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On the performance of affine policies for two-stage adaptive optimization: a geometric perspective

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Abstract

We consider two-stage adjustable robust linear optimization problems with uncertain right hand side \(\mathbf {b}\) belonging to a convex and compact uncertainty set \(\mathcal{U}\). We provide an a priori approximation bound on the ratio of the optimal affine (in \(\mathbf {b}\) ) solution to the optimal adjustable solution that depends on two fundamental geometric properties of \(\mathcal{U}\): (a) the “symmetry” and (b) the “simplex dilation factor” of the uncertainty set \(\mathcal{U}\) and provides deeper insight on the power of affine policies for this class of problems. The bound improves upon a priori bounds obtained for robust and affine policies proposed in the literature. We also find that the proposed a priori bound is quite close to a posteriori bounds computed in specific instances of an inventory control problem, illustrating that the proposed bound is informative.

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Acknowledgments

We would like to thank two anonymous referees for very thoughtful comments that have improved the paper and Dr. Angelos Georghiou for sharing with us his code and very helpful discussions.

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Authors and Affiliations

  1. Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Rm E40-147, Cambridge, MA, 02139, USA

    Dimitris Bertsimas

  2. Sloan School of Management, Massachusetts Institute of Technology, Rm E62-559, Cambridge, MA, 02139, USA

    Hoda Bidkhori

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  1. Dimitris Bertsimas
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  2. Hoda Bidkhori
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Correspondence to Dimitris Bertsimas.

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Bertsimas, D., Bidkhori, H. On the performance of affine policies for two-stage adaptive optimization: a geometric perspective. Math. Program. 153, 577–594 (2015). https://doi.org/10.1007/s10107-014-0818-5

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  • DOI: https://doi.org/10.1007/s10107-014-0818-5

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