R2 v2：用于双目标优化中更优基准测试的符合帕累托准则的R2指标
R2 v2: The Pareto-compliant R2 Indicator for Better Benchmarking in Bi-objective Optimization
AI总结：
本研究重新探究连续均匀切比雪夫效用函数分布下的R2指标，证明其符合帕累托准则，提出双目标问题下O(N log N)的高效计算与增量更新方法，为多目标优化基准测试提供高效替代指标。
摘要：
在多目标优化中，基于集合的质量指标是基准测试与性能评估的基石。它们通过将一组权衡解的质量降为一个标量数值来对其进行刻画。最常用的基于集合的度量指标之一是R2指标，它描述了解集在效用函数分布下对决策者的期望效用。通常，该指标通过对效用函数分布进行离散化来应用，由此得到的是弱帕累托合规的指标。因此，向解集中添加一个非支配解或支配解可能——但并非一定——提升该指标的取值。\n本文在（切比雪夫）效用函数服从连续均匀分布的前提下，重新研究了R2指标。我们详细分析了它的性质，证明这种连续变体确实是符合帕累托准则的——也就是说，任何有益的解都会提升该度量的取值。此外，我们提供了高效的计算流程：(a) 可在$\mathcal{O}(N \log N)$时间复杂度内计算双目标问题的该度量；(b) 当解被添加到当前解集（或从当前解集中移除）时，可对指标进行增量更新，无需重新计算整个集合的指标值。综上，本工作为超体积指标等当前最先进的帕累托合规一元性能指标体系补充了一种高效且极具潜力的替代方案。
英文摘要：
In multi-objective optimization, set-based quality indicators are a cornerstone of benchmarking and performance assessment. They capture the quality of a set of trade-off solutions by reducing it to a scalar number. One of the most commonly used set-based metrics is the R2 indicator, which describes the expected utility of a solution set to a decision-maker under a distribution of utility functions. Typically, this indicator is applied by discretizing the latter distribution, yielding a weakly Pareto-compliant indicator. In consequence, adding a nondominated or dominating solution to a solution set may -- but does not have to -- improve the indicator's value. In this paper, we reinvestigate the R2 indicator under the premise that we have a continuous, uniform distribution of (Tchebycheff) utility functions. We analyze its properties in detail, demonstrating that this continuous variant is indeed Pareto-compliant -- that is, any beneficial solution will improve the metric's value. Additionally, we provide efficient computational procedures that (a) compute this metric for bi-objective problems in $\mathcal O (N \log N)$, and (b) can perform incremental updates to the indicator whenever solutions are added to (or removed from) the current set of solutions, without needing to recompute the indicator for the entire set. As a result, this work contributes to the state-of-the-art Pareto-compliant unary performance metrics, such as the hypervolume indicator, offering an efficient and promising alternative.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
