Data-driven Multivariate Distributionally Robust Stochastic Dominance Optimization And Its Approximation

Stochastic dominance provides a powerful tool for the selection of risk assets,which originated in decision theory and is widely used in economics.Since the optimization problem with dominance constraints was proposed by Dentcheva and Ruszczy ′nski in 2003,stochastic dominance is introduced into optimization.As a model of risk aversion,the optimization model with stochastic dominance constraints has attracted the attention of domestic and foreign scholars,especially the second-order stochastic dominance problem attracts more attention.In the past ten years,many scholars have made profound research on this kind of problem from the aspects of optimality conditions,numerical schemes and practical applications.In this paper,we propose a stochastic optimization problem with multivariate distributionally robust stochastic dominance constraint and we focus on the solution method of the problem.The specific reaserch is as follows:In the first step,we first introduce the multivariate distributionally robust stochastic dominance optimization,the reaserch background and the domestic and foreign situation of it.Then the concepts of multivariate distributionally robust stochastic dominance problem are combed,such as second-order stochastic dominance,the uniform multivariate robust dominance condition,Lipschitz continuity and so on.After introducing the Wasserstein metric,we construct the ambiguity set by Wasserstein metric.In the second part,in order to solve the multivariate distributionally robust second-order stochastic domiance optimization problem proposed in this paper,we transform the multivariate stochastic dominance into univariate constraint by utilizing the linear scalarization function and then process the transformed robust model.Then after specifying an ambiguity set based on Wasserstein metric,we quantify the convergence of the ambiguity set to the true probability distribution as the sample size increases.Finally,we do the stability analysis,we prove the convergence of the related problems and consequently the optimal values and the optimal solutions of the correspongding problems.In the third part,this paper proposes the application of the problem in the portfolio.We apply the multivariate secondorder stochastic dominance problem presented in this paper to decision making problems.Under mild assumptions,the distributionally robust programming probelm over Wasserstein metric can actually be reformulated as a finite convex programming.At last part,some preliminary numerical test results are reported,which show the advantages of our model over the traditional SAA method.