New Results about Multi-band Uncertainty in Robust Optimization

Abstract

“The Price of Robustness” by Bertsimas and Sim [4] 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.

In this work, 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.

This work was partially supported by the German Federal Ministry of Education and Research (BMBF), project ROBUKOM, grant 03MS616E, and by the DFG Research Center Matheon - Mathematics for key technologies, project B23 - Robust optimization for network applications.