A Tractable Approach for designing Piecewise Affine Policies in Dynamic Robust Optimization

We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large. A significant challenge in designing a practical piecewise affine policy is constructing good pieces of the uncertainty set. Here we address this challenge by introducing a new framework in which the uncertainty set is “approximated” by a “dominating” simplex. The corresponding policy is then based on the map from the uncertainty set to the simplex. Although our piecewise affine policy has exponentially many pieces, it can be computed efficiently by solving a compact linear program. Furthermore, the performance of our policy is significantly better than the affine policy for many important uncertainty sets both theoretically and numerically. For instance, for hypersphere uncertainty set, our piecewise affine policy can be computed by an LP and gives a O(m)-approximation whereas the affine policy requires us to solve a second order cone program and has a worst-case performance bound of O( √ m). To the best of our knowledge, this is the first tractable approach for designing piecewise affine policies with significantly improved theoretical performance guarantees.