Robust Linear Optimization under General Norms

We propose a framework for robust modeling of linear programming problems using uncertainty sets described by an arbitrary norm. We explicitly characterize the robust counterpart as a convex optimization problem that involves the dual norm of the given norm. Under a Euclidean norm we recover the second order cone formulation in Ben-Tal and Nemirovski [1, 2], El Ghaoui et al. [8, 9], while under a particular D-norm we introduce we recover the linear programming formulation proposed in Bertsimas and Sim [6]. We also provide guarantees for constraint violation under general probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coefficients.