Intrinsic volumes of symmetric cones and applications in convex programming

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...

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Bibliographic Details
Published in:Mathematical programming 2015-02, Vol.149 (1-2), p.105-130
Main Authors: Amelunxen, Dennis, Bürgisser, Peter
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Complex numbers
Cones
Convex analysis
Full Length Paper
Gaussian
Integrals
Linear programming
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Matrices (mathematics)
Numerical Analysis
Optimization
Probability
Probability distribution
Semidefinite programming
Studies
Theoretical
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Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-013-0740-2