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

2025-10-09

AI Summary

Robust optimization often struggles to model dependency structures among uncertain parameters. Method: This paper proposes the “smooth uncertainty set”—a novel polyhedral uncertainty set that explicitly constrains pairwise parameter differences to encode dependencies; such difference bounds are derived from physical mechanisms or domain knowledge. To enable efficient solution, we design a compact reformulation strategy integrated with a column-generation algorithm, and equivalently recast the inner adversarial problem as a minimum-cost flow problem. Contribution/Results: This is the first work to incorporate pairwise difference constraints into uncertainty set construction, yielding strong physical interpretability and probabilistic guarantees. Experiments demonstrate that our model achieves solution quality comparable to ellipsoidal sets, while reducing computation time by 40–65% and memory usage by approximately 50%. Moreover, column generation significantly outperforms both cutting-plane methods and dual reformulations.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Proposes polyhedral uncertainty set for robust optimization
• Captures uncertain parameter dependencies via pairwise differences
• Develops specialized solution methods for optimization problems

Innovation

• Polyhedral uncertainty set constraining pairwise parameter differences
• Compact reformulations and column generation for optimization
• Adversarial problem reformulated as minimum-cost flow problem