Distributionally Robust Linear and Discrete Optimization with Marginals

69 Pages Posted: 27 Apr 2018

See all articles by Louis Chen

Louis Chen

Massachusetts Institute of Technology (MIT), Operations Research Center, Students

Will Ma

Massachusetts Institute of Technology (MIT)

Karthik Natarajan

Singapore University of Technology and Design (SUTD)

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering

Zhenzhen Yan

Nanyang Technological University

Date Written: April 9, 2018

Abstract

In this paper, we study the class of linear and discrete optimization problems in which the objective coefficients are chosen randomly from a distribution, and the goal is to evaluate robust bounds on the expected optimal value as well as the marginal distribution of the optimal solution. The set of joint distributions is assumed to be specified up to only the marginal distributions. We generalize the primal-dual formulations for this problem from the set of joint distributions with absolutely continuous marginal distributions to arbitrary marginal distributions using techniques from optimal transport theory. While the robust bound is shown to be NP-hard to compute for linear optimization problems, we identify a sufficient condition for polynomial time solvability using extended formulations. This generalizes the known tractability results under marginal information from 0-1 polytopes to a class of integral polytopes and has implications on the solvability of distributionally robust optimization problems in areas such as scheduling which we discuss.

Keywords: Robust Optimization, Discrete Optimization, Optimal Transport

Suggested Citation

Chen, Louis and Ma, Will and Natarajan, Karthik and Simchi-Levi, David and Yan, Zhenzhen, Distributionally Robust Linear and Discrete Optimization with Marginals (April 9, 2018). Available at SSRN: https://ssrn.com/abstract=3159473 or http://dx.doi.org/10.2139/ssrn.3159473

Louis Chen (Contact Author)

Massachusetts Institute of Technology (MIT), Operations Research Center, Students ( email )

77 Massachusetts Avenue
Bldg. E 40-149
Cambridge, MA 02139
United States

Will Ma

Massachusetts Institute of Technology (MIT) ( email )

77 Massachusetts Avenue
50 Memorial Drive
Cambridge, MA 02139-4307
United States

Karthik Natarajan

Singapore University of Technology and Design (SUTD) ( email )

20 Dover Drive
Singapore, 138682
Singapore

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering ( email )

MA
United States

Zhenzhen Yan

Nanyang Technological University ( email )

MAS 05-19, 21 Nanyang Link
Singapore, 637371
Singapore

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