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Consistency of statistical estimators of solutions to stochastic optimization problems

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  • Huynh Thi Hong Diem

    (University of Technology
    Vietnam National University)

Abstract

We consider the asymptotic behavior of the infimal values and the statistical estimators of the solutions to a general stochastic optimization problem. We establish the epi-convergence of the performance criteria of approximate problems when the approximate probability laws, obtained by sampling the values of the random variable, converge weakly and tightly. Based on this key convergence, consistency properties of the infimal values and the estimators of the solutions to the approximate problems are obtained. Applying these results and properties of epi/hypo-convergence of bifunctions to Lagrangians of stochastic mathematical programs, we obtain the consistency of the saddle points of approximate Lagrangians and hence the consistency of the optimal values and the estimators of the solutions of approximate mathematical programs and their dual programs.

Suggested Citation

  • Huynh Thi Hong Diem, 2022. "Consistency of statistical estimators of solutions to stochastic optimization problems," Journal of Global Optimization, Springer, vol. 83(4), pages 825-842, August.
    • Handle: RePEc:spr:jglopt:v:83:y:2022:i:4:d:10.1007_s10898-022-01125-3
    DOI: 10.1007/s10898-022-01125-3
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    References listed on IDEAS

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    1. Zvi Artstein & Sergiu Hart, 1981. "Law of Large Numbers for Random Sets and Allocation Processes," Mathematics of Operations Research, INFORMS, vol. 6(4), pages 485-492, November.
    2. Mihail Zervos, 1999. "On the Epiconvergence of Stochastic Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 495-508, May.
    3. Teemu Pennanen, 2005. "Epi-Convergent Discretizations of Multistage Stochastic Programs," Mathematics of Operations Research, INFORMS, vol. 30(1), pages 245-256, February.
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