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A Massively Parallel Algorithm for Nonlinear Stochastic Network Problems

Author

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  • Soren S. Nielsen

    (The University of Texas, Austin, Texas)

  • Stavros A. Zenios

    (University of Pennsylvania, Philadelphia, Pennsylvania)

Abstract

We develop an algorithm for solving nonlinear, two-stage stochastic problems with network recourse. The algorithm is based on the framework of row-action methods. The problem is formulated by replicating the first-stage variables and then adding nonanticipativity side constraints. A series of (independent) deterministic network problems are solved at each step of the algorithm, followed by an iterative step over the nonanticipativity constraints. The solution point of the iterates over the nonanticipativity constraints is obtained analytically. The row-action nature of the algorithm makes it suitable for parallel implementations. A data representation of the problem is developed that permits the massively parallel solution of all the scenario subproblems concurrently. The algorithm is implemented on a Connection Machine CM-2 with up to 32K processing elements and achieves computing rates of 276 MFLOPS. Very large problems—8,192 scenarios with a deterministic equivalent nonlinear program with 868,367 constraints and 2,474,017 variables—are solved within a few minutes. We report extensive numerical results regarding the effects of stochasticity on the efficiency of the algorithm.

Suggested Citation

  • Soren S. Nielsen & Stavros A. Zenios, 1993. "A Massively Parallel Algorithm for Nonlinear Stochastic Network Problems," Operations Research, INFORMS, vol. 41(2), pages 319-337, April.
    • Handle: RePEc:inm:oropre:v:41:y:1993:i:2:p:319-337
    DOI: 10.1287/opre.41.2.319
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    Citations

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    Cited by:

    1. Patriksson, Michael, 2008. "A survey on the continuous nonlinear resource allocation problem," European Journal of Operational Research, Elsevier, vol. 185(1), pages 1-46, February.
    2. De Waegenaere, A.M.B. & Wielhouwer, J.L., 2001. "A Partial Ranking Algorithm for Resource Allocation Problems," Other publications TiSEM 8b2e0185-36f9-43df-8a3d-d, Tilburg University, School of Economics and Management.
    3. Zenios, Stavros A. & Pinar, Mustafa C. & Dembo, Ron S., 1995. "A smooth penalty function algorithm for network-structured problems," European Journal of Operational Research, Elsevier, vol. 83(1), pages 220-236, May.
    4. Og[caron]uzsoy, Cemal Berk & Güven, Sibel, 2007. "Robust portfolio planning in the presence of market anomalies," Omega, Elsevier, vol. 35(1), pages 1-6, February.
    5. Mulvey, John M. & Rosenbaum, Daniel P. & Shetty, Bala, 1997. "Strategic financial risk management and operations research," European Journal of Operational Research, Elsevier, vol. 97(1), pages 1-16, February.
    6. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    7. Diana Barro & Elio Canestrelli, 2005. "Time and nodal decomposition with implicit non-anticipativity constraints in dynamic portfolio optimization," GE, Growth, Math methods 0510011, University Library of Munich, Germany.
    8. De Waegenaere, A.M.B. & Wielhouwer, J.L., 2001. "A Partial Ranking Algorithm for Resource Allocation Problems," Discussion Paper 2001-40, Tilburg University, Center for Economic Research.
    9. Wielhouwer, J.L., 2002. "Optimal tax depreciation and its effects on optimal firm investments," Other publications TiSEM 822d5150-6b3a-463a-8001-4, Tilburg University, School of Economics and Management.
    10. J. Gondzio, 1994. "Preconditioned Conjugate Gradients in an Interior Point Method for Two-stage Stochastic Programming," Working Papers wp94130, International Institute for Applied Systems Analysis.
    11. Taesung Hwang & Yanfeng Ouyang, 2015. "Urban Freight Truck Routing under Stochastic Congestion and Emission Considerations," Sustainability, MDPI, vol. 7(6), pages 1-16, May.
    12. Sodhi, ManMohan S. & Tang, Christopher S., 2009. "Modeling supply-chain planning under demand uncertainty using stochastic programming: A survey motivated by asset-liability management," International Journal of Production Economics, Elsevier, vol. 121(2), pages 728-738, October.
    13. Gaivoronski, Alexei A. & Stella, Fabio, 2003. "On-line portfolio selection using stochastic programming," Journal of Economic Dynamics and Control, Elsevier, vol. 27(6), pages 1013-1043, April.
    14. Siva Sankaran & Tung Bui, 2008. "An organizational model for transitional negotiations: concepts, design and applications," Group Decision and Negotiation, Springer, vol. 17(2), pages 157-173, March.

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