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Author: Römisch, Werner × Subject: Stochastic optimization ×

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    Conditioning of linear-quadratic two-stage stochastic optimization problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Emich, Konstantin; Henrion, René; Römisch, Werner
    In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.
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    Problem-based optimal scenario generation and reduction in stochastic programming
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Henrion, Réne; Römisch, Werner
    Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.