On the performance of affine policies for two-stage adaptive optimization: a geometric perspective

Abstract

We consider two-stage adjustable robust linear optimization problems with uncertain right hand side \(\mathbf {b}\) belonging to a convex and compact uncertainty set \(\mathcal{U}\). We provide an a priori approximation bound on the ratio of the optimal affine (in \(\mathbf {b}\) ) solution to the optimal adjustable solution that depends on two fundamental geometric properties of \(\mathcal{U}\): (a) the “symmetry” and (b) the “simplex dilation factor” of the uncertainty set \(\mathcal{U}\) and provides deeper insight on the power of affine policies for this class of problems. The bound improves upon a priori bounds obtained for robust and affine policies proposed in the literature. We also find that the proposed a priori bound is quite close to a posteriori bounds computed in specific instances of an inventory control problem, illustrating that the proposed bound is informative.