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Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone

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Abstract

We develop a practical semidefinite programming (SDP) facial reduction procedure that utilizes computationally efficient approximations of the positive semidefinite cone. The proposed method simplifies SDPs with no strictly feasible solution (a frequent output of parsers) by solving a sequence of easier optimization problems and could be a useful pre-processing technique for SDP solvers. We demonstrate effectiveness of the method on SDPs arising in practice, and describe our publicly-available software implementation. We also show how to find maximum rank matrices in our PSD cone approximations (which helps us find maximal simplifications), and we give a post-processing procedure for dual solution recovery that generally applies to facial-reduction-based pre-processing techniques. Finally, we show how approximations can be chosen to preserve problem sparsity.

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Notes

  1. This designation of primal (P) and dual (D), while standard in facial reduction literature, is opposite the convention used by semidefinite solvers such as SeDuMi. We will switch to the convention favored by solvers when we discuss our software implementation in Sect. 6.

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Acknowledgements

The authors thanks Mark Tobenkin for many helpful discussions, and Michael Posa and Anirudha Majumdar for testing early versions of the implemented algorithms. Michael Posa also provided the original (i.e. unreduced) SDP for Example 7.4. The authors thank Cynthia Vinzant for providing the Vámos matroid example \(\mathcal {V}_{10}\) and reference [14]. We thank Gábor Pataki and Johan Löfberg for helpful comments.

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  1. The Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Frank Permenter & Pablo Parrilo

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Permenter, F., Parrilo, P. Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone. Math. Program. 171, 1–54 (2018). https://doi.org/10.1007/s10107-017-1169-9

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