Abstract
Stochastic programming has extensive applications in practical problems such as production planning and portfolio selection. Typically, the model has very large size and some techniques are often used to exploit the special structure of the programs. It has been noticed that the coefficient matrix may not be of full rank in the well-known scenario formulation of stochastic programming; thus, the preprocessing is often necessary in developing rapid decomposition methods. In this paper, we propose a parallelizable preprocessing method, which exploits effectively the structure of the formulation. Although the underlying idea is simple, the method turns out to be very useful in practice, since it may help us to select the nonanticipativity constraints efficiently. Some numerical results are reported confirming the usefulness of the method.
Similar content being viewed by others
References
1. Birge, J. R., and Louveaux, F., Introduction to Stochastic Programming, Springer Verlag, New York, NY, 1997.
2. Gondzio, J., and Kouwenberg, R., High Performance Computing for Asset Liability Management, Operations Research, Vol. 49, pp. 879–891, 2001.
3. King, A. J., Stochastic Programming Problems: Examples from the Literature, Numerical Techniques for Stochastic Optimization, Edited by Y. Ermoliev and R. J. B. Wets, Springer Verlag, New York, NY, pp. 543–567, 1988.
4. Linderoth, J., and Wright, S., Decomposition Algorithms for Stochastic Programming on a Computational Grid, Computational Optimization and Applications, Vol. 24, pp. 207–250, 2003.
5. Ruszczyńaski, A., Decomposition Methods in Stochastic Programming, Mathematical Programming, Vol. 79, pp. 333–353, 1997.
6. Rockafellar, R. T., and Wets, R. J. B., Scenarios and Policy Aggregation in Optimization under Uncertainty, Mathematics of Operations Research, Vol. 16, pp. 119–147, 1991.
7. Chun, B. J., and Robinson, S. M., Scenario Analysis via Bundle Decomposition, Annals of Operations Research, Vol. 56, pp. 39–63, 1995.
8. Helgason, T., and Wallace, S. W., Approximate Scenario Solutions in the Progressive Hedging Algorithm, Annals of Operations Research, Vol. 31, pp. 425–444, 1991.
9. Mulvey, J. M., and Vladimirou, H., Applying the Progressive Hedging Algorithm to Stochastic Generalized Networks, Annals of Operations Research, Vol. 31, pp. 399–424, 1991.
10. Berkelaar, A., Dert, C., Oldenkamp, B., and Zhang, S., A Primal-Dual Decomposition-Based Interior-Point Approach to Two-Stage Stochastic Linear Programming, Operations Research, Vol. 50, pp. 904–915, 2002.
11. Birge, J. R., and Holmes, D. F., Efficient Solutions of Two-Stage Stochastic Linear Programs Using Interior-Point Methods, Computational Optimization and Applications, Vol. 1, pp. 245–276, 1992.
12. Birge, J. R., and Qi, L., Computing Block-Angular Karmarkar Projections with Applications to Stochastic Programming, Management Science, Vol. 34, pp. 1472–1479, 1988.
13. Jessup, E. R., Yang, D., and Zenios, S. A., Parallel Factorization of Structured Matrices Arising in Stochastic Programming, SIAM Journal on Optimization, Vol. 4, pp. 833–846, 1994.
14. Liu, X. W., and Sun, J., A New Decomposition Technique in Solving Multistage Stochastic Linear Programs by Infeasible Interior-Point Methods, Journal of Global Optimization, Vol. 28, pp. 197–215, 2004.
15. Liu, X. W., and Zhao, G. Y., A Decomposition Method Based on SQP for a Class of Multistage Stochastic Nonlinear Programs, SIAM Journal on Optimization, Vol. 14, pp. 200–222, 2003.
16. Lustig, I., Mulvey, J., and Carpenter, T., Formulating Two-Stage Stochastic Programs for Interior-Point Methods, Operations Research, Vol. 39, pp. 757–770, 1991.
17. Mulvey, J. M., and Ruszczyńaski, A., A New Scenario Decomposition Method for Large-Scale Stochastic Optimization, Operations Research, Vol. 43, pp. 477–490, 1995.
18. Yang, D., and Zenios, S. A., A Scalable Parallel Interior-Point Algorithm for Stochastic Linear Programming and Robust Optimization, Computational Optimization and Applications, Vol. 7, pp. 143–158, 1997.
19. Zhao, G. Y., A Log-Barrier Method with Benders Decomposition for Solving Two-Stage Stochastic Linear Programs, Mathematical Programming, Vol. 90, pp. 507–536, 2001.
20. Zhao, G. Y., A Lagrangian Dual Method with Self-Concordant Barrier for Multistage Stochastic Convex Nonlinear Programming, Mathematical Programming, Vol. 102, pp. 1–24, 2005.
21. Ruszczyńaski, A., A Regularized Decomposition Method for Minimizing a Sum of Polyhedral Functions, Mathematical Programming, Vol. 35, pp. 309–333, 1993.
22. ruszczyńaski, A., On Convergence of an Augmented Lagrangian Decomposition Method for Sparse Convex Optimization, Mathematics of Operations Research, Vol. 20, pp. 634–656, 1995.
23. Sun, J., and Liu, X. W., Scenario Formulation of Stochastic Linear Programs and the Homogeneous Self-Dual Interior-Point Method, INFORMS Journal on Computing, 2004 (to appear).
24. Fiacco, A. V., and Mccormick, G. P., Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, NY, 1968; reprinted by SIAM Publications, Philadelphia, Pennsylvania, 1990.
25. Kall, P., and Mayer, J., On Testing SLP Codes with SLP-IOR, Institute for Operations Research, University of Zurich, Zurich, Switzerland, 1998; see website http://www.unizh.ch/ior/Pages/Deutsch/Mitglieder/Kall/bib/ka-may-98a.pdf.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the Informatics Research Center for Development of Knowledge Society Infrastructure, Graduate School of Informatics, Kyoto University, Kyoto, Japan. The work of the first author was also supported in part by the National Science Foundation of China, Grant 10571039. The work of the second author was also supported in part by the Scientific Research Grant-in-Aid from the Japan Society for the Promotion of Science. The authors are grateful to the referees for careful reading of the paper and helpful comments.
This author’s work was done while he was visiting Kyoto University.
Rights and permissions
About this article
Cite this article
Liu, X.W., Fukushima, M. Parallelizable Preprocessing Method for Multistage Stochastic Programming Problems. J Optim Theory Appl 131, 327–346 (2006). https://doi.org/10.1007/s10957-006-9156-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-006-9156-y