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Bounds on linear PDEs via semidefinite optimization

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Abstract

Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this paper a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on linear functionals defined on solutions of linear partial differential equations. We apply the proposed method to examples of PDEs in one and two dimensions, with very encouraging results. We pay particular attention to a PDE with oblique derivative conditions, commonly arising in queueing theory. We also provide computational evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is, without them the bounds are weak.

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Authors and Affiliations

  1. Boeing Professor of Operations Research, Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E53-363, Cambridge, MA, 02142, USA

    Dimitris Bertsimas

  2. Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA

    Constantine Caramanis

Authors
  1. Dimitris Bertsimas
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  2. Constantine Caramanis
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Correspondence to Dimitris Bertsimas.

Additional information

Research supported by the SMA-MI Talliance.

Research partially supported by the SMA-MIT alliance and an NSF predoctoral fellowship.

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Bertsimas, D., Caramanis, C. Bounds on linear PDEs via semidefinite optimization. Math. Program. 108, 135–158 (2006). https://doi.org/10.1007/s10107-006-0702-z

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  • DOI: https://doi.org/10.1007/s10107-006-0702-z

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