The goal of multi-objective optimisation is to identify a collection of points which describe the best possible trade-offs between the multiple objectives. In order to solve this vector-valued optimisation problem, practitioners often appeal to the use of scalarisation functions in order to transform the multi-objective problem into a collection of single-objective problems. This set of scalarised problems can then be solved using traditional single-objective optimisation techniques. In this work, we formalise this convention into a general mathematical framework. We show how this strategy effectively recasts the original multi-objective optimisation problem into a single-objective optimisation problem defined over sets. An appropriate class of objective functions for this new problem are the R2 utilities, which are utility functions that are defined as a weighted integral over the scalarised optimisation problems. As part of our work, we show that these utilities are monotone and submodular set functions which can be optimised effectively using greedy optimisation algorithms. We then analyse the performance of these greedy algorithms both theoretically and empirically. Our analysis largely focusses on Bayesian optimisation, which is a popular probabilistic framework for black-box optimisation.

翻译：多目标优化的目标是识别一组能够描述多个目标间最优权衡的点集。为解决这一向量值优化问题，研究者通常采用标量化函数方法，将多目标问题转化为一系列单目标问题。这些标量化问题随后可使用传统单目标优化技术求解。本文将该惯例形式化为通用数学框架，论证该策略实质上是将原始多目标优化问题重新定义为定义在集合上的单目标优化问题。此类新问题的目标函数适宜类别为R2效用——即定义为标量化优化问题加权积分的效用函数。研究证明，这些效用具有单调性和子模性集合函数特性，可通过贪心优化算法高效求解。我们随后从理论与实证两个维度分析贪心算法的性能，主要聚焦于贝叶斯优化这一流行的黑箱优化概率框架。