A Coordinate-Free Condition Number for Convex Programming

We introduce and analyze a natural geometric version of Renegar’s condition number R, which we call Grassmann condition number, for the homogeneous convex feasibility problem associated with a regular cone C ⊆ R. Let Grn,m denote the Grassmann manifold of m-dimensional linear subspaces of R with the Riemannian distance metric dg. The set of ill-posed instances Σm ⊂ Grn,m consists of the linear subspaces W touching C. We define the Grassmann condition number C (W ) of an m-dimensional subspace W ∈ Grn,m as C (W ) := sin dg(W,Σm). We also provide other characterizations of C (W ) and prove that C (W ) ≤ R(A) ≤ C (W )κ(A), where W = imA , and where κ(A) = ‖A‖‖A†‖ denotes the matrix condition number. This extends work by Belloni and Freund in Math. Program. 119:95–107 (2009). Based on the Grassmann condition number, in a forthcoming paper, we shall provide, for the first time, a probabilistic analysis of Renegar’s condition number for an arbitrary regular cone C.