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，进化算法即使在目标函数评估存在噪声时，无需额外调整也能容忍一定程度的噪声。然而，在多目标优化中，这一问题尚完全未知。本研究首次对简单多目标进化算法（MOEA）在经典基准问题上的数学运行时间进行分析，考虑目标函数存在噪声的情况。我们证明：当逐位先验噪声率满足 \(p \le \alpha/n\)（其中 \(\alpha\) 为适当常数）时，未做任何抗噪声调整的简单多目标进化优化器（SEMO）能在时间复杂度 \(O(n^2\log n)\) 内找到OneMinMax基准问题的帕累托前沿，这与无噪声情况下的性能相当。考虑到该问题要求获得包含 \(n+1\) 个个体并覆盖帕累托前沿的种群，这一噪声鲁棒性令人惊讶地强（相比之下，当 \(p = \omega(\log(n)/n)\) 时，类似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出覆盖整个帕累托前沿的种群。值得注意的是，这一结论仅在目标值仅确定一次且算法此后使用该（可能含噪）目标值的情况下成立。我们证明：若每轮迭代都对所有解进行重新评估，则任何噪声率 \(p = \omega(\log(n)/n^2)\) 都将导致超多项式运行时间。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进行重新评估，且已知某些案例中若不重新评估解可能导致灾难性的性能损失。