Despite significant progress in the field of mathematical runtime analysis of multi-objective evolutionary algorithms (MOEAs), the performance of MOEAs on discrete many-objective problems is little understood. In particular, the few existing bounds for the SEMO, global SEMO, and SMS-EMOA algorithms on classic benchmarks are all roughly quadratic in the size of the Pareto front. In this work, we prove near-tight runtime guarantees for these three algorithms on the four most common benchmark problems OneMinMax, CountingOnesCountingZeros, LeadingOnesTrailingZeros, and OneJumpZeroJump, and this for arbitrary numbers of objectives. Our bounds depend only linearly on the Pareto front size, showing that these MOEAs on these benchmarks cope much better with many objectives than what previous works suggested. Our bounds are tight apart from small polynomial factors in the number of objectives and length of bitstrings. This is the first time that such tight bounds are proven for many-objective uses of these MOEAs. While it is known that such results cannot hold for the NSGA-II, we do show that our bounds, via a recent structural result, transfer to the NSGA-III algorithm.

翻译：尽管多目标进化算法（MOEAs）的数学运行时间分析领域取得了显著进展，但MOEAs在离散多目标问题上的性能仍鲜为人知。特别是，现有针对SEMO、全局SEMO和SMS-EMOA算法在经典基准问题上的少数界限，其复杂度与帕累托前沿大小大致呈二次关系。本研究证明，对于这三个算法在四种最常用基准问题（OneMinMax、CountingOnesCountingZeros、LeadingOnesTrailingZeros和OneJumpZeroJump）上，且针对任意目标数量，其运行时间保证均为近紧的。我们的界限仅与帕累托前沿大小呈线性依赖关系，表明这些MOEAs在这些基准问题上处理多目标问题的能力远优于先前研究所示。这些界限在目标数量和位串长度的小多项式因子范围内为紧的。这是首次针对这些MOEAs的多目标使用场景证明此类紧界。尽管已知此类结论不适用于NSGA-II，我们通过一项近期结构结果表明，我们的界限可迁移至NSGA-III算法。