Distributionally robust second-order stochastic dominance constrained optimization with Wasserstein ball

We consider a distributionally robust second-order stochastic dominance constrained optimization problem. We require the dominance constraints hold with respect to all probability distributions in a Wasserstein ball centered at the empirical distribution. We adopt the sample approximation approach to develop a linear programming formulation that provides a lower bound. We propose a novel split-and-dual decomposition framework which provides an upper bound. We establish quantitative convergency for both lower and upper approximations given some constraint qualification conditions. To efficiently solve the non-convex upper bound problem, we use a sequential convex approximation algorithm. Numerical evidences on a portfolio selection problem valid the convergency and effectiveness of the proposed two approximation methods.