Local search is a fundamental method in operations research and combinatorial optimisation. It has been widely applied to a variety of challenging problems, including multi-objective optimisation where multiple, often conflicting, objectives need to be simultaneously considered. In multi-objective local search algorithms, a common practice is to maintain an archive of all non-dominated solutions found so far, from which the algorithm iteratively samples a solution to explore its neighbourhood. A central issue in this process is how to explore the neighbourhood of a selected solution. In general, there are two main approaches: 1) systematic exploration and 2) random sampling. The former systematically explores the solution's neighbours until a stopping condition is met -- for example, when the neighbourhood is exhausted (i.e., the best improvement strategy) or once a better solution is found (i.e., first improvement). In contrast, the latter randomly selects and evaluates only one neighbour of the solution. One may think systematic exploration may be more efficient, as it prevents from revisiting the same neighbours multiple times. In this paper, however, we show that this may not be the case. We first empirically demonstrate that the random sampling method is consistently faster than the systematic exploration method across a range of multi-objective problems. We then give an intuitive explanation for this phenomenon using toy examples, showing that the superior performance of the random sampling method relies on the distribution of ``good neighbours''. Next, we show that the number of such neighbours follows a certain probability distribution during the search. Lastly, building on this distribution, we provide a theoretical insight for why random sampling is more efficient than systematic exploration, regardless of whether the best improvement or first improvement strategy is used.

翻译：局部搜索是运筹学与组合优化领域的基础方法，已广泛应用于各类复杂问题，包括需要同时考虑多个（通常相互冲突的）目标的多目标优化问题。在多目标局部搜索算法中，通常的做法是维护一个包含迄今发现的所有非支配解的存档，算法从该存档中迭代采样一个解以探索其邻域。这一过程的核心问题在于如何探索所选解的邻域。一般而言，存在两种主要方法：1）系统探索；2）随机抽样。前者系统性地探索解的邻域直至满足停止条件——例如当邻域被穷尽（即最优改进策略）或一旦找到更优解（即首次改进策略）。相比之下，后者仅随机选择并评估解的一个邻域点。人们可能认为系统探索效率更高，因为它避免了多次重复访问相同邻域点。然而，本文证明事实可能并非如此。我们首先通过实验证明，在一系列多目标问题上，随机抽样方法始终比系统探索方法更快。随后，我们通过简单示例对这一现象给出直观解释，表明随机抽样方法的优越性能依赖于“优质邻域点”的分布特性。接着，我们证明此类邻域点的数量在搜索过程中遵循特定的概率分布。最后，基于该分布，我们从理论上阐释了为何无论采用最优改进策略还是首次改进策略，随机抽样都比系统探索更高效。