Speeding Up the NSGA-II via Dynamic Population Sizes

Multi-objective evolutionary algorithms (MOEAs) are among the most widely and successfully applied optimizers for multi-objective problems. However, to store many optimal trade-offs (the Pareto optima) simultaneously, MOEAs are typically run with a large population of solution candidates. This slows down the algorithm and renders the choice of the population size a crucial design decision. In this work, we aim to overcome these difficulties by proposing the dynamic NSGA-II, a variant of the well-known NSGA-II that starts with a small initial population and doubles it after a user-specified number 𝜏 of function evaluations, up to a maximum size of 𝑁ₘₐₓ. We prove that the dynamic NSGA-II with optimal parameters computes the Pareto front of the OneMinMax benchmark of size 𝑛 with high probability in O(𝑛 log² 𝑛) function evaluations, which is considerably faster than the Θ(𝑛² log 𝑛) runtime of the static NSGA-II with optimal parameters. For the OneJumpZeroJump benchmark with gap size 𝑘, we show a runtime of O(𝑛^𝑘 log² 𝑛), improving upon the known runtime of Θ(𝑛^(𝑘 + 1)). We also propose a variant that uses the initial population size for a longer period and achieves slightly better performance. Finally, we show that a simple concurrent-run strategy turns our dynamic NSGA-II variants into parameter-less algorithms that exceed the above runtimes only by a logarithmic factor and hence still outperform the static NSGA-II by a factor of ͠Ω(𝑛).