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Convergence Properties of Two-Stage Stochastic Programming

Author

Listed:
  • L. Dai

    (Genuity)

  • C. H. Chen

    (University of Pennsylvania)

  • J. R. Birge

    (Northwestern University)

Abstract

This paper considers a procedure of two-stage stochastic programming in which the performance function to be optimized is replaced by its empirical mean. This procedure converts a stochastic optimization problem into a deterministic one for which many methods are available. Another strength of the method is that there is essentially no requirement on the distribution of the random variables involved. Exponential convergence for the probability of deviation of the empirical optimum from the true optimum is established using large deviation techniques. Explicit bounds on the convergence rates are obtained for the case of quadratic performance functions. Finally, numerical results are presented for the famous news vendor problem, which lends experimental evidence supporting exponential convergence.

Suggested Citation

  • L. Dai & C. H. Chen & J. R. Birge, 2000. "Convergence Properties of Two-Stage Stochastic Programming," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 489-509, September.
    • Handle: RePEc:spr:joptap:v:106:y:2000:i:3:d:10.1023_a:1004649211111
    DOI: 10.1023/A:1004649211111
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    References listed on IDEAS

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    1. Alan J. King & R. Tyrrell Rockafellar, 1993. "Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 148-162, February.
    2. Stephen M. Robinson, 1996. "Analysis of Sample-Path Optimization," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 513-528, August.
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    16. Shane S. Drew & Tito Homem-de-Mello, 2012. "Some Large Deviations Results for Latin Hypercube Sampling," Methodology and Computing in Applied Probability, Springer, vol. 14(2), pages 203-232, June.
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    18. Daniel Ralph & Huifu Xu, 2011. "Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 568-592, August.
    19. Zhenfang Liu & Yang Zhou & Gordon Huang & Bin Luo, 2019. "Risk Aversion Based Inexact Stochastic Dynamic Programming Approach for Water Resources Management Planning under Uncertainty," Sustainability, MDPI, vol. 11(24), pages 1-22, December.
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    21. P. Guo & G. Huang & L. He & H. Zhu, 2009. "Interval-parameter Two-stage Stochastic Semi-infinite Programming: Application to Water Resources Management under Uncertainty," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 23(5), pages 1001-1023, March.
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