Multipolar robust optimization

We consider linear programs involving uncertain parameters and propose a new tractable robust counterpart which contains and generalizes several other models including the existing Affinely Adjustable Robust Counterpart and the Fully Adjustable Robust Counterpart. It consists in selecting a set of poles whose convex hull contains some projection of the uncertainty set, and computing a recourse strategy for each data scenario as a convex combination of some optimized recourses (one for each pole). We show that the proposed multipolar robust counterpart is tractable and its complexity is controllable. Further, we show that under some mild assumptions, two sequences of upper and lower bounds converge to the optimal value of the fully adjustable robust counterpart. To illustrate the approach, a robust problem related to lobbying under some uncertain opinions of authorities is studied. Several numerical experiments are carried out showing the advantages of the proposed robustness framework and evaluating the benefit of adaptability.