@misc{PflugRuszczynskiSchultz1997, author = {Pflug, Georg Ch. and Ruszczynski, Andrzej and Schultz, R{\"u}diger}, title = {On the Glivenko-Cantelli Problem in Stochastic Programming: Mixed-Integer Linear Recourse}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2739}, number = {SC-97-04}, year = {1997}, abstract = {Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.}, language = {en} } @misc{CaroeeSchultz1996, author = {Car{\"o}e, Claus C. and Schultz, R{\"u}diger}, title = {Dual Decomposition in Stochastic Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2560}, number = {SC-96-46}, year = {1996}, abstract = {We present an algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation. The approach can be applied to multi-stage problems with mixed-integer variables in each time stage. \%We outline a branch-and-bound algorithm for obtaining primal feasible and \%possibly optimal solutions. Numerical experience is presented for some two-stage test problems.}, language = {en} } @misc{Schultz1996, author = {Schultz, R{\"u}diger}, title = {A Note on Preprocessing via Fourier-Motzkin Elimination in Two-Stage Stochastic Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2423}, number = {SC-96-32}, year = {1996}, abstract = {Preprocessing in two-stage stochastic programming is considered from the viewpoint of Fourier-Motzkin elimination. Although of exponential complexity in general, Fourier-Motzkin elimination is shown to provide valuable insights into specific topics such as solving integer recourse stochastic programs or verifying stability conditions. Test runs with the computer code PORTA [1994] are reported.}, language = {en} } @misc{PflugRuszczynskiSchultz1996, author = {Pflug, Georg Ch. and Ruszczynski, Andrzej and Schultz, R{\"u}diger}, title = {On the Glivenko-Cantelli Problem in Stochastic Programming: Linear Recourse and Extensions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2448}, number = {SC-96-34}, year = {1996}, abstract = {Integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure. Under fairly broad conditions, it is proved that uniform convergence of empirical approximations of the right hand sides of the constraints implies uniform convergence of the optimal values in the linear and convex case.}, language = {en} } @misc{Schultz1995, author = {Schultz, R{\"u}diger}, title = {Discontinuous Optimization Problems in Stochastic Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1861}, number = {SC-95-20}, year = {1995}, abstract = {Integer stochastic linear programming is considered from the viewpoint of discontinuous optimization. After reviewing solution approaches via mollifier subgradients and decomposition we outline how to base a solution method on efficient pointwise calculation of the objective employing computer algebra.}, language = {en} } @misc{Schultz1995, author = {Schultz, R{\"u}diger}, title = {Strong Convexity in Stochastic Programs with Complete Recourse II: Partially Random Right-Hand Side}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1874}, number = {SC-95-21}, year = {1995}, abstract = {We establish a verifiable sufficient condition for strong convexity of the expected recourse as a function of the tender variable in a two-stage stochastic program with linear recourse. Generalizing a former result where all components of the second-stage right-hand side vector were random we treat the case where only a subvector of the right-hand side is random. As prerequisite, a refined analysis of the polyhedral complex of lineality regions of the second-stage value function is carried out. The sufficient condition for strong convexity allows to widen the class of recourse models for which certain quantitative results on stability and asymptotic convergence of optimal solutions are valid.}, language = {en} } @misc{SchultzStougieVlerk1995, author = {Schultz, R{\"u}diger and Stougie, Leen and Vlerk, Maarten H. van der}, title = {Solving Stochastic Programs with Complete Integer Recourse: a framework using Gr{\"o}bner Bases}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1890}, number = {SC-95-23}, year = {1995}, abstract = {In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Gr{\"o}bner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the integer expected recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest possible improvements to economize on the number of function evaluations needed.}, language = {en} } @misc{CaroeeSchultz1998, author = {Car{\"o}e, Claus C. and Schultz, R{\"u}diger}, title = {A Two-Stage Stochastic Program for Unit Commitment Under Uncertainty in a Hydro-Thermal Power System}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3543}, number = {SC-98-11}, year = {1998}, abstract = {We develop a two-stage stochastic programming model with integer first-stage and mixed-integer recourse for solving the unit commitment problem in power generation in the presence of uncertainty of load profiles. The solution methodology rests on a novel scenario decomposition method for stochastic integer programming. This method combines Lagrangian relaxation of non-anticipativity constraints with branch-and-bound. It can be seen as a decomposition algorithm for large-scale mixed-integer linear programs with block-angular structure. With realistic data from a German utility we validate our model and carry out test runs. Sizes of these problems go up to 20.000 integer and 150.000 continuous variables together with up to 180.000 constraints.}, language = {en} } @article{KochSchmidtHilleretal.2020, author = {Koch, Thorsten and Schmidt, Martin and Hiller, Benjamin and Pfetsch, Marc and Geißler, Bj{\"o}rn and Henrion, Ren{\´e} and Joormann, Imke and Martin, Alexander and Morsi, Antonio and R{\"o}misch, Werner and Schewe, Lars and Schultz, R{\"u}diger}, title = {Capacity Evaluation for Large-Scale Gas Networks}, volume = {35}, journal = {German Success Stories in Industrial Mathematics}, isbn = {978-3-030-81454-0}, doi = {10.1007/978-3-030-81455-7}, pages = {23 -- 28}, year = {2020}, language = {en} } @article{PfetschFuegenschuhGeissleretal.2014, author = {Pfetsch, Marc and F{\"u}genschuh, Armin and Geißler, Bj{\"o}rn and Geißler, Nina and Gollmer, Ralf and Hiller, Benjamin and Humpola, Jesco and Koch, Thorsten and Lehmann, Thomas and Martin, Alexander and Morsi, Antonio and R{\"o}vekamp, Jessica and Schewe, Lars and Schmidt, Martin and Schultz, R{\"u}diger and Schwarz, Robert and Schweiger, Jonas and Stangl, Claudia and Steinbach, Marc and Vigerske, Stefan and Willert, Bernhard}, title = {Validation of Nominations in Gas Network Optimization: Models, Methods, and Solutions}, journal = {Optimization Methods and Software}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2014.888426}, year = {2014}, abstract = {In this article we investigate methods to solve a fundamental task in gas transportation, namely the validation of nomination problem: Given a gas transmission network consisting of passive pipelines and active, controllable elements and given an amount of gas at every entry and exit point of the network, find operational settings for all active elements such that there exists a network state meeting all physical, technical, and legal constraints. We describe a two-stage approach to solve the resulting complex and numerically difficult feasibility problem. The first phase consists of four distinct algorithms applying linear, and methods for complementarity constraints to compute possible settings for the discrete decisions. The second phase employs a precise continuous programming model of the gas network. Using this setup, we are able to compute high quality solutions to real-world industrial instances that are significantly larger than networks that have appeared in the mathematical programming literature before.}, language = {en} }