The (environmental) selection procedure in Evolutionary Multiobjective Optimization Algorithms (EMOAs) can be interpreted as a subset selection problem, where the goal is to determine a subset of a given size that maximizes a quality indicator. The hypervolume indicator possesses desirable theoretical properties (e.g. monotonicity properties) that make it well-suited for indicator-based selection, and SMS-EMOA is a good example of this. In such a case, selection is viewed as a Hypervolume Subset Selection Problem (HSSP) that consists of selecting a subset of $k$ points from a set of $n$ points that maximizes the hypervolume indicator. However, apart from SMS-EMOA that considers the case of $k=n-1$, HSSP-based selection in EMOAs with more than 2 objectives and $k<n-1$ is solved with approximation (greedy) algorithms. Although a few exact algorithms exist to compute the HSSP for these cases, faster algorithms are required to make its integration in EMOAs practical. This paper proposes a new integer linear programming formulation of the HSSP, named LHSSP, that relies on a decomposition of the dominated region based on the multivariate Empirical Cumulative Distribution Function (ECDF). It is shown that, under this formulation, only part of the dominated region needs to be modeled to obtain an optimal solution to the HSSP. A new algorithm is proposed, named LayersC, which exploits this observation through incremental computations. Experimental studies with 3 objectives show that this algorithm considerably speeds up the computation of the HSSP in comparison to state-of-the-art algorithms, and makes HSSP-based selection in EMOAs amenable.

进化多目标优化算法 (EMOA) 中的（环境）选择程序可以解释为子集选择问题，其目标是确定一个给定大小的子集，使质量指标最大化。hypervolume 指标具有理想的理论特性（例如单调性特性），使其非常适合基于指标的选择，SMS-EMOA 就是一个很好的例子。在这种情况下，选择被视为一个超体积子集选择问题 ( HSSP )，它包括选择$克$ 从一组点 $n$最大化 hypervolume 指标的点。然而，除了考虑以下情况的 SMS-EMOA$克=n-1$,具有 2 个以上目标的EMOA 中基于HSSP的选择和$克<n-1$用近似（贪婪）算法解决。尽管存在一些精确的算法来计算这些情况下的HSSP，但需要更快的算法才能使其在 EMOA 中的集成变得可行。本文提出了一种新的HSSP整数线性规划公式，称为LHSSP，它依赖于基于多元经验累积分布函数 (ECDF) 的支配区域的分解。结果表明，在这个公式下，只需要对主导区域的一部分进行建模以获得HSSP的最优解。提出了一种新算法，命名为LayersC，它通过增量计算利用这一观察结果。具有 3 个目标的实验研究表明，与最先进的算法相比，该算法大大加快了HSSP的计算速度，并使EMOA 中基于HSSP的选择变得可行。