  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 a mapping 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 given the dominating simplex. Furthermore, we can find the
dominating simplex in a closed form if the uncertainty set satisfies some
symmetries and can be computed using a MIP in general. The performance of our
policy is significantly better than the affine policy for many important
uncertainty sets, such as ellipsoids and norm-balls, both theoretically and
numerically. For instance, for hypersphere uncertainty set, our piecewise
affine policy can be computed by an LP and gives a $O(m^{1/4})$-approximation
whereas the affine policy requires us to solve a second order cone program and
has a worst-case performance bound of $O(\sqrt m)$.
