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An extension of Ordered Weighted Averaging over intervals with application to optimization under risk

Werner Baak, Marc Goerigk, Adam Kasperski, Paweł Zieliński

TL;DR

This work extends the Ordered Weighted Averaging framework from finite discrete scenarios to the continuous realm through distortion risk measures, enabling optimization with interval-based uncertainty in linear objectives. It formalizes the continuous extension, establishes key theoretical properties (coherence, convergence, and bounds), and analyzes the optimization problem under risk where costs are interval-uncertain and, in a central case, independent and uniformly distributed. The authors develop both sampling-based and deterministic approximation methods, prove #P-hardness for exact evaluation in the uniform case, and provide approximation guarantees tied to the distortion function’s shape; computational tests on knapsack-like instances demonstrate practical efficiency of the simple $\mathbf{c}^Q$-based approach, with more sophisticated methods offering gains for larger problem sizes. Overall, the paper delivers a rigorous bridge between OWA and distortion risk measures, with concrete algorithms and empirical validation for risk-aware optimization under interval uncertainty.

Abstract

The Ordered Weighted Averaging (OWA) operator is a traditional and commonly used criterion for aggregating discrete values of uncertain quantities. In this paper, it is shown that the discrete OWA naturally extends to the continuous case by using the concept of a distortion risk measure. It is shown how to apply the distortion risk measure to optimization problems with a linear objective function, whose coefficients are random variables with continuous distribution functions supported on intervals. The case where these coefficients are independent, uniformly distributed random variables is explored in more detail. The computational complexity of the resulting optimization problem is analyzed, and solution methods with approximation guarantees are proposed. These methods are also verified through computational experiments.

An extension of Ordered Weighted Averaging over intervals with application to optimization under risk

TL;DR

This work extends the Ordered Weighted Averaging framework from finite discrete scenarios to the continuous realm through distortion risk measures, enabling optimization with interval-based uncertainty in linear objectives. It formalizes the continuous extension, establishes key theoretical properties (coherence, convergence, and bounds), and analyzes the optimization problem under risk where costs are interval-uncertain and, in a central case, independent and uniformly distributed. The authors develop both sampling-based and deterministic approximation methods, prove #P-hardness for exact evaluation in the uniform case, and provide approximation guarantees tied to the distortion function’s shape; computational tests on knapsack-like instances demonstrate practical efficiency of the simple -based approach, with more sophisticated methods offering gains for larger problem sizes. Overall, the paper delivers a rigorous bridge between OWA and distortion risk measures, with concrete algorithms and empirical validation for risk-aware optimization under interval uncertainty.

Abstract

The Ordered Weighted Averaging (OWA) operator is a traditional and commonly used criterion for aggregating discrete values of uncertain quantities. In this paper, it is shown that the discrete OWA naturally extends to the continuous case by using the concept of a distortion risk measure. It is shown how to apply the distortion risk measure to optimization problems with a linear objective function, whose coefficients are random variables with continuous distribution functions supported on intervals. The case where these coefficients are independent, uniformly distributed random variables is explored in more detail. The computational complexity of the resulting optimization problem is analyzed, and solution methods with approximation guarantees are proposed. These methods are also verified through computational experiments.

Paper Structure

This paper contains 18 sections, 21 theorems, 82 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discrete OWA operator and distortion risk measure
  3. Discrete OWA operator
  4. Distortion risk measure - an extension of the OWA operator to the continuous case
  5. Properties of distortion risk measure
  6. Optimization under risk
  7. Optimization problem with a distortion risk measure
  8. Solution algorithms
  9. The problem with independent uniformly distributed cost coefficients
  10. Motivation
  11. The computational complexity of the problem
  12. Solution algorithms
  13. Computational tests
  14. Setup
  15. Instance generation
  16. ...and 3 more sections

Key Result

Corollary 1

If $Q(t)=1-Q(1-t)$ for $t\in [0,1]$ and $Q$ is continuously differentiable, then the weight function $w$ is symmetric, $w(t)=w(1-t)$ for $t\in [0,1]$, and $\rho_Q(X)=\rho_w(X)=\mathbb{E}[X]$.

Figures (5)

  • Figure 1: The BUM function $Q_p(t)$ and the corresponding nonincreasing weight function $w(t)=\frac{d Q(t)}{dt}$.
  • Figure 2: Comparison of VaR profiles for different solutions.
  • Figure 3: The BUM function $Q(t)=\frac{1}{1+\text{e}^{-M(t-\alpha)}}$ for $M=100$, $\alpha=0.3$, and the corresponding weight function $w(t)=\frac{d Q(t)}{d t}$.
  • Figure 4: Average ratio of objective value to objective value of sampling with $K=10$.
  • Figure 5: Average computation times.

Theorems & Definitions (39)

  • Definition 1: D90SBR10
  • proof
  • proof
  • Corollary 1
  • Theorem 1: Convergence of Discrete OWA
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • Lemma 2
  • ...and 29 more