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Inference of statistical bounds for multistage stochastic programming problems

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  • Alexander Shapiro

Abstract

We discuss in this paper statistical inference of sample average approximations of multistage stochastic programming problems. We show that any random sampling scheme provides a valid statistical lower bound for the optimal (minimum) value of the true problem. However, in order for such lower bound to be consistent one needs to employ the conditional sampling procedure. We also indicate that fixing a feasible first-stage solution and then solving the sampling approximation of the corresponding (T−1)-stage problem, does not give a valid statistical upper bound for the optimal value of the true problem. Copyright Springer-Verlag 2003

Suggested Citation

  • Alexander Shapiro, 2003. "Inference of statistical bounds for multistage stochastic programming problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(1), pages 57-68, September.
    • Handle: RePEc:spr:mathme:v:58:y:2003:i:1:p:57-68
    DOI: 10.1007/s001860300280
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    Citations

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    Cited by:

    1. Sha, Yue & Zhang, Junlong & Cao, Hui, 2021. "Multistage stochastic programming approach for joint optimization of job scheduling and material ordering under endogenous uncertainties," European Journal of Operational Research, Elsevier, vol. 290(3), pages 886-900.
    2. D. Kuhn, 2009. "Convergent Bounds for Stochastic Programs with Expected Value Constraints," Journal of Optimization Theory and Applications, Springer, vol. 141(3), pages 597-618, June.
    3. Xiaomin Xi & Ramteen Sioshansi, 2016. "A dynamic programming model of energy storage and transformer deployments to relieve distribution constraints," Computational Management Science, Springer, vol. 13(1), pages 119-146, January.
    4. Baker, Erin & Solak, Senay, 2011. "Climate change and optimal energy technology R&D policy," European Journal of Operational Research, Elsevier, vol. 213(2), pages 442-454, September.
    5. Arnab Bhattacharya & Jeffrey P. Kharoufeh & Bo Zeng, 2023. "A Nonconvex Regularization Scheme for the Stochastic Dual Dynamic Programming Algorithm," INFORMS Journal on Computing, INFORMS, vol. 35(5), pages 1161-1178, September.
    6. Shin, Youngchul & Lee, Sangyoon & Moon, Ilkyeong, 2020. "Robust multiperiod inventory model considering trade-in program and refurbishment service: Implications to emerging markets," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 138(C).
    7. Gauthier Maere d’Aertrycke & Alexander Shapiro & Yves Smeers, 2013. "Risk exposure and Lagrange multipliers of nonanticipativity constraints in multistage stochastic problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 393-405, June.
    8. Boris Defourny & Damien Ernst & Louis Wehenkel, 2013. "Scenario Trees and Policy Selection for Multistage Stochastic Programming Using Machine Learning," INFORMS Journal on Computing, INFORMS, vol. 25(3), pages 488-501, August.
    9. Senay Solak & Gustaf Solveling & John-Paul B. Clarke & Ellis L. Johnson, 2018. "Stochastic Runway Scheduling," Transportation Science, INFORMS, vol. 52(4), pages 917-940, August.
    10. Rocha, Paula & Kuhn, Daniel, 2012. "Multistage stochastic portfolio optimisation in deregulated electricity markets using linear decision rules," European Journal of Operational Research, Elsevier, vol. 216(2), pages 397-408.
    11. Georg Pflug & Alois Pichler, 2015. "Dynamic generation of scenario trees," Computational Optimization and Applications, Springer, vol. 62(3), pages 641-668, December.
    12. Xiaotie Chen & David L. Woodruff, 2023. "Software for Data-Based Stochastic Programming Using Bootstrap Estimation," INFORMS Journal on Computing, INFORMS, vol. 35(6), pages 1218-1224, November.
    13. Agnieszka Konicz & David Pisinger & Alex Weissensteiner, 2015. "Optimal annuity portfolio under inflation risk," Computational Management Science, Springer, vol. 12(3), pages 461-488, July.
    14. Eyyüb Y. Kıbış & İ. Esra Büyüktahtakın & Robert G. Haight & Najmaddin Akhundov & Kathleen Knight & Charles E. Flower, 2021. "A Multistage Stochastic Programming Approach to the Optimal Surveillance and Control of the Emerald Ash Borer in Cities," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 808-834, May.
    15. Solak, Senay & Clarke, John-Paul B. & Johnson, Ellis L. & Barnes, Earl R., 2010. "Optimization of R&D project portfolios under endogenous uncertainty," European Journal of Operational Research, Elsevier, vol. 207(1), pages 420-433, November.

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