Multivariate robust stochastic dominance and resulting risk-averse optimization

By utilizing the min-biaffine scalarization function, we introduce a multivariate robust second-order stochastic dominance concept to flexibly compare random vectors. We discuss the basic properties of the dominance relation, and relate the multivariate robust second-order stochastic dominance to a functional which is continuous and subdifferentiable everywhere. We study the stochastic optimization problem with multivariate robust secondorder stochastic dominance constraints, reformulate the constraints using the introduced functional and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moment information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem and establish its qualitative stability under Kantorovich metric and pseudo metric. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.

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