, no such study exists for the NSGA-II so far. In this work,
we show that mathematical runtime analyses are feasible also for the NSGA-II.
As particular results, we prove that with a population size four times larger
than the size of the Pareto front, the NSGA-II with two classic mutation
operators and four different ways to select the parents satisfies the same
asymptotic runtime guarantees as the SEMO and GSEMO algorithms on the basic
OneMinMax and LeadingOnesTrailingZeros benchmarks. However, if the population
size is only equal to the size of the Pareto front, then the NSGA-II cannot
efficiently compute the full Pareto front: for an exponential number of
iterations, the population will always miss a constant fraction of the Pareto
front. Our experiments confirm the above findings.
