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Intrinsic volumes of symmetric cones and applications in convex programming

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Abstract

We express the probability distribution of the solution of a random (standard Gaussian) instance of a convex cone program in terms of the intrinsic volumes and curvature measures of the reference cone. We then compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions in terms of integrals related to Mehta’s integral. In particular, we obtain a closed formula for the probability that the solution of a random (standard Gaussian) semidefinite program has a certain rank.

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Notes

  1. In fact, intrinsic volumes are usually defined for intersections of convex cones with the unit sphere. We adopt the conical viewpoint for technical reasons, and also adopt a convenient shift in the indices of the intrinsic volumes compared to [5, 18, 19, 36].

  2. Some authors differentiate between faces and exposed faces, cf. for example [35]. We do not make this distinction as for the cones we are interested in both notions coincide.

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Acknowledgments

We thank Michael B. McCoy for pointing out that almost sure nonvanishing is the only assumption on \(b\) that is needed in a standard Gaussian (CP). We are grateful to the anonymous referees for comments that led to a more structured presentation. This work has been supported by the grants AM 386/1-1 and BU 1371/2-2 of the German Research Foundation (DFG).

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  1. School of Mathematics, The University of Manchester, Manchester, UK

    Dennis Amelunxen

  2. Institut für Mathematik, Technische Universität Berlin, Sekretariat MA 3-2, Straße des 17. Juni 136, 10623 , Berlin, Germany

    Peter Bürgisser

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  1. Dennis Amelunxen
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Correspondence to Dennis Amelunxen.

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Amelunxen, D., Bürgisser, P. Intrinsic volumes of symmetric cones and applications in convex programming. Math. Program. 149, 105–130 (2015). https://doi.org/10.1007/s10107-013-0740-2

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