New results about multi-band uncertainty in Robust Optimization

TL;DR

This paper extends Robust Optimization by introducing a multi-band uncertainty model, providing a more detailed representation of uncertainty, and demonstrating its computational tractability and practical effectiveness through wireless network design case studies.

Contribution

It develops a general multi-band uncertainty set for Robust Optimization, proves the compactness of the robust LP formulation, and shows efficient separation via min-cost flow, enhancing modeling of real-world uncertainties.

Findings

Robust counterpart remains a compact LP formulation.

Separation of robustness constraints can be efficiently solved as a min-cost flow.

The approach improves robustness modeling in wireless network design.

Abstract

"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.