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，即使不做额外调整，进化算法也能容忍目标函数评估中的一定噪声。然而，对于多目标优化而言，这一问题尚未得到充分理解。本文首次对简单多目标进化算法(MOEA)在经典基准测试中目标函数存在噪声时的运行时进行了数学分析。我们证明，当位级先验噪声率满足$p \le \alpha/n$（$\alpha$为适当常数）时，无需任何噪声应对调整的简单多目标优化器(SEMO)即可在$O(n^2\log n)$时间内找到OneMinMax基准测试的帕累托前沿，这与无噪声情形相同。考虑到问题是要获得由$n+1$个个体组成的见证帕累托前沿的种群，这表现出令人惊讶的强鲁棒性（相比之下，当$p = \omega(\log(n)/n)$时，较为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种群。有趣的是，这一结果仅在以下条件下成立：每个解的目标值仅被确定一次，此后算法将基于这个可能带有噪声的目标值继续运行。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，通常推荐在意适应度重要性时重新评估解，且存在已知案例表明，不对解进行重新评估可能导致灾难性性能损失。