Probabilistic Analysis of the Grassmann Condition Number

We perform an average analysis of the Grassmann condition number C (A) for the homogeneous convex feasibility problem ∃x ∈ C \ 0 : Ax = 0, where C ⊂ R may be any regular cone. This in particular includes the cases of linear programming, second-order programming, and semidefinite programming. We thus give the first average analysis of convex programming, which is not restricted to linear programming. The Grassmann condition number is a geometric version of Renegar’s condition number, which we have introduced recently in [arXiv:1105.4049v1]. In this work we use techniques from spherical convex geometry and differential geometry. Among other things, we will show that if the entries of A ∈ Rm×n are chosen i.i.d. standard normal, then for any regular cone C we have E[ln C (A)] < 1.5 ln(n) + 1.5. AMS subject classifications: 90C25, 90C22, 90C31, 52A22, 52A55, 60D05