Finite adaptability in two-stage robust optimization: asymptotic optimality and tractability

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.