Many methods for performing multi-objective optimisation of computationally expensive problems have been proposed recently. Typically, a probabilistic surrogate for each objective is constructed from an initial dataset. The surrogates can then be used to produce predictive densities in the objective space for any solution. Using the predictive densities, we can compute the expected hypervolume improvement (EHVI) due to a solution. Maximising the EHVI, we can locate the most promising solution that may be expensively evaluated next. There are closed-form expressions for computing the EHVI, integrating over the multivariate predictive densities. However, they require partitioning the objective space, which can be prohibitively expensive for more than three objectives. Furthermore, there are no closed-form expressions for a problem where the predictive densities are dependent, capturing the correlations between objectives. Monte Carlo approximation is used instead in such cases, which is not cheap. Hence, the need to develop new accurate but cheaper approximation methods remains. Here we investigate an alternative approach toward approximating the EHVI using Gauss-Hermite quadrature. We show that it can be an accurate alternative to Monte Carlo for both independent and correlated predictive densities with statistically significant rank correlations for a range of popular test problems.

翻译：最近提出了许多方法来对计算费用昂贵的问题进行多客观优化。 通常, 每个目标的概率替代模型是从最初的数据集中构建的。 然后, 代孕器可以用来在任何解决方案的客观空间中产生预测密度。 使用预测密度, 我们可以计算出由于一个解决方案而预期的高容量改进( EHVI ) 。 最大化 EHVI, 我们可以找到最有希望的解决方案, 可能在接下来进行昂贵的评估。 计算 EHVI 时有封闭式的表达式, 将多变量预测密度整合在一起。 但是, 它们需要分割目标空间, 而这对三个以上的目标来说可能代价太高。 此外, 在预测密度依赖的问题上, 也没有封闭式表达方式, 捕捉目标之间的关联。 蒙特卡洛 近距离被使用, 而不是廉价的。 因此, 需要开发新的准确但更便宜的近似方法。 我们在这里研究一种替代方法, 用高频度- HVI 和高频度水平的汇率对比测试来对 EHVI 进行匹配 。