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Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario
sets. The relevant distances of (multivariate) probability
distributions for deriving quantitative stability results for such stochastic programs are $\mathcal{B}$-discrepancies, where the class $\mathcal{B}$ of Borel sets depends on their structural properties.
Hence, the optimal scenario reduction problem for such models is stated with respect to $\mathcal{B}$-discrepancies. In this paper,
upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop
heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or
Kolmogorov metric) in some detail and provide some numerical experience.
Polyhedral discrepancies are relevant for the quantitative stability
of mixed-integer two-stage and chance constrained stochastic programs. We
study the problem of optimal scenario reduction for a discrete probability
distribution with respect to certain polyhedral discrepancies and develop
algorithms for determining the optimally reduced distribution approximately.
Encouraging numerical experience for optimal scenario reduction is provided.