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| Content Provider | Springer Nature Link |
|---|---|
| Author | Bürgisser, Peter Amelunxen, Dennis |
| Copyright Year | 2013 |
| Abstract | We analyze the probability that a random m-dimensional linear subspace of $\mathbb{R}^{n}$ both intersects a regular closed convex cone $C\subseteq\mathbb{R}^{n}$ and lies within distance α of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number for the homogeneous convex feasibility problem ∃x∈C∖0:Ax=0. The Grassmann condition number is a geometric version of Renegar’s condition number, which we have introduced recently in Amelunxen and Bürgisser (SIAM J. Optim. 22(3):1029–1041 (2012)). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of $A\in\mathbb {R}^{m\times n}$ are chosen i.i.d. standard normal, then for any regular cone C, we have . The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds. |
| Ending Page | 51 |
| Page Count | 49 |
| Starting Page | 3 |
| File Format | |
| ISSN | 16153375 |
| e-ISSN | 16153383 |
| Journal | Foundations of Computational Mathematics |
| Issue Number | 1 |
| Volume Number | 15 |
| Language | English |
| Publisher | Springer US |
| Publisher Date | 2013-11-14 |
| Publisher Place | Boston |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Semidefinite programming Economics general Tube formulas Spherically convex sets Geometric probability and stochastic geometry Linear and Multilinear Algebras, Matrix Theory Perturbation Random convex sets and integral geometry Convex programming Average analysis Grassmann manifold Numerical Analysis Computer Science Condition number Spherical and hyperbolic convexity Applications of Mathematics Math Applications in Computer Science Sensitivity, stability, parametric optimization |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics |
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