Piecewise static policies for two-stage adjustable robust linear optimization

In this paper, we consider two-stage adjustable robust linear optimization problems under uncertain constraints and study the performance of piecewise static policies. These are a generalization of static policies where we divide the uncertainty set into several pieces and specify a static solution for each piece. We show that in the worst-case, there is no piecewise static policy with a polynomial number of pieces that has a significantly better performance than an optimal static policy. This is quite surprising as piecewise static policies are significantly more general than static policies. The proof is based on a combinatorial argument and the structure of piecewise static policies.