  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.
