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A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization

  • Aharon Ben-Tal
  • Omar El Housni
  • Vineet GoyalEmail author
Full Length Paper Series A
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Abstract

We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right-hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large. A significant challenge in designing a practical piecewise affine policy is constructing good pieces of the uncertainty set. Here we address this challenge by introducing a new framework in which the uncertainty set is “approximated” by a “dominating” simplex. The corresponding policy is then based on a mapping from the uncertainty set to the simplex. Although our piecewise affine policy has exponentially many pieces, it can be computed efficiently by solving a compact linear program given the dominating simplex. Furthermore, we can find the dominating simplex in a closed form if the uncertainty set satisfies some symmetries and can be computed using a MIP in general. We would like to remark that our policy is an approximate piecewise-affine policy and is not necessarily a generalization of the class of affine policies. Nevertheless, the performance of our policy is significantly better than the affine policy for many important uncertainty sets, such as ellipsoids and norm-balls, both theoretically and numerically. For instance, for hypersphere uncertainty set, our piecewise affine policy can be computed by an LP and gives a \(O(m^{1/4})\)-approximation whereas the affine policy requires us to solve a second order cone program and has a worst-case performance bound of \(O(\sqrt{m})\).

Mathematics Subject Classification

90C39 90C47 49K35 

Notes

Acknowledgements

O. El Housni and V. Goyal are supported by NSF Grants CMMI 1201116 and CMMI 1351838.

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Copyright information

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

Authors and Affiliations

  1. 1.Industrial Engineering and ManagementTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.CentER, Tilburg UniversityTilburgThe Netherlands
  3. 3.Industrial Engineering and Operations ResearchColumbia UniversityNew YorkUSA

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