Abstract |
We analyze the probability that a random $m$-dimensional linear subspace of $IR^n$ both intersects a regular closed convex cone $C\subseteq IR^n$ 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 $CG(A)$ for the homogeneous convex feasibility problem $\exists x\in C\setminus0:Ax=0$. The Grassmann condition number is a geometric version of Renegar's condition number, that we have introduced recently in [SIOPT 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 $A\in IR^m× n$ are chosen i.i.d. standard normal, then for any regular cone $C$, we have $ IE[łn CG(A)]<1.5łn(n)+1.5$. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds. |