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.

该研究探讨了一种使用标量化函数来转化多目标优化问题为单一目标优化问题的方法，其中使用了一种称为 R2 效用函数的目标函数解决了这些问题，并且该函数还可以利用贪心算法进行有效优化。