Evolutionary algorithms (EAs) are widely used for multi-objective optimization due to their population-based nature. Traditional multi-objective EAs (MOEAs) generate a large set of solutions to approximate the Pareto front, leaving a decision maker (DM) with the task of selecting a preferred solution. However, this process can be inefficient and time-consuming, especially when there are many objectives or the subjective preferences of DM is known. To address this issue, interactive MOEAs (iMOEAs) combine decision making into the optimization process, i.e., update the population with the help of the DM. In contrast to their wide applications, there has existed only two pieces of theoretical works on iMOEAs, which only considered interactive variants of the two simple single-objective algorithms, RLS and (1+1)-EA. This paper provides the first running time analysis (the essential theoretical aspect of EAs) for practical iMOEAs. Specifically, we prove that the expected running time of the well-developed interactive NSGA-II (called R-NSGA-II) for solving the OneMinMax and OneJumpZeroJump problems is $O(n \log n)$ and $O(n^k)$, respectively, which are all asymptotically faster than the traditional NSGA-II. Meanwhile, we present a variant of OneMinMax, and prove that R-NSGA-II can be exponentially slower than NSGA-II. These results provide theoretical justification for the effectiveness of iMOEAs while identifying situations where they may fail. Experiments are also conducted to validate the theoretical results.

翻译：进化算法（EAs）因其基于种群的特点，被广泛用于多目标优化。传统多目标进化算法（MOEAs）生成大量解集以近似帕累托前沿，留给决策者（DM）选择偏好解的任务。然而，当目标数量较多或已知DM的主观偏好时，这一过程可能效率低下且耗时。为解决此问题，交互式MOEAs（iMOEAs）将决策过程融入优化中，即借助DM更新种群。尽管应用广泛，但目前仅有两项关于iMOEAs的理论工作，且仅考虑了两种简单单目标算法RLS和(1+1)-EA的交互变体。本文首次对实用iMOEAs进行了运行时间分析（EAs的核心理论方面）。具体而言，我们证明了成熟的交互式NSGA-II（称为R-NSGA-II）在求解OneMinMax和OneJumpZeroJump问题时的期望运行时间分别为$O(n \log n)$和$O(n^k)$，这些均渐进快于传统NSGA-II。同时，我们提出了OneMinMax的一个变体，并证明R-NSGA-II可能比NSGA-II慢指数级。这些结果在理论上验证了iMOEAs的有效性，同时指出了其可能失效的场景。还进行了实验以验证理论结果。