Optimization Problems with Second Order Stochastic Dominance Constraints: Duality, Compact Formulations, and Cut Generation Methods

For stochastic optimization problems with second order stochastic dominance constraints we develop a new form of the duality theory featuring measures on the product of the probability space and the real line. We present two formulations involving small numbers of variables and exponentially many constraints: primal and dual. The dual formulation reveals connections between dominance constraints, generalized transportation problems, and the theory of measures with given marginals. Both formulations lead to two classes of cutting plane methods. Finite convergence of both methods is proved in the case of finitely many events. Numerical results for a portfolio problem are provided.