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.

翻译：鲁棒优化是在不确定性条件下优化函数的成熟框架。该问题的本质目标是确定一组输入，其输出既对决策者具有吸引力，又能对问题中的潜在不确定性保持鲁棒性。本文研究了该问题的多目标情形。我们发现绝大多数鲁棒多目标算法依赖于两个关键操作：鲁棒化与标量化。鲁棒化指用于处理问题中不确定性的策略；标量化指将各目标相对重要性编码为标量奖励的过程。由于这些操作不一定满足交换律，其执行顺序会影响最终确定的解集与决策结果。本文旨在系统阐述不同顺序产生的效应，特别强调何时应选择特定顺序。在分析过程中，我们展示了如何将多种现有风险概念整合到鲁棒多目标优化问题的建模与求解中。此外，我们还基于"先鲁棒化后标量化"的方法论，从原理上阐释了如何定义鲁棒帕累托前沿与鲁棒性能度量。为验证这些新概念的有效性，我们基于真实世界数据集提出了两个具有启发性的案例研究。