Stochastic programming, Chance constraints, Two-stage, Mixed-integer, Scenario reduction, Discrepancy, Kolmogorov metric,

We study the quantitative stability of linear multistage stochastic programs under perturbations of the underlying stochastic processes. It is shown that the optimal values behave Lipschitz continuously with respect to an L p -distance. In order to establish continuity of the recourse function with respect to the current state of the stochastic process, we assume continuity of the conditional distributions in terms of a Fortet-Mourier metric. The main stability result holds for nonanticipative approximations of the underlying process and thus represents a rigorous justification of established discretization techniques.AMS 2000 subject classification: 90C15, 90C31

We give a new and comparably short proof of Gittins' index theorem for dynamic allocation problems of the multi-armed bandit type in continuous time under minimal assumptions. This proof gives a complete characterization of optimal allocation strategies as those policies which follow the current leader among the Gittins indices while ensuring that a Gittins index is at an all-time low whenever the associated project is not worked on exclusively. The main tool is a representation property of Gittins index processes which allows us to show that these processes can be chosen to be pathwise lower semi-continuous from the right and quasi-lower semi-continuous from the left. Both regularity properties turn out to be crucial for our characterization and the construction of optimal allocation policies.

Abstract. 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.

We study an approach for the evaluation of approximation and solution methods for multistage linear stochastic programs by measuring the performance of the obtained solutions on a set of out-of-sample scenarios. The main point of the approach is to restore the feasibility of solutions to an approximate problem along the out-of-sample scenarios. For this purpose, we consider and compare different feasibility and optimality based projection methods. With this at hand, we study the quality of solutions to different test models based on classical as well as recombining scenario trees.

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