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Discrete approximation and quantification in distributionally robust optimization

Discrete approximation and quantification in distributionally robust optimization
Discrete approximation and quantification in distributionally robust optimization
Discrete approximation of probability distributions is an important topic in stochastic programming. In this paper, we extend the research on this topic to distributionally robust optimization (DRO), where discretization is driven by either limited availability of empirical data (samples) or a computational need for improving numerical tractability. We start with a one-stage DRO where the ambiguity set is defined by generalized prior moment conditions and quantify the discrepancy between the discretized ambiguity set and the original one by employing the Kantorovich/Wasserstein metric. The quantification is achieved by establishing a new form of Hoffman’s lemma for moment problems under a general class of metrics—namely, ζ-structures. We then investigate how the discrepancy propagates to the optimal value in one-stage DRO and discuss further the multistage DRO under nested distance. The technical results lay down a theoretical foundation for various discrete approximation schemes to be applied to solve one-stage and multistage distributionally robust optimization problems.
0364-765X
1-19
Liu, Yongchao
e7721a8a-028e-42b2-ac67-e30a0d3a2cf7
Pichler, Alois
78070747-244a-48df-a775-7b1a46fe7b14
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Liu, Yongchao
e7721a8a-028e-42b2-ac67-e30a0d3a2cf7
Pichler, Alois
78070747-244a-48df-a775-7b1a46fe7b14
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5

Liu, Yongchao, Pichler, Alois and Xu, Huifu (2018) Discrete approximation and quantification in distributionally robust optimization. Mathematics of Operations Research, 1-19. (doi:10.1287/moor.2017.0911).

Record type: Article

Abstract

Discrete approximation of probability distributions is an important topic in stochastic programming. In this paper, we extend the research on this topic to distributionally robust optimization (DRO), where discretization is driven by either limited availability of empirical data (samples) or a computational need for improving numerical tractability. We start with a one-stage DRO where the ambiguity set is defined by generalized prior moment conditions and quantify the discrepancy between the discretized ambiguity set and the original one by employing the Kantorovich/Wasserstein metric. The quantification is achieved by establishing a new form of Hoffman’s lemma for moment problems under a general class of metrics—namely, ζ-structures. We then investigate how the discrepancy propagates to the optimal value in one-stage DRO and discuss further the multistage DRO under nested distance. The technical results lay down a theoretical foundation for various discrete approximation schemes to be applied to solve one-stage and multistage distributionally robust optimization problems.

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LPX-MOR-17_final - Accepted Manuscript
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Accepted/In Press date: 8 September 2017
e-pub ahead of print date: 14 September 2018

Identifiers

Local EPrints ID: 414753
URI: http://eprints.soton.ac.uk/id/eprint/414753
ISSN: 0364-765X
PURE UUID: 51df7ef1-b602-4418-a31f-bbe5ca906717
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

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Date deposited: 10 Oct 2017 16:31
Last modified: 17 Dec 2019 01:49

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