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We consider convex optimization problems with $k$th order stochastic dominance constraints for $k\ge 2$. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets in terms of a suitably selected probability metrics.Moreover, we provide conditions ensuring that the optimal value function is Hadamard directionally differentiable. Finally, we discuss some implications of the results for empirical (Monte Carlo,
sample average) approximations of dominance constrained optimization models.
Mixed-integer two-stage stochastic programs with fixed recourse matrix, random recourse costs, technology matrix, and right-hand sides are considered. Quantitative continuity properties of its optimal value and solution set are derived when the underlying probability distribution is perturbed with respect to an appropriate probability metric.
By extending the stability analysis of [17] for multistage stochastic programs we show that their solution sets behave stable with respect to the sum of an Lr-distance and a filtration distance. Based on such stability results we suggest a scenario tree generation method for the (multivariate) stochastic input process. It starts with a fan of individual scenarios and consists of a recursive deletion and branching procedure which is controlled by bounding the approximation error. Some numerical experience for generating scenario trees in electricity portfolio management is reported.