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 extension of this problem from a computational standpoint. 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 marginalise over the uncertainty in the problem. Whilst scalarisation refers to the procedure that is used to encode the relative importance of each objective. 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. This work aims to give an exposition on the philosophical differences between these two operations and 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 easily 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 numerical case studies which are based on real-world data sets.

翻译：鲁棒优化是在不确定性下优化函数的成熟框架。该问题的根本目标是识别一组输入，其输出既满足决策者的期望，又对问题中的潜在不确定性具有鲁棒性。本文从计算角度研究该问题的多目标扩展。我们指出，大多数鲁棒多目标算法依赖两个关键操作：鲁棒化（robustification）与标量化（scalarisation）。鲁棒化指用于边际化问题不确定性的策略，而标量化指编码各目标相对重要性的过程。由于这些操作未必满足交换律，其执行顺序会影响所识别的解及最终决策。本文旨在阐释这两种操作之间的哲学差异，并指出何时应优先选择某种顺序。在分析中，我们展示了如何将现有多种风险概念轻松集成到鲁棒多目标优化问题的规范与求解中。此外，我们还演示了如何基于"鲁棒化与标量化"方法论原则性地定义鲁棒帕累托前沿（robust Pareto front）及鲁棒性能度量。为验证这些新思想的有效性，我们给出了两个基于真实数据集的启发性数值案例研究。