Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our “robustify and scalarise” methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.

中文翻译：

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基于标量化的风险概念，实现稳健的多目标优化

稳健优化是一个成熟的框架，用于在存在不确定性的情况下优化功能。这个问题的内在目标是确定一组输入，这些输入的输出既是决策者所希望的，同时又对问题中的潜在不确定性具有鲁棒性。在这项工作中，我们研究了这个问题的多目标案例。我们发现，大多数稳健的多目标算法都依赖于两个关键作：稳健化和标量化。稳健性是指用于解释问题不确定性的策略。标量化是指用于将每个目标的相对重要性编码为标量值奖励的过程。由于这些作不一定是可交换的，因此执行它们的顺序会影响确定的结果解决方案和做出的最终决策。这项工作的目的是对这些不同排序的影响进行全面阐述，特别是强调何时应该选择一种排序而不是另一种排序。作为我们分析的一部分，我们展示了有多少现有风险概念可以集成到稳健的多目标优化问题的规范和解决方案中。除此之外，我们还演示了如何主要定义稳健的帕累托前沿的概念和基于我们的 “稳健化和缩放 ”方法的稳健性能指标。为了说明这些新想法的有效性，我们提出了两个基于真实数据集的有见地的案例研究。