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Smooth Uncertainty Sets: Dependence of Uncertain Parameters via a Simple Polyhedral Set

Noam Goldberg, Michael Poss, Shimrit Shtern

TL;DR

This paper addresses robust optimization under dependent uncertainty by introducing the smooth uncertainty set $\mathcal{U}_{\mathcal{S}}$ that constrains pairwise differences of uncertain parameters via a graph $G=(V,E,\gamma)$. It develops probabilistic guarantees linking this set to traditional ellipsoidal bounds, and provides practical solution techniques, including exact compact reformulations for structured cases, a column-generation algorithm for large polytopes, and a minimum-cost-flow reformulation of the adversarial problem. Empirical results across transshipment, energy data, and robust shortest path demonstrate that $\mathcal{U}_{\mathcal{S}}$ delivers comparable or better performance to ellipsoidal sets while offering significant runtime and memory benefits, and that column generation can outperform classic dualization approaches. The work highlights a scalable, interpretable framework for encoding dependencies in uncertain parameters, with broad potential impact in engineering applications where physics or domain knowledge dictate smoothness or proximity across parameters.

Abstract

We propose a novel polyhedral uncertainty set for robust optimization, termed the smooth uncertainty set, which captures dependencies of uncertain parameters by constraining their pairwise differences. The bounds on these differences may be dictated by the underlying physics of the problem and may be expressed by domain experts. When correlations are available, the bounds can be set to ensure that the associated probabilistic constraints are satisfied for any given probability. We explore specialized solution methods for the resulting optimization problems, including compact reformulations that exploit special structures when they appear, a column generation algorithm, and a reformulation of the adversarial problem as a minimum-cost flow problem. Our numerical experiments, based on problems from literature, illustrate (i) that the performance of the smooth uncertainty set model solution is similar to that of the ellipsoidal uncertainty model solution, albeit, it is computed within significantly shorter running times, and (ii) our column-generation algorithm can outperform the classical cutting plane algorithm and dualized reformulation, respectively in terms of solution time and memory consumption.

Smooth Uncertainty Sets: Dependence of Uncertain Parameters via a Simple Polyhedral Set

TL;DR

This paper addresses robust optimization under dependent uncertainty by introducing the smooth uncertainty set that constrains pairwise differences of uncertain parameters via a graph . It develops probabilistic guarantees linking this set to traditional ellipsoidal bounds, and provides practical solution techniques, including exact compact reformulations for structured cases, a column-generation algorithm for large polytopes, and a minimum-cost-flow reformulation of the adversarial problem. Empirical results across transshipment, energy data, and robust shortest path demonstrate that delivers comparable or better performance to ellipsoidal sets while offering significant runtime and memory benefits, and that column generation can outperform classic dualization approaches. The work highlights a scalable, interpretable framework for encoding dependencies in uncertain parameters, with broad potential impact in engineering applications where physics or domain knowledge dictate smoothness or proximity across parameters.

Abstract

We propose a novel polyhedral uncertainty set for robust optimization, termed the smooth uncertainty set, which captures dependencies of uncertain parameters by constraining their pairwise differences. The bounds on these differences may be dictated by the underlying physics of the problem and may be expressed by domain experts. When correlations are available, the bounds can be set to ensure that the associated probabilistic constraints are satisfied for any given probability. We explore specialized solution methods for the resulting optimization problems, including compact reformulations that exploit special structures when they appear, a column generation algorithm, and a reformulation of the adversarial problem as a minimum-cost flow problem. Our numerical experiments, based on problems from literature, illustrate (i) that the performance of the smooth uncertainty set model solution is similar to that of the ellipsoidal uncertainty model solution, albeit, it is computed within significantly shorter running times, and (ii) our column-generation algorithm can outperform the classical cutting plane algorithm and dualized reformulation, respectively in terms of solution time and memory consumption.

Paper Structure

This paper contains 17 sections, 20 theorems, 36 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Probability Bounds
  3. Constructing $\mathcal{U}_{\mathcal{S}}$ for General Random Variables
  4. Constructing $\mathcal{U}_{\mathcal{S}}$ for Normally Distributed Uncertainty
  5. Compact Reformulations and Computational Methods
  6. Exact Compact Reformulations
  7. Column Generation
  8. The Constraint Generation Subproblem
  9. Evaluating the Conservatism of $\mathcal{U}_{\mathcal{S}}$
  10. Experiments using Probabilistic Bounds
  11. Evaluation of Uncertainty Sets on Energy Data
  12. Robust Shortest Path
  13. Numerical Study of the Column Generation Algorithm
  14. Summary and Conclusion
  15. Experiments' Details
  16. ...and 2 more sections

Key Result

Lemma 1

Let $A\in \mathbb{R}^{k\times n}$ and $\mathcal{U}(b)=\{\delta\in\mathbb{R}^n:A\delta\leq b\}$, and let $b^\mathcal{E}\in\mathbb{R}^n$. If $b_i^{{\mathcal{E}}}=\sqrt{\Omega}\|A_i\Sigma^{1/2}\|$ for all $i\in[k]$, then $\mathcal{U}_\mathcal{E}\subseteq \mathcal{U}(b^{{\mathcal{E}}})$ and therefore $\

Figures (5)

  • Figure 1: Different uncertainty sets for Example \ref{['example:normal']} with $n=5$ and $p=0.01$.
  • Figure 2: Mean and maximum of costs \ref{['eq:simulated_cost']}.
  • Figure 3: Size-probability tradeoff comparison between $\mathcal{U}_{\mathcal{S}}$ and $\mathcal{U}_\mathcal{E}$ uncertainty sets on energy consumption data in China.
  • Figure 4: Out-of-sample trade-off of normalized worst-case and mean path travel times on the Chicago traffic morning dataset Chassein2019. Travel times for each source-destination pair are normalized with respect to the mean nominal model travel time for that pair.
  • Figure 5: Out-of-sample trade-off of normalized worst-case and mean path length on the Chicago traffic morning dataset Chassein2019 for particular choices of $\lambda'$ and $\lambda$.

Theorems & Definitions (32)

  • Remark 1
  • Lemma 1
  • proof
  • Corollary 1
  • Corollary 2
  • Lemma 2: Multivariate Chebyshev Inequality
  • Proposition 3
  • Example 1
  • Lemma 4: bertsimas2021probabilistic
  • Lemma 5
  • ...and 22 more