The goal of multi-objective optimisation is to identify the Pareto front surface which is the set obtained by connecting the best trade-off points. Typically this surface is computed by evaluating the objectives at different points and then interpolating between the subset of the best evaluated trade-off points. In this work, we propose to parameterise the Pareto front surface using polar coordinates. More precisely, we show that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction. We then use this representation in order to rigorously develop the theory and applications of stochastic Pareto front surfaces. In particular, we derive many Pareto front surface statistics of interest such as the expectation, covariance and quantiles. We then discuss how these can be used in practice within a design of experiments setting, where the goal is to both infer and use the Pareto front surface distribution in order to make effective decisions. Our framework allows for clear uncertainty quantification and we also develop advanced visualisation techniques for this purpose. Finally we discuss the applicability of our ideas within multivariate extreme value theory and illustrate our methodology in a variety of numerical examples, including a case study with a real-world air pollution data set.

翻译：多目标优化的目标是识别帕累托前沿曲面，即通过连接最优权衡点所获得的集合。通常，该曲面通过在不同点评估目标函数，然后对最优评估权衡点子集进行插值来计算。在本研究中，我们提出使用极坐标参数化帕累托前沿曲面。更准确地说，我们证明任何帕累托前沿曲面均可等价地用一个标量值长度函数表示，该函数返回沿任意正径向的投影长度。随后，我们利用该表示法严格发展了随机帕累托前沿曲面的理论与应用。特别地，我们推导出许多关注的帕累托前沿曲面统计量，如期望、协方差和分位数。接着，我们讨论如何在实验设计场景中实际应用这些统计量，其目标在于同时推断并利用帕累托前沿曲面分布以做出有效决策。我们的框架支持清晰的不确定性量化，并为此开发了先进的可视化技术。最后，我们探讨了所提方法在多变量极值理论中的适用性，并通过多种数值算例（包括一个真实世界空气污染数据集的案例研究）阐释了我们的方法论。