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Formulating Two-Stage Stochastic Programs for Interior Point Methods

Author

Listed:
  • Irvin J. Lustig

    (Princeton University, Princeton, New Jersey)

  • John M. Mulvey

    (Princeton University, Princeton, New Jersey)

  • Tamra J. Carpenter

    (Princeton University, Princeton, New Jersey)

Abstract

This paper describes an approach for modeling two-stage stochastic programs that yields a form suitable for interior point algorithms. A staircase constraint structure is created by replacing first stage variables with sparse “split variables” in conjunction with side-constraints. Dense columns are thereby eliminated. The resulting model is larger than traditional stochastic programs, but computational savings are substantial—over a tenfold improvement for the problems tested. A series of experiments with stochastic networks drawn from financial planning demonstrates the attained efficiencies. Comparisons with MINOS and the dual block angular stochastic programming model are provided as benchmarks. The split variable approach is applicable to general two-stage stochastic programs and other dual block angular models.

Suggested Citation

  • Irvin J. Lustig & John M. Mulvey & Tamra J. Carpenter, 1991. "Formulating Two-Stage Stochastic Programs for Interior Point Methods," Operations Research, INFORMS, vol. 39(5), pages 757-770, October.
    • Handle: RePEc:inm:oropre:v:39:y:1991:i:5:p:757-770
    DOI: 10.1287/opre.39.5.757
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    Cited by:

    1. T. Glenn Bailey & Paul A. Jensen & David P. Morton, 1999. "Response surface analysis of two‐stage stochastic linear programming with recourse," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(7), pages 753-776, October.
    2. Hong‐Chih Huang, 2010. "Optimal Multiperiod Asset Allocation: Matching Assets to Liabilities in a Discrete Model," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 77(2), pages 451-472, June.
    3. 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.
    4. Nie, S. & Li, Y.P. & Liu, J. & Huang, Charley Z., 2017. "Risk management of energy system for identifying optimal power mix with financial-cost minimization and environmental-impact mitigation under uncertainty," Energy Economics, Elsevier, vol. 61(C), pages 313-329.
    5. Messina, E. & Mitra, G., 1997. "Modelling and analysis of multistage stochastic programming problems: A software environment," European Journal of Operational Research, Elsevier, vol. 101(2), pages 343-359, September.
    6. Meszaros, Csaba, 1997. "The augmented system variant of IPMs in two-stage stochastic linear programming computation," European Journal of Operational Research, Elsevier, vol. 101(2), pages 317-327, September.
    7. Castro, Jordi & Escudero, Laureano F. & Monge, Juan F., 2023. "On solving large-scale multistage stochastic optimization problems with a new specialized interior-point approach," European Journal of Operational Research, Elsevier, vol. 310(1), pages 268-285.
    8. M. Alvarez & C. Cuevas & L. Escudero & J. Escudero & C. García & F. Prieto, 1994. "Network planning under uncertainty with an application to hydropower generation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 2(1), pages 25-58, June.
    9. Arjan Berkelaar & Cees Dert & Bart Oldenkamp & Shuzhong Zhang, 2002. "A Primal-Dual Decomposition-Based Interior Point Approach to Two-Stage Stochastic Linear Programming," Operations Research, INFORMS, vol. 50(5), pages 904-915, October.
    10. C.H. Rosa & A. Ruszczynski, 1994. "On Augmented Lagrangian Decomposition Methods for Multistage Stochastic Programs," Working Papers wp94125, International Institute for Applied Systems Analysis.
    11. 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.
    12. X. W. Liu & M. Fukushima, 2006. "Parallelizable Preprocessing Method for Multistage Stochastic Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 327-346, December.
    13. Suvrajeet Sen & Lihua Yu & Talat Genc, 2006. "A Stochastic Programming Approach to Power Portfolio Optimization," Operations Research, INFORMS, vol. 54(1), pages 55-72, February.
    14. P. Beraldi & D. Conforti & A. Violi, 2009. "SICOpt: Solution Approach for Nonlinear Integer Stochastic Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 17-36, October.
    15. ZhenFang Liu & GuoHe Huang, 2009. "Dual-Interval Two-Stage Optimization for Flood Management and Risk Analyses," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 23(11), pages 2141-2162, September.
    16. Maqsood, Imran & Huang, Guo H. & Scott Yeomans, Julian, 2005. "An interval-parameter fuzzy two-stage stochastic program for water resources management under uncertainty," European Journal of Operational Research, Elsevier, vol. 167(1), pages 208-225, November.
    17. A. Ruszczynski, 1993. "Interior Point Methods in Stochastic Programming," Working Papers wp93008, International Institute for Applied Systems Analysis.
    18. Vladimirou, Hercules & Zenios, Stavros A., 1997. "Stochastic linear programs with restricted recourse," European Journal of Operational Research, Elsevier, vol. 101(1), pages 177-192, August.
    19. 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.
    20. Sanjay Mehrotra & M. Gokhan Ozevin, 2009. "Decomposition Based Interior Point Methods for Two-Stage Stochastic Convex Quadratic Programs with Recourse," Operations Research, INFORMS, vol. 57(4), pages 964-974, August.
    21. Kavinesh J. Singh & Andy B. Philpott & R. Kevin Wood, 2009. "Dantzig-Wolfe Decomposition for Solving Multistage Stochastic Capacity-Planning Problems," Operations Research, INFORMS, vol. 57(5), pages 1271-1286, October.

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