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Robust Min-Max (Regret) Optimization using Ordered Weighted Averaging

Werner Baak, Marc Goerigk, Adam Kasperski, Paweł Zieliński

TL;DR

A novel variant of ordered weighted averaging (OWA) for optimization problems that generalizes the classic OWA approach, which includes robust min-max optimization as a special case, as well as min-max regret optimization.

Abstract

In decision-making under uncertainty, several criteria have been studied to aggregate the performance of a solution over multiple possible scenarios. This paper introduces a novel variant of ordered weighted averaging (OWA) for optimization problems. It generalizes the classic OWA approach, which includes robust min-max optimization as a special case, as well as min-max regret optimization. We derive new complexity results for this setting, including insights into the inapproximability and approximability of this problem. In particular, we provide stronger positive approximation results that asymptotically improve the previously best-known bounds for the classic OWA approach. In computational experiments, we evaluate the quality of the proposed methods and compare the proposed setting with classic OWA and min-max regret approaches.

Robust Min-Max (Regret) Optimization using Ordered Weighted Averaging

TL;DR

A novel variant of ordered weighted averaging (OWA) for optimization problems that generalizes the classic OWA approach, which includes robust min-max optimization as a special case, as well as min-max regret optimization.

Abstract

In decision-making under uncertainty, several criteria have been studied to aggregate the performance of a solution over multiple possible scenarios. This paper introduces a novel variant of ordered weighted averaging (OWA) for optimization problems. It generalizes the classic OWA approach, which includes robust min-max optimization as a special case, as well as min-max regret optimization. We derive new complexity results for this setting, including insights into the inapproximability and approximability of this problem. In particular, we provide stronger positive approximation results that asymptotically improve the previously best-known bounds for the classic OWA approach. In computational experiments, we evaluate the quality of the proposed methods and compare the proposed setting with classic OWA and min-max regret approaches.
Paper Structure (23 sections, 26 theorems, 38 equations, 8 figures, 3 tables)

This paper contains 23 sections, 26 theorems, 38 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation
  4. Robust linear programming with the OWA criterion
  5. Robust combinatorial problems with the OWA criterion
  6. Some hardness results
  7. Approximation algorithms based on $p$-norm minimization
  8. Application to the matroidal, the shortest path, and the minimum Steiner tree problems
  9. The case of $\pmb{b}=\pmb{0}$
  10. The case of $\textcolor{black}{\pmb{b}>\pmb{0}}$
  11. Scenario aggregation
  12. Experiments
  13. Experiment 1: Performance of greedy algorithm
  14. Setup
  15. Results
  16. ...and 8 more sections

Key Result

Proposition 1

For every $\pmb{u},\pmb{v}\in \mathbb{R}^K_{+}$ and $p,q\in \overline{\mathbb{R}}_{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$, the inequality holds.

Figures (8)

  • Figure 1: A sample shortest path problem with $K=4$ cost scenarios.
  • Figure 2: The function $\rho(p)$ for $\pmb{w}=(0.32, 0.22, 0.12, 0.12, 0.12, 0.1, 0,0,0,0)$.
  • Figure 3: The ratio $\min\{w_1K, \sqrt{K}, \frac{\log K}{w_1}\}$ for $K=100$, depending on $w_1$.
  • Figure 4: Preference vectors $\pmb{w}$ of dimension $K=50$ for different $\alpha$-values.
  • Figure 5: Performance of greedy Algorithm for OWA.
  • ...and 3 more figures

Theorems & Definitions (37)

  • Proposition 1: Hölder's inequality
  • Proposition 2: Chebyshev's sum inequality
  • Proposition 3: Rearrangement inequality
  • Proposition 4
  • Definition 1: YA88
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7: CGKZ20
  • ...and 27 more