Abstract
Stochastic programming usually represents uncertainty discretely by means of a scenario tree. This representation leads to an exponential growth of the size of stochastic mathematical problems when better accuracy is needed. Trying to solve the problem as a whole, considering all scenarios together, yields to huge memory requirements that surpass the capabilities of current computers. Thus, decomposition algorithms are employed to divide the problem into several smaller subproblems and to coordinate their solution in order to obtain the global optimum. This paper analyzes several decomposition strategies based on the classical Benders decomposition algorithm, and applies them in the emerging computational grid environments. Most decomposition algorithms are not able to take full advantage of all the computing power available in a grid system because of unavoidable dependencies inherent to the algorithms. However, a special decomposition method presented in this paper aims at reducing dependency among subproblems, to the point where all the subproblems can be sent simultaneously to the grid. All algorithms have been tested in a grid system, measuring execution times required to solve standard optimization problems and a real-size hydrothermal coordination problem. Numerical results are shown to confirm that this new method outperforms the classical ones when used in grid computing environments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abrahamson, P. G. (1983). A nested decomposition approach for solving staircase linear programs. In Systems Optimization Laboratory. Department of Operations Research, Stanford University, SOL 83-4, June 1983.
Anstreicher, K., Brixius, N., Goux, J.-P., & Linderoth, J. T. (2002). Solving large quadratic assignment problems on computational grids. Mathematical Programming, Series B, 91, 563–588.
Benders, J. F. (1962). Partition procedures for solving mixed variables programming problems. Numerische Mathematik, 4, 238–252.
Birge, J. R., Dempster, M. A., Gassmann, H. I., Gunn, E. A., King, A. J., & Wallace, S. (1987). A standard input format for multiperiod stochastic linear programs. COAL (Mathematical Programming Society, Committee on Algorithms) Newsletter, 17.
Birge, J. R., Donohue, C. J., Holmes, D. F., & Svintsitski, O. G. (1996). A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs. Mathematical Programming, 75, 327–352.
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. Berlin: Springer.
Bixby, R. E., & Martin, A. (2000). Parallelizing the dual simplex method. Informs Journal on Computing, 12(A), 45–56.
Boduroglu, I. (1997). Scalable massively parallel simplex algorithms for block-structured LP problems. Ph.D. Thesis, Columbia University, New York.
Cerisola, S., & Ramos, A. (2000). Node aggregation in stochastic nested Benders decomposition applied to hydrothermal coordination. In 6th International conference on probabilistic methods applied to power systems (PMAPS), Madeira, Portugal, 2000.
Chen, Q., Ferris, M. C., & Linderoth, J. T. (2001). FATCOP 2.0: Advanced Features in an Opportunistic Mixed Integer PRogramming solver. Annals of Operations Research, 103, 17–32.
Conejo, A. J., Castillo, E., Mínguez, R., & García-Bertrand, R. (2006). Decomposition techniques in mathematical programming. Engineering and science applications. Berlin: Springer.
Dempster, M. A. H., & Thompson, R. T. (1998). Parallelization and aggregation of nested Benders decomposition. Annals of Operations Research, 81, 163–188.
Dye, S. (2003). Subtree decomposition for multistage stochastic programs. In 18th international symposium on mathematical programming, Humboldt University Berlin, Germany, 2003.
Entriken, R. (1996). Parallel decomposition: Results for staircase linear programs. SIAM Journal on Optimization, 6(4), 961–977.
Foster, I., & Kesselman, C. (1997). Globus: a metacomputing infrastructure toolkit. International Journal of Supercomputer Applications, 11, 115–128.
Flynn, M. J. (1972). Some computer organizations and their effectiveness. IEEE Transactions on Computers, C-21(9), 948–960.
Goux, J.-P., & Leyffer, S. (2002). Solving large MINLPs on computational grids. Optimization and Engineering, 3, 327–354.
Iamnitchi, A., & Foster, I. T. (2000). A problem-specific fault-tolerance mechanism for asynchronous, distributed systems. In International conference on parallel processing (pp. 4–14), 2000.
Johnson, E. E. (1988). Completing an MIMD multiprocessor taxonomy. Computer Architecture News, 16, 44–47.
Karypis, G., Gupta, A., & Kumar, V. (1994).A parallel formulation of interior point algorithms (Technical Report 94-20). Computer Science Department, University of Minnesota, Minneapolis.
Linderoth, J., & Wright, S. J. (2003). Decomposition algorithms for stochastic programming on a computational grid. Computational Optimization and Applications, 24, 207–250. Special Issue on Stochastic Programming.
Luther, A., Buyya, R., Ranjan, R., & Venugopal, S. (2005). Peer-to-peer grid computing and a .NET-based Alchemi framework. In High performance computing: paradigm and infrastructure. New York: Wiley.
Mulvey, J. M., & Ruszczyński, A. (1995). A new scenario decomposition method for large-scale stochastic optimization. Operations Research 43(3), 477–490.
Murphy, L., Contreras, J., & Wu, F. F. (1995). A decomposition-coordination approach for large-scale optimization. In Proceedings SIAM conference on parallel processing for scientific computing (pp. 78–83), 1995.
Nielsen, S. S., & Zenios, S. A. (1997). Scalable parallel benders decomposition for stochastic linear programming. Parallel Computing, 23(8), 1069–1088.
Nogales, F. J., Prieto, F. J., & Conejo, A. J. (2003). A decomposition methodology applied to the multi-area optimal power flow problem. Annals of Operations Research, 120, 99–116.
Rockafellar, R. T., & Wets, R. J.-B. (1991). Scenarios and policy aggregation in optimization under uncertainty. Mathematics of Operations Research, 16, 119–147.
Rosa, C., & Ruszczyński, A. (1994). On augmented Lagrangian decomposition methods for multistage stochastic program (Tech. Report WP-94-125). International Institute for Applied Systems Analysis.
Seymour, K., YarKhan, A., Agrawal, S., & Dongarra, J. (2005). NetSolve: grid enabling scientific computing environments. In Grid computing and new frontiers of high performance processing. Amsterdam: Elsevier.
Shamir, R. (1987). The efficiency of the simplex method: a survey. Management Science, 33(3), 301–334.
Shu, W., & Wu, M.-Y. (1993). Sparse implementation of revised simplex algorithms on parallel computers. In The sixth SIAM conference on parallel processing for scientific computing (pp. 501–509), 1993.
SIPLIB (2004). A stochastic integer programming test problem library. http://www2.isye.gatech.edu/~sahmed/siplib/.
Stone, H. S., Chen, T. C., Flynn, M. J., Fuller, S. H., Lane, W. G., Loomis, Jr., H. H., McKeeman, W. M., Magleby, K. B., Matick, R. E., Sites, R., & Whitney, T. M. (1975). Introduction to computer architecture. Chicago: Science Research Associates.
Thain, D., Tannenbaum, T., & Livny, M. (2005). Distributed computing in practice: the Condor experience. Concurrency—Practice and Experience, 17, 323–356.
Trienekens, H. W. J. M., & de Bruin, A. (1992). Toward a taxonomy of parallel branch and bound algorithms (Report EUR-CS-92-01). Erasmus University Rotterdam, Rotterdam.
Van Slyke, R. M., & Wets, R. (1969). L-shaped linear programs with applications to optimal control and stochastic programming. SIAM Journal on Applied Mathematics, 17, 638–663.
Wright, S. J. (2001). Solving optimization problems on computational grids. Optima, 65, 8–13
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Latorre, J.M., Cerisola, S., Ramos, A. et al. Analysis of stochastic problem decomposition algorithms in computational grids. Ann Oper Res 166, 355–373 (2009). https://doi.org/10.1007/s10479-008-0476-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-008-0476-1