Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric

summary:Optimization problems with stochastic dominance constraints are helpful to many real-life applications. We can recall e. g., problems of portfolio selection or problems connected with energy production. The above mentioned constraints are very suitable because they guarantee a solution fulfilling partial order between utility functions in a given subsystem $ {\cal U} $ of the utility functions. Especially, considering $ {\cal U} := {\cal U}_{1} $ (where ${\cal U}_{1} $ is a system of non decreasing concave nonnegative utility functions) we obtain second order stochastic dominance constraints. Unfortunately it is also well known that these problems are rather complicated from the theoretical and the numerical point of view. Moreover, these problems goes to semi-infinite optimization problems for which Slater's condition is not necessary fulfilled. Consequently it is suitable to modify the constraints. A question arises how to do it. The aim of the paper is to suggest one of the possibilities how to modify the original problem with an "estimation" of a gap between the original and a modified problem. To this end the stability results obtained on the base of the Wasserstein metric corresponding to ${\cal L}_{1}$ norm are employed. Moreover, we mention a scenario generation and an investigation of empirical estimates. At the end attention will be paid to heavy tailed distributions