Mathematical Runtime Analysis for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)
This provides foundational theoretical insights for practitioners using NSGA-II in optimization, though it is incremental as it extends existing analyses to a more complex algorithm.
The authors tackled the lack of mathematical runtime analysis for the widely used NSGA-II multi-objective evolutionary algorithm, proving that with a population size four times the Pareto front size, it matches asymptotic runtime guarantees of simpler algorithms on benchmarks, but fails to compute the full Pareto front efficiently with a smaller population size.
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.