Abstract
Let us start our discussion with a famous textbook example: the newsboy problem. The story goes like this. Every morning a newsboy has to decide how many newspapers to buy from a newspaper publisher. Let us assume that the publisher sells the newspaper to the boy at the price of $2 each paper, and the boy then sells the newspaper along the street at the price of $5 per copy. In the end of the day, the newsboy may return any unsold copies to the publisher at $1 for each copy. The profit of the newsboy in this business depends, obviously, on the success of the sales and his initial decision on the order quantity. Unfortunately, the problem is that, as it is always the case, one cannot really predict the future with certainty. To make our analysis simple, let us further assume that there are only two possible scenarios: (1) the newsboy can sell 100 copies a day, or, (2) in the case of a boring day, he can only sell 50 copies. Furthermore, let us assume that the chance for a day with some exciting news is lower.
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
E.D. Andersen, and K.D. Andersen, The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm, High Performance Optimization Techniques, pp. 197–232, eds. J.B.G. Frenk, K. Roos, T. Terlaky and S. Zhang, Kluwer Academic Publishers, 1999.
K.A. Ariyawansa, and P.L. Jiang, Polynomial cutting plane algorithms for two-stage stochastic linear programs based on ellipsoids, volumetric centers and analytic centers, Working Paper, Department of Pure and Applied Mathematics, Washington State University, Pullman, USA, 1996.
O. Bahn, O. du Merle, J.-L. Goffin, and J.P. Vial, A cutting plane method from analytic centers for stochastic programming, Mathematical Programming, 69 (1995), 45–73.
A.J. Berger, J.M. Mulvey and A. Ruszczynski, An extension of the DQA algorithm to convex stochastic programs, SIAM Journal on Optimization, 4 (1994), 735–753.
A. Berkelaar, C. Dert, C. Oldenkamp, and S. Zhang, A Primal-Dual Decomposition-Based Interior Point Approach to Two-Stage Stochastic Linear Programming, Report 9918/A, Econometric Institute, Erasmus University Rotterdam, 1999. (To appear in Operations Research).
A. Berkelaar, R. Kouwenberg, and S. Zhang, A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming, Technical Report SEEM2000–07, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, 2000. ( Submitted for publication).
J.R. Birge, C.J. Donohue, D.F. Holmes, and O.G. Svintsitski, A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs, Mathematical Programming, 75 (1996), 327–352.
J.R. Birge, and D.F. Holmes, Efficient solution of two-stage stochastic linear programs using interior point methods, Computational Optimization and Applications, 1 (1992), 245–276.
J.R. Birge, and F. Louveaux, Introduction to Stochastic Programming, Springer, New York, 1997.
J.R. Birge, and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science, 34 (1988), 1472–1479.
I.C. Choi, and D. Goldfarb, Exploiting special structure in a primal-dual path-following algorithm, Mathematical Programming, 58 (1993), 33–52.
J. Czyzyk, R. Fourer and S. Mehrotra, Using a massively parallel processor to solve large sparse linear programs by an interior-point method, SIAM Journal on Scientific Computing, 19 (1988), 553–565.
E. Fragnière, J. Gondzio, and J.-P. Vial, Building and Solving Large-scale Stochastic Programs on an Affordable Distributed Computing System, forthcoming in Annals of Operations Research, 1998.
A.J. Goldman and A.W. Tucker, Polyhedral convex cones, in H.W. Kuhn and A.W. Tucker eds., Linear Inequalities and Related Systems, Princeton University Press, New Jersey, 19–40, 1956.
J. Gondzio, and R. Kouwenberg, High Performance Computing for Asset Liability Management, forthcoming in Operations Research, 2000.
E.R. Jessup, D. Yang, and S.A. Zenios, Parallel factorization of structured matrices arising in stochastic programming, SIAM Journal on Optimization, 4 (1994), 833–846.
P. Kall, and S.W. Wallace, Stochastic Programming, John Wiley and Sons, Chichester, 1994.
I.J. Lustig, J.M. Mulvey and T.J. Carpenter, The formulation of stochastic programs for interior point methods, Operations Research, 39 (1991), 757–770.
S. Mizuno, M.J. Todd, and Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Mathematics of Operations Research, 18 (1993), 964–981.
J.M. Mulvey, and A. Ruszczynski, A new scenario decomposition method for large-scale stochastic optimization, Operations Research, 43 (1995), 477–490.
R.T. Rockafellar, and R.J.-B. Wets, Scenarios and policy aggregation in optimization under uncertainty, Mathematics of Operations Research, 16 (1991), 119–147.
C. Roos, T. Terlaky, and J.-Ph. Vial, Theory and Algorithms for Linear Optimization, John Wiley & Sons, chapter 19, 1997.
R. Van Slyke, and R.J.-B. Wets, L-shaped linear programs with applications to optimal control and stochastic linear programs, SIAM Journal on Applied Mathematics, 17 (1969), 638–663.
X. Xu, P.F. Hung, and Y. Ye, A simplified homogeneous self-dual linear programming algorithm and its implementation, Annals of Operations Research, 62 (1996), 151–171.
D. Yang, and S.A. Zenios, A Scalable parallel interior point algorithm for stochastic linear programming and robust optimization, Computational Optimization and Applications, 7 (1997), 143–158.
Y. Ye, and K. Anstreicher, On quadratic and 0(./L) convergence of a predictor-corrector algorithm for LCP, Mathematical Programming, 62 (1993), 537–551.
Y. Ye, M.J. Todd, and S. Mizuno, An O(/L)-iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research, 19 (1994), 53–67.
G. Zhao, A log-barrier method with Benders decomposition for solving two-stage stochastic linear programs, Mathematical Programming, 90 (2001), 507–536.
Rights and permissions
Copyright information
© 2003 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Zhang, S. (2003). An Interior-Point Approach to Multi-Stage Stochastic Programming. In: Stochastic Modeling and Optimization. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21757-4_5
Download citation
DOI: https://doi.org/10.1007/978-0-387-21757-4_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3065-1
Online ISBN: 978-0-387-21757-4
eBook Packages: Springer Book Archive