[Paper Review] Robust Optimization under Multi-band Uncertainty - Part I: Theory

Christina Büsing, Fabio D’Andreagiovanni|arXiv (Cornell University)|Jan 12, 2013

Risk and Portfolio Optimization22 references17 citations

TL;DR

This paper introduces a novel multi-band uncertainty model for Robust Optimization, extending the classical Bertsimas-Sim framework by partitioning uncertainty into multiple asymmetric bands with individual deviation limits. The approach enables more accurate modeling of real-world asymmetric and non-uniform data distributions, particularly in network design, and establishes a compact robust counterpart with probabilistic bounds using moment generating functions and Hoeffding's inequality.

ABSTRACT

The classical single-band uncertainty model introduced by Bertsimas and Sim has represented a breakthrough in the development of tractable robust counterparts of Linear Programs. However, adopting a single deviation band may be too limitative in practice: in many real-world problems, observed deviations indeed present asymmetric distributions over asymmetric ranges, so that getting a higher modeling resolution by partitioning the band into multiple sub-bands is advisable. The critical aim of our work is to close the knowledge gap on the adoption of multi-band uncertainty in Robust Optimization: a general definition and intensive theoretical study of a multi-band model are actually still missing. Our new developments have been also strongly inspired and encouraged by our industrial partners, interested in getting a better modeling of arbitrary shaped distributions, built on historical data about the uncertainty affecting the considered real-world problems.

Motivation & Objective

• Address the lack of theoretical foundation for multi-band uncertainty models in Robust Optimization, which better capture asymmetric and non-uniform real-world data distributions.
• Respond to industrial demand—particularly from telecom partners like Nokia Siemens Networks and BT—for refined uncertainty modeling in optical and telecommunication network design.
• Develop a general theoretical framework for multi-band uncertainty that extends the classical single-band robust optimization model of Bertsimas and Sim.
• Enable higher modeling resolution by allowing multiple deviation bands with individual Γ parameters, improving robustness over single-band models.
• Establish a compact robust counterpart formulation that preserves the tractability of the original MILP while incorporating complex uncertainty structures.

Proposed method

• Define a multi-band uncertainty set where each coefficient is bounded within multiple disjoint intervals around its nominal value, each with its own maximum deviation.
• Introduce individual Γ parameters for each band to control the number of coefficients allowed to deviate within that band, enabling asymmetric modeling.
• Derive a probabilistic bound on the moment generating function of uncertain coefficients using convexity and Hoeffding’s inequality.
• Use Hoeffding’s inequality to bound the difference between the sample mean and true mean of uncertain coefficients, providing a confidence interval for the actual mean.
• Construct a tight upper bound on the expected exponential moment of the uncertain coefficient-weighted sum, which is essential for deriving the robust counterpart.
• Formulate a compact robust counterpart using the derived moment generating function bound, ensuring the solution remains robust under the multi-band uncertainty set with high probability.

Experimental results

Research questions

• RQ1 How can the classical single-band robust optimization model be generalized to handle asymmetric and non-uniform uncertainty distributions through multiple deviation bands?
• RQ2 What theoretical properties does a multi-band uncertainty model possess, and how can it be embedded into a compact robust counterpart formulation?
• RQ3 Can probabilistic bounds on the moment generating function be derived for multi-band uncertainty to ensure solution robustness with high confidence?
• RQ4 How can the uncertainty model be calibrated using historical data to reflect arbitrary-shaped distributions observed in real-world applications?
• RQ5 What is the impact of using multiple bands with individual Γ values on the tractability and solution quality of robust optimization problems in network design?

Key findings

• The proposed multi-band uncertainty model provides a more accurate representation of asymmetric and non-uniform data distributions compared to the classical single-band model.
• A compact robust counterpart is derived that maintains the tractability of the original MILP while incorporating multiple deviation bands with individual Γ parameters.
• Probabilistic bounds on the moment generating function are established using convexity and Hoeffding’s inequality, enabling the construction of a robust counterpart with high confidence.
• The method achieves a confidence level of at least $\prod_{j\in J}(1 - \beta_{ij})$ for the derived bounds, where $\beta_{ij}$ controls the probability of mean deviation.
• The approach enables tighter and more realistic uncertainty sets by modeling inner-band behavior rather than relying solely on extreme values, reducing over-conservatism.
• The framework is validated through industrial collaboration and is applicable to real-world problems such as optical network design with non-symmetric traffic uncertainty.