The goal of multi-objective optimization 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 optimization problem, practitioners often appeal to the use of scalarization functions in order to transform the multi-objective problem into a collection of single-objective problems. This set of scalarized problems can then be solved using traditional single-objective optimization techniques. In this work, we formalise this convention into a general mathematical framework. We show how this strategy effectively recasts the original multi-objective optimization problem into a single-objective optimization problem defined over sets. An appropriate class of objective functions for this new problem is the R2 utility function, which is defined as a weighted integral over the scalarized optimization problems. We show that this utility function is a monotone and submodular set function, which can be optimised effectively using greedy optimization algorithms. We analyse the performance of these greedy algorithms both theoretically and empirically. Our analysis largely focusses on Bayesian optimization, which is a popular probabilistic framework for black-box optimization.

翻译：多目标优化的目标是找到一组能够描述多个目标之间最佳权衡的点集。为解决此类向量值优化问题，实践者常借助标量化函数将多目标问题转化为一系列单目标问题，进而通过传统单目标优化技术求解。本研究将该范式形式化为通用数学框架，论证此策略如何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。针对该新问题，R2效用函数构成了一类适当的目标函数——该函数定义为对标量化优化问题的加权积分。我们证明该效用函数具有单调性和子模性，可通过贪心优化算法高效求解，并从理论与实证两个维度分析贪心算法的性能。分析重点聚焦贝叶斯优化，这是当前黑箱优化领域广泛使用的概率框架。