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Exploiting partial correlations in distributionally robust optimization

  • Divya PadmanabhanEmail author
  • Karthik Natarajan
  • Karthyek Murthy
Full Length Paper Series A

Abstract

In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set \(\{(\mathbf x , \mathbf x {} \mathbf x '): \mathbf x \in \mathcal {X}\}\) where \(\mathcal {X}\) is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in the newsvendor problem, project evaluation and review technique networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled. We also discuss extensions where the random coefficients are assumed to be non-negative and additional overlapping correlation information is available.

Mathematics Subject Classification

90-02 (operations research and mathematical programming. research exposition) 

Notes

Acknowledgements

The research of the first and the second author was partially supported by the MOE Academic Research Fund Tier 2 Grant T2MOE1706, “On the Interplay of Choice, Robustness and Optimization in Transportation” and the SUTD-MIT International Design Center Grant IDG21700101 on “Design of the Last Mile Transportation System: What Does the Customer Really Want?”. The authors would like to thank Teo Chung-Piaw (NUS) for providing some useful references on this research. We would also like to thank the editor Alper Atamturk, the two anonymous reviewers and the associate editor for the detailed perusal of the paper and providing useful suggestions towards improving the paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Singapore University of Technology and DesignSingaporeSingapore

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