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Stochastic Integer Programming: Limit Theorems and Confidence Intervals

Department of Mathematics, Humboldt-University Berlin, Unter den Linden 6, 10099 Berlin, Germany
Department of Mathematics, Humboldt-University Berlin, Unter den Linden 6, 10099 Berlin, Germany
eichhorn{at}math.hu-berlin.de, http://www.math.hu-berlin.de/eichhorn
romisch{at}math.hu-berlin.de, http://www.math.hu-berlin.de/romisch

We consider empirical approximations (sample average approximations) of two-stage stochastic mixed-integer linear programs and derive central limit theorems for the objectives and optimal values. The limit theorems are based on empirical process theory and the functional delta method. We also show how these limit theorems can be used to derive confidence intervals for optimal values via resampling methods (bootstrap, subsampling).

Key Words: stochastic programming; mixed-integer optimization; stability; sample average approximation; empirical process; Donsker class; delta method; Hadamard directional differentiability; bootstrap; subsampling
History: Received: January 6, 2005; revision received: August 10, 2005;