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识别一组描述多个目标之间最佳权衡的点。为解决这一向量值优化问题，实践者常借助标量化函数，将多目标问题转化为一组单目标问题，随后利用传统单目标优化技术求解该标量化问题集。本研究将此惯例形式化为通用数学框架，阐明该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。针对这一新问题，R2效用函数被提出作为合适的目标函数类别，该函数定义为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。我们从理论和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——一种流行的黑盒优化概率框架。