Towards Running Time Analysis of Interactive Multi-objective Evolutionary Algorithms

Abstract: 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.