2016/04/15 by Pena, Javier, Roshchina, Vera
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1604.04637
We give new insight into the Grassmann condition of the conic feasibility problem x ∈ L ∩ K ∖\0\. 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 this problem 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 the above problem and a natural measure of the least violated trial solution to its alternative feasibility problem. We also show a tight relationship between the Grassmann and Renegar's condition measures, and between the Grassman measure and a symmetry measure of the above feasibility problem. Our approach can be readily specialized to a canonical norm in V induced by K, a prime example being the one-norm for the non-negative orthant. For this special case we show that the Grassmann distance ill-posedness of is equivalent to a measure of the most interior solution to the above conic feasibility problem.