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Stochastic optimization problems with nonlinear dependence on a probability measure via the Wasserstein metric

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  • Vlasta Kaňková

    (Institute of Information Theory and Automation)

Abstract

Nonlinear dependence on a probability measure has recently been encountered with increasing intensity in stochastic optimization. This type of dependence corresponds to many situations in applications; it can appear in problems static (one-stage), dynamic with finite (multi-stage) or infinite horizon, and single- and multi-objective ones. Moreover, the nonlinear dependence can appear not only in the objective functions but also in the constraint sets. In this paper, we will consider static one-objective problems in which the nonlinear dependence appears in the objective function and may also appear in the constraint sets. In detail, we consider “deterministic” constraint sets, whose dependence on the probability measure is nonlinear, constraint sets determined by second-order stochastic dominance, and sets given by mean-risk problems. The last mentioned instance means that the constraint set corresponds to solutions which guarantee acceptable values of both criteria. To obtain relevant assertions, we employ the stability results given by the Wasserstein metric, based on the $$ {{\mathcal {L}}}_{1} $$ L 1 norm. We mainly focus on the case in which a solution has to be obtained on the basis of the data and of investigating a relationship between the original problem and its empirical version.

Suggested Citation

  • Vlasta Kaňková, 2024. "Stochastic optimization problems with nonlinear dependence on a probability measure via the Wasserstein metric," Journal of Global Optimization, Springer, vol. 90(3), pages 593-617, November.
    • Handle: RePEc:spr:jglopt:v:90:y:2024:i:3:d:10.1007_s10898-024-01380-6
    DOI: 10.1007/s10898-024-01380-6
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    References listed on IDEAS

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    1. Walter J. Gutjahr & Alois Pichler, 2016. "Stochastic multi-objective optimization: a survey on non-scalarizing methods," Annals of Operations Research, Springer, vol. 236(2), pages 475-499, January.
    2. repec:czx:journl:v:19:y:2012:i:29:id:195 is not listed on IDEAS
    3. Vlasta Kaňková, 2013. "Risk Measures in Optimization Problems via Empirical Estimates," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 7(3), pages 162-177, November.
    4. Walter Gutjahr & Alois Pichler, 2016. "Stochastic multi-objective optimization: a survey on non-scalarizing methods," Annals of Operations Research, Springer, vol. 236(2), pages 475-499, January.
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