The non-dominated sorting genetic algorithm~II (NSGA-II) is the most popular multi-objective optimization heuristic. Recent mathematical runtime analyses have detected two shortcomings in discrete search spaces, namely, that the NSGA-II has difficulties with more than two objectives and that it is very sensitive to the choice of the population size. To overcome these difficulties, we analyze a simple tie-breaking rule in the selection of the next population. Similar rules have been proposed before, but have found only little acceptance. We prove the effectiveness of our tie-breaking rule via mathematical runtime analyses on the classic OneMinMax, LeadingOnesTrailingZeros, and OneJumpZeroJump benchmarks. We prove that this modified NSGA-II can optimize the three benchmarks efficiently also for many objectives, in contrast to the exponential lower runtime bound previously shown for OneMinMax with three or more objectives. For the bi-objective problems, we show runtime guarantees that do not increase when moderately increasing the population size over the minimum admissible size. For example, for the OneJumpZeroJump problem with representation length $n$ and gap parameter $k$, we show a runtime guarantee of $O(\max\{n^{k+1},Nn\})$ function evaluations when the population size is at least four times the size of the Pareto front. For population sizes larger than the minimal choice $N = \Theta(n)$, this result improves considerably over the $\Theta(Nn^k)$ runtime of the classic NSGA-II.

翻译：非支配排序遗传算法II（NSGA-II）是最流行的多目标优化启发式算法。最近的数学运行时分析揭示了其在离散搜索空间中的两个缺陷：一是处理超过两个目标时存在困难，二是对种群规模的选择极为敏感。为克服这些缺陷，我们分析了在下一代种群选择中引入简单平局决胜规则的机制。类似规则虽曾被提出，但鲜少被采纳。通过对经典基准测试问题OneMinMax、LeadingOnesTrailingZeros和OneJumpZeroJump进行数学运行时分析，我们证明了该平局决胜规则的有效性。研究表明，改进后的NSGA-II能够高效优化这三个基准问题，即便在多目标场景下亦如此——这与先前针对三个及以上目标的OneMinMax问题所证明的指数级运行时下界形成鲜明对比。对于双目标问题，我们证明了当种群规模适度超过最小允许规模时，运行时保证不会增加。例如，对于表示长度为$n$、间隙参数为$k$的OneJumpZeroJump问题，当种群规模至少为帕累托前沿规模的4倍时，我们证明了$O(\max\{n^{k+1},Nn\})$函数评估的运行时保证。相较于经典NSGA-II在最小选择$N = \Theta(n)$时的$\Theta(Nn^k)$运行时，该结果在种群规模大于最小选择时展现出显著改进。