Abstract
Stochastic programs are proper tools for real-time optimization if real-time features arise due to lack of data information at the moment of decision. The paper’s focus is at two-stage linear stochastic programs involving integer requirements. After a discussion of basic structural properties, two decomposition approaches are developed. While the first approach is directed to decomposition of the stochastic program itself, the second deals with decomposition of the related Graver test set.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B. Bank, R. Mandel: Parametric Integer Optimization, Akademie-Verlag, Berlin 1988.
J. R. Birge, F. Louveaux: Introduction to Stochastic Programming, Springer, New York, 1997.
C. E. Blair, R. G. Jeroslow: The value function of a mixed integer program: I, Discrete Mathematics 19 (1977), 121–138.
B. Buchberger: History and basic features of the critical-pair/completion procedure, Journal of Symbolic Computation, 2 (1987), 3–38.
C. C. Caroe, R. Schultz: Dual decomposition in stochastic integer programming, Operations Research Letters 24 (1999), 37–45.
C. C. Car0e, R. Schultz: A two-stage stochastic program for unit commitment under uncertainty in a hydro-thermal system, Schwerpunktprogramm “Echtzeit-Optimierung großer Systeme” der Deutschen Forschungsgemeinschaft, Preprint 98–13, 1998, revised 1999.
C. C. Caroe, J. Tind: L-shaped decomposition of two-stage stochastic programs with integer recourse, Mathematical Programming 83 (1998), 451–464.
D. Cox, J. Little, D. O’Shea: Ideals, Varieties, Algorithms, Springer-Verlag, 1992.
Using the CPLEX Callable Library, CPLEX Optimization, Inc. 1998.
J. E. Graver: On the foundation of linear and integer programming I, Mathematical Programming, 9 (1975), 207–226.
R. Hemmecke: On the positive sum property of Graver test sets, Preprint SM-DU-468, University of Duisburg, 2000, available for download1.
R. Hemmecke: Homepage on test sets, URL: http://www.testsets.de.
R. Hemmecke, R. Schultz: Decomposition of Test Sets in Stochastic Integer Programming, Preprint SM-DU-475, University of Duisburg, 2000, available for download2.
J. B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimzation Algorithms, Springer-Verlag, Berlin 1993.
P. Kail, S. W. Wallace: Stochastic Programming, Wiley, Chichester, 1994.
K. C. Kiwiel: Proximity control in bundle methods for convex nondifferentiable opti-mization, Mathematical Programming 46 (1990), 105–122.
K. C. Kiwiel: User’s Guide for NOA 2.0/3.0: A Fortran Package for Convex Nondifferentiable Optimization, Systems Research Institute, Polish Academy of Sciences, Warsaw, 1994.
D. Maclagan: Antichains of monomial ideals are finite, Electronic preprint, University of California at Berkeley, 1999, available for download3.
D. Maclagan: Structures on sets of monomial ideals, PhD thesis, University of California at Berkeley, 2000.
G. L. Nemhauser, L. A. Wolsey: Integer and Combinatorial Optimization, Wiley, New York, 1988.
A. Prékopa: Stochastic Programming, Kluwer, Dordrecht, 1995.
A. Ruszczyriski: Some advances in decomposition methods for stochastic linear pro-gramming, Annals of Operations Research 85 (1999), 153–172.
W. Römisch, R. Schultz: Multistage stochastic integer programs: an introduction, this volume.
A. Schrijver: Theory of Linear and Integer Programming, Wiley, Chichester, 1986.
R. Schultz: Continuity properties of expectation functions in stochastic integer program-ming, Mathematics of Operations Research 18 (1993), 578–589.
R. Schultz: On structure and stability in stochastic programs with random technology matrix and complete integer recourse, Mathematical Programming 70 (1995), 73–89.
R. Schultz: Some aspects of stability in stochastic programming, Annals of Operations Research 100 (2001), (to appear).
R. Schultz, L. Stougie, M. H. van der Vlerk: Solving stochastic programs with integer recourse by enumeration: A framework using Gröbner basis reductions, Mathematical Programming 83 (1998), 229–252.
B. Sturmfels: Gröbner Bases and Convex Polytopes, American Mathematics Society, Providence, Rhode Island, 1995.
B. Sturmfels, R. R. Thomas: Variation of cost functions in integer programming, Math-ematical Programming, 77 (1997), 357–387.
R. R. Thomas: A geometric Buchberger algorithm for integer programming, Mathematics of Operations Research, 20 (1995), 864–884.
R. M. van Slyke, R. J.-B. Wets: L-shaped linear programs withapplication to optimal control and stochastic programming, SI AM Journal of Applied Mathematics 17 (1969), 638–663.
R. Weismantel: Test sets of integer programs, Mathematical Methods of Operations Re-search, 47 (1998), 1–37.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hemmecke, R., Schultz, R. (2001). Decomposition Methods for Two-Stage Stochastic Integer Programs. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds) Online Optimization of Large Scale Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04331-8_30
Download citation
DOI: https://doi.org/10.1007/978-3-662-04331-8_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07633-6
Online ISBN: 978-3-662-04331-8
eBook Packages: Springer Book Archive