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Multi-objective optimisation via the R2 utilities

Ben Tu, Nikolas Kantas, Robert M. Lee, Behrang Shafei

TL;DR

This paper unifies scalarisation and utility approaches to multi-objective optimisation through the R2 utilities, a family defined by $U(Y)=\mathbb{E}_{p(\boldsymbol{\theta})}[\max_{\mathbf{y}\in Y} s_{\boldsymbol{\theta}}(\mathbf{y})]$. It proves that R2 utilities are monotone and submodular, enabling greedy optimisation and establishing performance guarantees, including extensions to Bayesian optimisation via AEUI acquisition functions. The framework subsumes popular metrics such as hypervolume, IGD, and D1 as special cases, and provides a principled basis for designing preference-aware and adaptive strategies. The authors provide theoretical bounds, empirical validation on Bayesian setups, and discuss practical extensions like preference elicitation, constraints, and robustness for real-world decision problems.

Abstract

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.

Multi-objective optimisation via the R2 utilities

TL;DR

This paper unifies scalarisation and utility approaches to multi-objective optimisation through the R2 utilities, a family defined by . It proves that R2 utilities are monotone and submodular, enabling greedy optimisation and establishing performance guarantees, including extensions to Bayesian optimisation via AEUI acquisition functions. The framework subsumes popular metrics such as hypervolume, IGD, and D1 as special cases, and provides a principled basis for designing preference-aware and adaptive strategies. The authors provide theoretical bounds, empirical validation on Bayesian setups, and discuss practical extensions like preference elicitation, constraints, and robustness for real-world decision problems.

Abstract

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.
Paper Structure (82 sections, 10 theorems, 55 equations, 16 figures, 1 algorithm)

This paper contains 82 sections, 10 theorems, 55 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Set Pareto domination
  5. Reformulating the multi-objective optimisation problem
  6. Scalarisation perspective
  7. Scalarised optimisation problems
  8. Utility perspective
  9. Utility optimisation problem
  10. Qualitative aspects
  11. Other useful properties
  12. R2 Utilities
  13. R2 metric
  14. Qualitative aspects
  15. Useful properties
  16. ...and 67 more sections

Key Result

Proposition 2.1

Consider an objective function $f:\mathbb{X} \rightarrow \mathbb{R}^M$ and a scalarisation function $s: \mathbb{R}^M \rightarrow \mathbb{R}$. If the scalarisation function is strictly or strongly monotonically increasing over the feasible objective space, then the solutions in $X^*_s = \mathop{\math

Figures (16)

  • Figure 1: A comparison of the domination regions based on the standard Pareto partial ordering over vectors and some popular scalarisation functions in the two-objective setting.
  • Figure 2: A comparison of the aspects that determine the quality of a Pareto front approximation.
  • Figure 3: A comparison of the improvement regions based on the standard Pareto partial ordering over sets and some different R2 utilities in the two-objective setting. For the IGD+ utility, we set $p=2$ and $q=1$, whilst for the D1 utility we set $\mathbf{w} = (1/M, \dots, 1/M)$.
  • Figure 4: A visual comparison of the approximate greedy strategy with different number of Monte Carlo samples on the hypersphere problem (\ref{['eg:hypersphere']}).
  • Figure 5: A performance comparison of the approximate greedy strategy when varying the number of Monte Carlo samples and the number of objectives on the hypersphere problem (\ref{['eg:hypersphere']}). The mean and two standard deviations of the log utility regret, $\log(U(f(\mathbb{X})) - U(f(X_N)))$, obtained from 100 independent runs, are plotted. In each experiment, the Pareto front and the standard R2 utility were approximated using $10^5$ points and samples.
  • ...and 11 more figures

Theorems & Definitions (29)

  • Definition 2.1: Pareto domination
  • Definition 2.2: Pareto optimality
  • Definition 2.3: Dominated region
  • Definition 2.4: Set Pareto domination
  • Remark 2.1
  • Definition 2.5: Monotonicity
  • Proposition 2.1: Monotonicity implies optimality
  • Example 2.1: Popular scalarisation functions
  • Remark 2.2: Chebyshev scalarisation
  • Remark 2.3: Objective transformations
  • ...and 19 more