New results about multi-band uncertainty in Robust Optimization

“The Price of Robustness” by Bertsimas and Sim represented a breakthrough in the development of a tractable robust counterpart of Linear Programming Problems. However, the central modeling assumption that the deviation band of each uncertain parameter is single may be too limitative in practice: experience indeed suggests that the deviations distribute also internally to the single band, so that getting a higher resolution by partitioning the band into multiple sub-bands seems advisable. The critical aim of our work is to close the knowledge gap about the adoption of a multi-band uncertainty set 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, which have been interested in getting a better modeling of arbitrary distributions, built on historical data of the uncertainty affecting the considered real-world problems. In this paper, we study the robust counterpart of a Linear Programming Problem with uncertain coefficient matrix, when a multi-band uncertainty set is considered. We first show that the robust counterpart corresponds to a compact LP formulation. Then we investigate the problem of separating cuts imposing robustness and we show that the separation can be efficiently operated by solving a min-cost flow problem. Finally, we test the performance of our new approach to Robust Optimization on realistic instances of a Wireless Network Design Problem subject to uncertainty.

💡 Research Summary

The paper addresses a fundamental limitation of the classic Bertsimas‑Sim robust optimization framework, namely the assumption that each uncertain coefficient is confined to a single deviation band. In many real‑world applications—especially those involving historical data such as wireless network design, power systems, or logistics—the actual distribution of deviations is more complex, often exhibiting multiple modes, asymmetry, or heavy tails. To capture this richer structure, the authors introduce a Multi‑Band Uncertainty Set (MBUS). For each coefficient (a_{ij}) they define (K) sub‑bands (B_{k}^{ij}) with associated deviation magnitudes (\delta_{ij}^{k}) and band‑specific budget parameters (\Gamma_{k}) that limit how many coefficients may take a deviation from each band. The formal definition is:

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