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Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods

Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods
Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods
A key step in solving minimax distributionally robust optimization (DRO) problems is to reformulate the inner maximization w.r.t. probability measure as a semiinfinite programming problem through Lagrange dual. Slater type conditions have been widely used for strong duality (zero dual gap) when the ambiguity set is defined through moments. In this paper, we investigate effective ways for verifying the Slater type conditions and introduce other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem. Moreover, we propose two discretization schemes for solving the DRO with one for the dualized DRO and the other directly through the ambiguity set of the DRO. In the absence of strong duality, the discretization scheme via Lagrange duality may provide an upper bound for the optimal value of the DRO whereas the direct discretization approach provides a lower bound. Two cutting plane schemes are consequently proposed: one for the discretized dualized DRO and the other for the minimax DRO with discretized ambiguity set. Convergence analysis is presented for the approximation schemes in terms of the optimal value, optimal solutions and stationary points. Comparative numerical results are reported for the resulting algorithms.
0025-5610
489-529
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Liu, Yongchao
e7721a8a-028e-42b2-ac67-e30a0d3a2cf7
Sun, Hailin
e45868ea-33dd-4e81-b4e3-e03cbf0399a7
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Liu, Yongchao
e7721a8a-028e-42b2-ac67-e30a0d3a2cf7
Sun, Hailin
e45868ea-33dd-4e81-b4e3-e03cbf0399a7

Xu, Huifu, Liu, Yongchao and Sun, Hailin (2018) Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods. Mathematical Programming, 169 (2), 489-529. (doi:10.1007/s10107-017-1143-6).

Record type: Article

Abstract

A key step in solving minimax distributionally robust optimization (DRO) problems is to reformulate the inner maximization w.r.t. probability measure as a semiinfinite programming problem through Lagrange dual. Slater type conditions have been widely used for strong duality (zero dual gap) when the ambiguity set is defined through moments. In this paper, we investigate effective ways for verifying the Slater type conditions and introduce other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem. Moreover, we propose two discretization schemes for solving the DRO with one for the dualized DRO and the other directly through the ambiguity set of the DRO. In the absence of strong duality, the discretization scheme via Lagrange duality may provide an upper bound for the optimal value of the DRO whereas the direct discretization approach provides a lower bound. Two cutting plane schemes are consequently proposed: one for the discretized dualized DRO and the other for the minimax DRO with discretized ambiguity set. Convergence analysis is presented for the approximation schemes in terms of the optimal value, optimal solutions and stationary points. Comparative numerical results are reported for the resulting algorithms.

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DRO-Matrix-R2-7-Jan-2017 - Accepted Manuscript
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Accepted/In Press date: 25 March 2017
e-pub ahead of print date: 12 April 2017
Published date: 1 June 2018
Organisations: Operational Research

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Local EPrints ID: 407211
URI: http://eprints.soton.ac.uk/id/eprint/407211
ISSN: 0025-5610
PURE UUID: 97d44cf6-2083-4c97-8210-62c96421219c
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

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Date deposited: 01 Apr 2017 01:07
Last modified: 27 Jan 2020 13:39

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Contributors

Author: Huifu Xu ORCID iD
Author: Yongchao Liu
Author: Hailin Sun

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