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Consistency of statistical estimators of solutions to stochastic optimization problems

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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.

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Acknowledgements

The author acknowledges the support of time and facilities from Ho Chi Minh City University of Technology (HCMUT), VNU, HCM for this study. A significant part of the paper was completed during a scientific stay of the author at Vietnam Institute for Advanced Study in Mathematics (VIASM), whose support and hospitality are gratefully appreciated. The author is indebted to Professor Phan Quoc Khanh for his suggestion of the research topic and many important discussions and suggestions during her work on the paper. She is especially thankful to the editor and the two anonymous reviewers for their valuable comments and suggestions which were so crucial for revising the paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Technology, Ho Chi Minh City, Vietnam

    Huynh Thi Hong Diem

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Huynh Thi Hong Diem

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  1. Huynh Thi Hong Diem
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Correspondence to Huynh Thi Hong Diem.

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Diem, H.T.H. Consistency of statistical estimators of solutions to stochastic optimization problems. J Glob Optim 83, 825–842 (2022). https://doi.org/10.1007/s10898-022-01125-3

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  • DOI: https://doi.org/10.1007/s10898-022-01125-3

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