In single-objective optimization, it is well known that evolutionary algorithms also without further adjustments can tolerate a certain amount of noise in the evaluation of the objective function. In contrast, this question is not at all understood for multi-objective optimization. In this work, we conduct the first mathematical runtime analysis of a simple multi-objective evolutionary algorithm (MOEA) on a classic benchmark in the presence of noise in the objective functions. We prove that when bit-wise prior noise with rate $p \le \alpha/n$, $\alpha$ a suitable constant, is present, the \emph{simple evolutionary multi-objective optimizer} (SEMO) without any adjustments to cope with noise finds the Pareto front of the OneMinMax benchmark in time $O(n^2\log n)$, just as in the case without noise. Given that the problem here is to arrive at a population consisting of $n+1$ individuals witnessing the Pareto front, this is a surprisingly strong robustness to noise (comparably simple evolutionary algorithms cannot optimize the single-objective OneMax problem in polynomial time when $p = \omega(\log(n)/n)$). Our proofs suggest that the strong robustness of the MOEA stems from its implicit diversity mechanism designed to enable it to compute a population covering the whole Pareto front. Interestingly this result only holds when the objective value of a solution is determined only once and the algorithm from that point on works with this, possibly noisy, objective value. We prove that when all solutions are reevaluated in each iteration, then any noise rate $p = \omega(\log(n)/n^2)$ leads to a super-polynomial runtime. This is very different from single-objective optimization, where it is generally preferred to reevaluate solutions whenever their fitness is important and where examples are known such that not reevaluating solutions can lead to catastrophic performance losses.

翻译：在单目标优化中，众所周知，即使不做进一步调整，进化算法也能容忍目标函数评估中的一定噪声。然而，对于多目标优化，这一问题尚未得到根本理解。本文首次对简单多目标进化算法（MOEA）在目标函数存在噪声的经典基准问题上进行了数学运行时分析。我们证明，当以概率$p \le \alpha/n$（$\alpha$为适当常数）存在逐位先验噪声时，未经任何噪声调整的简单多目标优化器（SEMO）能在时间$O(n^2\log n)$内找到OneMinMax基准的帕累托前沿，这与无噪声情形相同。考虑到此处问题是获得由$n+1$个个体组成的群体以展示帕累托前沿，这表明噪声鲁棒性出奇地强（而相比之下，当$p = \omega(\log(n)/n)$时，同样简单的进化算法无法在多项式时间内优化单目标OneMax问题）。我们的证明表明，MOEA的强鲁棒性源于其隐式多样性机制——该机制旨在使其能够计算覆盖整个帕累托前沿的群体。有趣的是，这一结果仅在解的客观值仅被确定一次且算法此后基于该（可能含噪声的）客观值运行时成立。我们证明，当所有解在每次迭代中被重新评估时，任何噪声率$p = \omega(\log(n)/n^2)$都会导致超多项式运行时。这与单目标优化截然不同——在单目标优化中，通常更倾向于在解的适应度重要时对其重新评估，且已知存在不重新评估解会导致灾难性性能损失的案例。