Abstract
A scenario-based decomposition algorithm is proposed for large stochastic pro-grams. The subproblem clusters consisting of separable quadratic programs are solved by means of a nonlinear interior point algorithm. Critical implementation issues are analyzed, including restarting and alternative splitting strategies. The approach is suited to a distributed multicomputer such as a network of workstations. Testing with several large LPs (117,000 constraints and 276,000 variables) shows the efficiency of the concepts.
Supported in part by the National Science Foundation CCR-9102660 and U.S.Air Force AFOSR-91-0359. Also acknowledgement are Scientific Conputing Associates who supplied the network-Linda sorfware.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. Beguelin, J. Dongarra, A. Geist, R. Manchek, and V. Sunderam (1991), “A Users’ Guide to PVM Parallel Virtual Machine,” Technical Report ÓRNL/TM- 11826, Engineering Physics and Mathematics Division, Mathematical Sciences Section, Oak Ridge National Laboratory, Oak Ridge, Tennessee.
A. J. Berger, J. M. Mulvey, and A. Ruszczynski (1993), “A Distributed Scenario Decomposition Algorithm for Large Stochastic Programs,” Technical Report SOR 93 - 2, Department of Civil Engineering and Operations Research, Princeton University, Princeton.
D. P. Bertsekas (1982), Constrained Optimization and Lagrange Multiplier Methods, Academic Press.
J. R. Birge, M. A. H. Dempster, H. Gassmann, E. Gunn, A. J. King, S. W. Wallace (1987), “A Standard Input Format For Multiperiod Stochastic Linear Programs,” Working Paper WP-87-118, International Institute for Applied Systems Analysis, Laxenburg, Austria.
D. R. Cariño, T. Kent, D. H. Myers, C. Stacy, M. Sylvanus, A. L. Turner, K. Watanabe, and W. T. Ziemba (May 1993), “Russell-Yasuda Kasai model: An asset/liability model for a Japanese insurance company using multistage stochastic programming,” Presented at the ORS A/TIMS Joint National Meeting in Chicago.
N. Carriero and D. Gelernter (1989), “Linda in Context,” Commun. ACM 32, 444–458.
G. B. Dantzig and G. Infanger (1991), “Large-Scale Stochastic Linear Programs: Importance Sampling and Benders Decomposition,” Report SOL 91 - 4, Department of Operations Research, Stanford University.
Yu. Ermoliev and R. J.-B. Wets, eds (1988), Numerical Methods in Stochastic Programming, Springer-Verlag, Berlin.
R. M. Freund (1991), “A Potential Reduction Function Algorithm for Solving a Linear Program from an Infeasible Warm Start,” Mathematical Programming 51, 441–466.
R. R. Grauer and N. H. Hakansson (1986), “A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills, with and without Small Stocks,” Journal of Business 59 (2).
J. L. Higle and S. Sen (1991), “Stochastic Decomposition: An Algorithm for Stage Linear Programs with Recourse,” Mathematics of Operations Research 16, 650–669.
M. Kojima, N. Megiddo, and S. Mizuno (December 1991), “A Primal-Dual Exterior Point Algorithm for Linear Programming,” Research Report RJ 8500, IBM Almadén Research Center, San Jose, CA 95120-6099, USA.
M. Kusy, and W. Ziemba (1986), “A Bank Asset Liability Model,” Operations Research 34, 356–376.
I. J. Lustig, R. E. Marsten and D. F. Shanno (1992), “On Implementing Mehrotra’s Predictor-Corrector Interior Point Method for Linear Programming,” SIAM J. Optimization 2, 435–449.
I. J. Lustig, J. M. Mulvey, and T. J. Carpenter (1991), “Formulating Two-Stage Stochastic Programs for Interior Point Methods,” Operations Research 39, 757–770.
J. M. Mulvey (1992), “Incorporating Transaction Costs in Models for Asset Allocation,” in: Financial Optimization, S. Zenios (ed. ), Cambridge University Press.
J. M. Mulvey and A. Ruszczyñski (1992), “A Diagonal Quadratic Approximation Method for Large Scale Linear Programs,” Operations Research Letters 12, 205–215.
J. M. Mulvey and A. Ruszczynski (1992), “A New Scenario Decomposition Method for Large-Scale Stochastic Optimization,” Technical Report SOR 91–19, Department of Civil Engineering and Operations Research, Princeton University, Princeton.
J. M. Mulvey and H. Vladimirou (1992), “Stochastic Network Programming for Financial Planning Problems,” Management Science 38, 1642–1664.
G. Stephanopoulos and W. Westerberg (1975), “The Use of Hestenes’ Method of Multipliers to resolve Dual Gaps in Engineering System Optimization,” Journal of Optimization Theory and Applications 15, 285–309.
V. S. Sunderam (1990), “PVM: A Framework for Parallel Distributed Computing,” Concurrency: Practice and Experience 2, 315–339.
R. J. Vanderbei and T. J. Carpenter (1991), “Symmetric Indefinite Systems for Interior-Point Methods,” Technical Report SOR 91 - 7, Department of Civil Engineering and Operations Research, Princeton University, Princeton.
R. J. Vanderbei (1991), “Symmetric Quasi-Definite Matrices,” Technical Report SOR 91–10, Department of Civil Engineering and Operations Research, Princeton University, Princeton.
R. J. Vanderbei (1992), “LOQO User’s Manual,” Technical Report SOR 92 - 5, Department of Civil Engineering and Operations Research, Princeton University, Princeton.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Kluwer Academic Publishers
About this chapter
Cite this chapter
Berger, A.J., Mulvey, J.M., Ruszczyński, A. (1994). Restarting Strategies for the DQA Algorithm. In: Hager, W.W., Hearn, D.W., Pardalos, P.M. (eds) Large Scale Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3632-7_1
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3632-7_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3634-1
Online ISBN: 978-1-4613-3632-7
eBook Packages: Springer Book Archive