2023/05/09 by Safia Kedad‐Sidhoum, Kedad-Sidhoum, Safia, A. S. Medvedev +3
Decision Sciences · Engineering · #90C17 #Advanced Control Systems Optimization #FOS: Mathematics #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design #Stability and Control of Uncertain Systems
paper · pdf · doi:10.48550/arxiv.2305.05399
openalex publication_date 2023/05/09 · openalex created_date 2023/05/12 · openalex updated_date 2026/07/28
Two-stage robust optimization is a fundamental paradigm for modeling and solving optimization problems with uncertain parameters. A now classical method within this paradigm is finite adaptability, introduced by Bertsimas and Caramanis (IEEE Transactions on Automatic Control, 2010). It consists in restricting the recourse to a finite number k of possible values. In this work, we point out that the continuity assumption they stated to ensure the convergence of the method when k goes to infinity is not correct, and we propose an alternative assumption for which we prove the desired convergence. Bertsimas and Caramanis also established that finite adaptability is NP-hard, even in the special case when k=2, the variables are continuous, and only specific parameters are subject to uncertainty. We provide a theorem showing that this special case becomes polynomial when the uncertainty set is a polytope with a bounded number of vertices, and we extend this theorem for k=3 as well. On our way, we establish new geometric results on coverings of polytopes with convex sets, which might be interesting for their own sake.