Structure and Stability in Two-Stage Stochastic Programming

Please always quote using this URN: urn:nbn:de:0297-zib-1819
  • {\begin{footnotesize} This thesis is concerned with structural properties and the stability behaviour of two-stage stochastic programs. Chapter~1 gives an introduction into stochastic programming and a summary of the main results of the thesis. In Chapter~2 we present easily verifiable sufficient conditions for the strong convexity of the expected-recourse function in a stochastic program with linear complete recourse. Different levels of randomness in the data are considered. We start with models where only the right-hand side of the constraints is random and extend these results to the situation where also the technology matrix contains random entries. The statements on strong convexity imply new stability estimates for sets of optimal solutions when perturbing the underlying probability measure. We work out Hölder estimates (in terms of the $\mbox{L}_1$-Wasserstein distance) for optimal solution sets to linear recourse models with random technology matrix. In Chapter~3 ({\it joint work with Werner Römisch, Berlin}) we are aiming at the Lipschitz stability of optimal solution sets to linear recourse models with random right-hand side. To this end , we first adapt the distance notion for the underlying probability measures to the structure of the model and derive a Lipschitz estimate for optimal solutions based on that distance. Here, the strong convexity established in Chapter~2 turns out as an essential assumption. For applications, however, a Lipschitz estimate with respect to a more accesssible probability distance is desirable. Structural properties of the expected-recourse function finally permit such an estimate in terms of the Kolmogorov-Smirnov distance of linear transforms of the underlying measures. The general analysis is specified to estimation via empirical measures. We obtain a law of iterated logarithm, a large deviation estimate and an estimate for the asymptotic distribution of optimal solution sets. Chapters~4 and~5 deal with two-stage linear stochastic programs where integrality constraints occur in the second stage. In Chapter~4 we study basic continuity properties of the expected-recourse function for models with random right-hand side and random technology matrix. The joint continuity with respect to the decision variable and the underlying probability measure leads to qualitative statements on the stability of local optimal values and local optimal solutions. In Chapter~5 we demonstrate that a variational distance of probability measures based on a suitable Vapnik-\v{C}ervonenkis class of Borel sets leads to convergence rates of the Hölder type for the expected recourse as a function of the underlying probability measure. The rates carry over to the convergence of local optimal values. As an application we again consider estimation via empirical measures. Beside qualitative asymptotic results for optimal values and optimal solutions we obtain a law of iterated logarithm for optimal values. \end{footnotesize}}

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Author:Rüdiger Schultz
Document Type:Habilitation
Date of first Publication:1995/05/29
Series (Serial Number):ZIB-Report (SC-95-15)