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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.
We consider multistage stochastic optimization models containing nonconvex constraints, e.g.,
due to logical or integrality requirements. We study three variants of Lagrangian relaxations and of the corresponding
decomposition schemes, namely, scenario, nodal and geographical decomposition. Based on
convex equivalents for the Lagrangian duals, we compare the duality gaps for these decomposition schemes.
The first main result states that scenario decomposition provides a smaller or equal duality gap than nodal
decomposition. The second group of results concerns large stochastic optimization models with loosely coupled
components. The results provide conditions implying relations between the duality gaps of geographical
decomposition and the duality gaps for scenario and nodal decomposition, respectively.
Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints
(2007)
We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type.
Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional
derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the
dominance constraints.