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Convergence Analysis on Data-Driven Fortet-Mourier Metrics with Applications in Stochastic Optimization

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  • Zhiping Chen

    (School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China)

  • He Hu

    (School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China)

  • Jie Jiang

    (School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China)

Abstract

Fortet-Mourier (FM) probability metrics are important probability metrics, which have been widely adopted in the quantitative stability analysis of stochastic programming problems. In this study, we contribute to different types of convergence assertions between a probability distribution and its empirical distribution when the deviation is measured by FM metrics and consider their applications in stochastic optimization. We first establish the quantitative relation between FM metrics and Wasserstein metrics. After that, we derive the non-asymptotic moment estimate, asymptotic convergence, and non-asymptotic concentration estimate for FM metrics, which supplement the existing results. Finally, we apply the derived results to four kinds of stochastic optimization problems, which either extend the present results to more general cases or provide alternative avenues. All these discussions demonstrate the motivation as well as the significance of our study.

Suggested Citation

  • Zhiping Chen & He Hu & Jie Jiang, 2022. "Convergence Analysis on Data-Driven Fortet-Mourier Metrics with Applications in Stochastic Optimization," Sustainability, MDPI, vol. 14(8), pages 1-29, April.
    • Handle: RePEc:gam:jsusta:v:14:y:2022:i:8:p:4501-:d:790587
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    References listed on IDEAS

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    1. Svetlozar T. Rachev & Werner Römisch, 2002. "Quantitative Stability in Stochastic Programming: The Method of Probability Metrics," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 792-818, November.
    2. Erick Delage & Yinyu Ye, 2010. "Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems," Operations Research, INFORMS, vol. 58(3), pages 595-612, June.
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