Multiobjective Optimization Using the R2 Utility

Abstract

SIAM Review, Volume 67, Issue 2, Page 213-255, May 2025.
Abstract.The goal of multiobjective optimization is to identify a collection of points which describe the best possible trade-offs among 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 multiobjective 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 paper, we formalize this convention into a general mathematical framework. We show how this strategy effectively recasts the original multiobjective optimization problem into a single-objective optimization problem defined over sets. An appropriate class of objective functions for this new problem is that of the R2 utilities, which are utility functions that are defined as a weighted integral over the scalarized optimization problem. As part of our work, we show that these utilities are monotone and submodular set functions that can be optimized effectively using greedy optimization algorithms. We then analyze the performance of these greedy algorithms both theoretically and empirically. Our analysis largely focuses on Bayesian optimization, which is a popular probabilistic framework for black-box optimization.