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Scalarisation-based risk concepts for robust multi-objective optimisation

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

TL;DR

The paper tackles robust multi-objective optimisation under uncertainty by formalising two scalarisation-based strategies: RTS (robustify then scalarise) and STR (scalarise then robustify). It introduces robust Pareto fronts via polar parameterisation and develops risk-statistic front concepts to interpret trade-offs under uncertainty, along with robust performance metrics such as robust R2 utilities, robust IGD, and robust hypervolume. The authors demonstrate the approach on real datasets (cake optimization) and a distributionally robust rocket injector problem, showing how RTS/STR yield distinct robust solutions and frontiers. The work provides a general, flexible framework that subsumes many existing methods and offers practical tools for robust MOOs in engineering and data-driven settings.

Abstract

Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our ``robustify and scalarise'' methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.

Scalarisation-based risk concepts for robust multi-objective optimisation

TL;DR

The paper tackles robust multi-objective optimisation under uncertainty by formalising two scalarisation-based strategies: RTS (robustify then scalarise) and STR (scalarise then robustify). It introduces robust Pareto fronts via polar parameterisation and develops risk-statistic front concepts to interpret trade-offs under uncertainty, along with robust performance metrics such as robust R2 utilities, robust IGD, and robust hypervolume. The authors demonstrate the approach on real datasets (cake optimization) and a distributionally robust rocket injector problem, showing how RTS/STR yield distinct robust solutions and frontiers. The work provides a general, flexible framework that subsumes many existing methods and offers practical tools for robust MOOs in engineering and data-driven settings.

Abstract

Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our ``robustify and scalarise'' methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.
Paper Structure (59 sections, 12 theorems, 83 equations, 11 figures)

This paper contains 59 sections, 12 theorems, 83 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Structure of the paper
  4. Preliminaries
  5. Multi-objective optimisation
  6. Scalarisation perspective
  7. Risk functionals
  8. Univariate risk functionals
  9. Multivariate risk functionals
  10. Robust multi-objective optimisation
  11. Robustify then scalarise
  12. Union robust optimisation problem
  13. Scalarise then robustify
  14. STR length scalarised problem
  15. Discussion
  16. ...and 44 more sections

Key Result

Theorem 2.1

miettinen1998 Consider a bounded objective function $g: \mathbb{X} \rightarrow \mathbb{R}^M$, and a reference vector $\boldsymbol{\eta} \in \cap_{\mathbf{x} \in \mathbb{X}} \mathbb{D}_{\prec\mathrel{\mkern-5mu}\prec}[\{g(\mathbf{x})\}]$ which is strongly dominated by the whole feasible objective spa Namely we can choose $\boldsymbol{\lambda} = (g(\mathbf{z}) - \boldsymbol{\eta}) / ||g(\mathbf{z})

Figures (11)

  • Figure 1: An illustration of a length scalarised problem in $M=2$ dimensions.
  • Figure 2: An illustration of some univariate and multivariate risk functionals. On the left plots, we consider two univariate probability distributions and plot its associated risk-adjusted values. On the right plots, we consider two uniform distributed random variables and plot its associated risk-adjusted sets. In these multivariate plots, the circles are used to denote the component-wise risk functionals (\ref{['eg:multivariate_component_wise']}), whilst the lines are used to denote the output of the set-valued risk functionals (\ref{['eg:multivariate_var', 'eg:multivariate_dist_robust', 'eg:multivariate_pareto_statistics']}).
  • Figure 3: An illustration of the upper set and lower set ordered optimisation problem \ref{['eqn:robust_moo_set']} based on five output sets. On the left, we highlight the corresponding robust points; on the right, we illustrate how one can solve for these points via the scalarisation-based approaches.
  • Figure 4: We illustrate the polar parameterisation (\ref{['thm:polar_parameterisation']}) and the Pareto front surface statistics (\ref{['prop:rho_front_statistics']}) for a simple two-dimensional example. On the left, we plot the polar parameterisation of the Pareto front surface from \ref{['fig:chebyshev_intuition']}. On the remaining plots, we draw the corresponding risk statistics associated with a finite collection of Pareto front surfaces and the partially coherent univariate risk functionals listed in \ref{['sec:univariate_risk_functionals']}.
  • Figure 5: An illustration of the different Pareto front surfaces that can be computed from an $M=2$ dimensional objective function. On the left, we plot the output sets $f(\mathbf{x}, \Xi) \subset \mathbb{R}^M$ for each input $\mathbf{x} \in \mathbb{X}$, which are all assumed to be uniformly distributed in some ellipsoid. On the remaining plots, we draw the Pareto front surfaces associated with the Pareto front statistics, RTS fronts and STR fronts for the different risk functionals described in \ref{['sec:risk_functionals']}.
  • ...and 6 more figures

Theorems & Definitions (48)

  • Definition 2.1: Pareto domination
  • Definition 2.2: Domination region
  • Definition 2.3: Pareto optimality
  • Example 2.1: Lp scalarisation function
  • Theorem 2.1: Weak solutions
  • Example 2.2: Extreme cases
  • Example 2.3: Expectation
  • Example 2.4: Value-at-risk
  • Example 2.5: Distributionally robust
  • Definition 2.4: Partial coherency
  • ...and 38 more