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Distributions and bootstrap for data-based stochastic programming

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Abstract

In the context of optimization under uncertainty, we consider various combinations of distribution estimation and resampling (bootstrap and bagging) for obtaining samples used to estimate a confidence interval for an optimality gap. This paper makes three experimental contributions to on-going research in data driven stochastic programming: (a) most of the combinations of distribution estimation and resampling result in algorithms that have not been published before, (b) within the algorithms, we describe innovations that improve performance, and (c) we provide open-source software implementations of the algorithms. Among others, three important conclusions can be drawn: using a smoothed point estimate for the optimality gap for the center of the confidence interval is preferable to a purely empirical estimate, bagging often performs better than bootstrap, and smoothed bagging sometimes performs better than bagging based directly on the data.

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References

  • Anitescu M, Petra C (2011) Higher-order confidence intervals for stochastic programming using bootstrapping. Technical report, Citeseer

  • Bayraksan G, Morton DP (2011) A sequential sampling procedure for stochastic programming. Oper Res 59(4):898–913

    Article  Google Scholar 

  • Bayraksan G, Pierre-Louis P (2012) Fixed-width sequential stopping rules for a class of stochastic programs. SIAM J Optim 22(4):1518–1548

    Article  Google Scholar 

  • Birge JR, Louveaux F (2011) Introduction to stochastic programming. Springer, Berlin

    Book  Google Scholar 

  • boot-sp (2023) Boot-sp software. https://github.com/boot-sp/boot-sp

  • Breiman L (1996) Bagging predictors. Machine Learn 24:123–140

    Article  Google Scholar 

  • Bühlmann P, Yu B (2002) Analyzing bagging. Ann Stat 30(4):927–961

    Article  Google Scholar 

  • Chen X, Woodruff DL (2023) Software for data-based stochastic programming using bootstrap estimation. INFORMS J Comp 35(6):1218–1224

    Article  Google Scholar 

  • Davison AC, Hinkley DV (1997) Bootstrap methods and their application. Number 1. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • De Angelis D, Young GA (1992) Smoothing the bootstrap. Int Stat Rev/Revue Internationale de Statistique 45–56

  • De Matos VL, Morton DP, Finardi EC (2017) Assessing policy quality in a multistage stochastic program for long-term hydrothermal scheduling. Ann Oper Res 253:713–731

    Article  Google Scholar 

  • Diciccio TJ, Romano JP (1988) A review of bootstrap confidence intervals. J R Stat Soc Ser B Stat Methodol 50(3):338–354

    Article  Google Scholar 

  • Efron B (1981) Nonparametric estimates of standard error: the jackknife, the bootstrap and other methods. Biometrika 68(3):589–599

    Article  Google Scholar 

  • Efron B (1982). The jackknife, the bootstrap and other resampling plans. SIAM

  • Eichhorn A, Römisch W (2007) Stochastic integer programming: limit theorems and confidence intervals. Math Oper Res 32(1):118–135

    Article  Google Scholar 

  • Fuentes R, Lillo-Banuls A (2015) Smoothed bootstrap malmquist index based on dea model to compute productivity of tax offices. Expert Syst Appl 42(5):2442–2450

    Article  Google Scholar 

  • Gregory KB, Lahiri SN, Nordman DJ (2018) A smooth block bootstrap for quantile regression with time series. Ann Stat 46(3):1138–1166

    Article  Google Scholar 

  • Hall P, DiCiccio TJ, Romano JP (1989) On smoothing and the bootstrap. Ann Stat 692–704

  • Higle JL, Sen S (1991) Stochastic decomposition: an algorithm for two-stage linear programs with recourse. Math Oper Res 16(3):650–669

    Article  Google Scholar 

  • Huh WT, Levi R, Rusmevichientong P, Orlin JB (2011) Adaptive data-driven inventory control with censored demand based on Kaplan–Meier estimator. Oper Res 59(4):929–941

    Article  Google Scholar 

  • King AJ, Wallace SW (2012) Modeling with stochastic programming. Springer, Berlin

  • Knueven B, Mildebrath D, Muir C, Siirola J, Woodruff D, Watson J-P (2020) mpi-sppy: optimization under uncertainty for pyomo. Technical report, National Renewable Energy Lab.(NREL), Golden, CO (United States)

  • Lam H, Qian H (2018a) Assessing solution quality in stochastic optimization via bootstrap aggregating. In: 2018 Winter Simulation Conference (WSC). IEEE, pp 2061–2071

  • Lam H, Qian H (2018b) Bounding optimality gap in stochastic optimization via bagging: statistical efficiency and stability. arXiv preprint arXiv:1810.02905

  • Lee S, Cho S (2001) Smoothed bagging with kernel bandwidth selectors. Neural Process Lett 14:157–168

    Article  Google Scholar 

  • Li Y, Wang Y-G (2008) Smooth bootstrap methods for analysis of longitudinal data. Stat Med 27(7):937–953

    Article  Google Scholar 

  • Linderoth J, Shapiro A, Wright S (2006) The empirical behavior of sampling methods for stochastic programming. Ann Oper Res 142(1):215–241

    Article  Google Scholar 

  • Mak W-K, Morton DP, Wood RK (1999) Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper Res Lett 24(1–2):47–56

    Article  Google Scholar 

  • Mentch L, Hooker G (2016) Quantifying uncertainty in random forests via confidence intervals and hypothesis tests. J Mach Learn Res 17(1):841–881

    Google Scholar 

  • Parpas P, Ustun B, Webster M, Tran QK (2015) Importance sampling in stochastic programming: A Markov chain Monte Carlo approach. INFORMS J Comput 27(2):358–377

    Article  Google Scholar 

  • Prékopa A (2013). Stochastic programming, vol 324. Springer, Berlin

  • Raviv Y, Intrator N (1996) Bootstrapping with noise: an effective regularization technique. Connect Sci 8(3–4):355–372

    Article  Google Scholar 

  • Royset JO, Wets RJ-B (2015) Fusion of hard and soft information in nonparametric density estimation. Eur J Oper Res 247(2):532–547. https://doi.org/10.1016/j.ejor.2015.06.034

    Article  Google Scholar 

  • Ruszczyński A, Shapiro A (2003) Stochastic programming models. Handb Oper Res Management Sci 10:1–64

    Google Scholar 

  • Scott DW (2015) Multivariate density estimation: theory, practice, and visualization. Wiley

  • Shao J, Tu D (2012) The jackknife and bootstrap. Springer, Berlin

    Google Scholar 

  • Shapiro A (1991) Asymptotic analysis of stochastic programs. Ann Oper Res 30:169–186

    Article  Google Scholar 

  • Shapiro A (2003) Statistical inference of multistage stochastic programming problems. Math Methods Oper Res 58:57–68

    Article  Google Scholar 

  • Silverman B, Young G (1987) The bootstrap: To smooth or not to smooth? Biometrika 74(3):469–479

    Article  Google Scholar 

  • Vaagen H, Wallace SW (2008) Product variety arising from hedging in the fashion supply chains. Int J Prod Econ 114(2):431–455

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, UC Davis, Davis, CA, USA

    Xiaotie Chen

  2. Graduate School of Management, UC Davis, Davis, CA, USA

    David L. Woodruff

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  1. Xiaotie Chen
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  2. David L. Woodruff
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Correspondence to David L. Woodruff.

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Chen, X., Woodruff, D.L. Distributions and bootstrap for data-based stochastic programming. Comput Manag Sci 21, 33 (2024). https://doi.org/10.1007/s10287-024-00512-3

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