Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Random Pareto front surfaces

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

TL;DR

It is shown that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction.

Abstract

The goal of multi-objective optimisation is to identify the Pareto front surface which is the set obtained by connecting the best trade-off points. Typically this surface is computed by evaluating the objectives at different points and then interpolating between the subset of the best evaluated trade-off points. In this work, we propose to parameterise the Pareto front surface using polar coordinates. More precisely, we show that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction. We then use this representation in order to rigorously develop the theory and applications of stochastic Pareto front surfaces. In particular, we derive many Pareto front surface statistics of interest such as the expectation, covariance and quantiles. We then discuss how these can be used in practice within a design of experiments setting, where the goal is to both infer and use the Pareto front surface distribution in order to make effective decisions. Our framework allows for clear uncertainty quantification and we also develop advanced visualisation techniques for this purpose. Finally we discuss the applicability of our ideas within multivariate extreme value theory and illustrate our methodology in a variety of numerical examples, including a case study with a real-world air pollution data set.

Random Pareto front surfaces

TL;DR

It is shown that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction.

Abstract

The goal of multi-objective optimisation is to identify the Pareto front surface which is the set obtained by connecting the best trade-off points. Typically this surface is computed by evaluating the objectives at different points and then interpolating between the subset of the best evaluated trade-off points. In this work, we propose to parameterise the Pareto front surface using polar coordinates. More precisely, we show that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction. We then use this representation in order to rigorously develop the theory and applications of stochastic Pareto front surfaces. In particular, we derive many Pareto front surface statistics of interest such as the expectation, covariance and quantiles. We then discuss how these can be used in practice within a design of experiments setting, where the goal is to both infer and use the Pareto front surface distribution in order to make effective decisions. Our framework allows for clear uncertainty quantification and we also develop advanced visualisation techniques for this purpose. Finally we discuss the applicability of our ideas within multivariate extreme value theory and illustrate our methodology in a variety of numerical examples, including a case study with a real-world air pollution data set.
Paper Structure (69 sections, 13 theorems, 81 equations, 18 figures, 1 table)

This paper contains 69 sections, 13 theorems, 81 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structure and contributions
  3. Preliminaries
  4. Nomenclature
  5. Polar parameterisation
  6. Polar coordinates
  7. Polar surfaces
  8. Order-preserving transformations
  9. Utility functions
  10. Length-based R2 utilities
  11. Hypervolume indicator
  12. General properties
  13. Loss functions
  14. Frontier loss
  15. Frontier distance
  16. ...and 54 more sections

Key Result

Theorem 3.1

For any bounded set of vectors $A \subset \mathbb{R}^M$ and reference vector $\boldsymbol{\eta} \in \mathbb{R}^M$, if the corresponding Pareto front surface is not empty $\mathcal{Y}_{\boldsymbol{\eta}}^{\textnormal{int}}[A] \neq \emptyset$, then it admits the following polar parameterisation: where $\ell_{\boldsymbol{\eta}, \boldsymbol{\lambda}}[\mathcal{Y}_{\boldsymbol{\eta}}^{\textnormal{int}}

Figures (18)

  • Figure 1: An illustration of the different Pareto fronts in $M=2$ dimensions.
  • Figure 2: An illustration of the polar parameterisation result in $M=2$ dimensions.
  • Figure 3: An illustration of the length-based scalarisation functions in $M=2$ dimensions.
  • Figure 4: An illustration of the polar parameterisation associated with a set of two points.
  • Figure 5: An illustration of some algebraic operations on the space of Pareto front surfaces in $M=2$ dimensions. For these examples, we set the scalar multipliers to be $\alpha=2$ and $\epsilon \in \{0.0, 0.2, 0.4, 0.6, 0.8, 1.0\}$.
  • ...and 13 more figures

Theorems & Definitions (36)

  • Definition 2.1: Pareto domination
  • Definition 2.2: Domination region
  • Definition 2.3: Pareto optimality
  • Definition 2.4: Interpolated weak optimality
  • Remark 2.1: Reference vector
  • Definition 3.1: Polar surfaces
  • Theorem 3.1: Polar parameterisation
  • Remark 3.1: Singleton front
  • Remark 3.2: Conceptual idea
  • Remark 3.3: Other representations
  • ...and 26 more