On the Grassmann condition number

We give new insight into the Grassmann condition of the conic feasibility problem find x ∈ L ∩K \ {0}. (1) Here K ⊆ V is a regular convex cone and L ⊆ V is a linear subspace of the finite dimensional Euclidean vector space V . The Grassmann condition of (1) is the reciprocal of the distance from L to the set of ill-posed instances in the Grassmann manifold where L lives. We consider a very general distance in the Grassmann manifold defined by two possibly different norms in V . We establish the equivalence between the Grassmann distance to ill-posedness of (1) and a natural measure of the least violated trial solution to the alternative to (1). We also show a tight relationship between the Grassmann and Renegar’s condition measures, and between the Grassmann measure and a symmetry measure of (1). Our approach can be readily specialized to a canonical norm in V induced by K, a prime example being the one-norm for the nonnegative orthant. For this special case we show that the Grassmann distance ill-posedness of (1) is equivalent to a measure of the most interior solution to (1). ∗Tepper School of Business, Carnegie Mellon University, USA, [email protected] †School of Science, RMIT University, AUSTRALIA, [email protected]