An approximation scheme for stochastic programs with second order dominance constraints

It is well known that second order dominance relation between two random variables can be described by a system of stochastic semi-infinite inequalities indexed by $\mathcal R$, the set of all real number. In this paper, we show the index set can be reduced to the support set of the dominated random variable strengthening a similar result established by Dentcheva and Ruszczyński [9] for discrete random variables. Viewing the semi-infinite constraints as an extreme robust risk measure, we relax it by replacing it with entropic risk measure and regarding the latter as an approximation of the former in an optimization problem with second order dominance constraints.To solve the entropic approximation problem, we apply the well knownsample average approximation method to discretizeit. Detailed analysis is given to quantify both the entropic approximation and sample average approximationfor various statistical quantities including the optimal value, the optimal solutions and the stationary points obtained fromsolving the sample average approximated problem.The numerical scheme provides an alternative to the mainstream numerical methods for this important class of stochastic programs.