- On the Grassmann condition number Javier Pena (jfpandrew.cmu.edu) Vera Roshchina (vera.roshchinagmail.com) Abstract: We give new insight into the Grassmann condition of the conic feasibility problem $x \in L \cap K \setminus\{0\}.$ Here $K\subseteq V$ is a regular convex cone and $L\subseteq V$ 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. Keywords: Grassmann manifold, distance to ill-posedness, condition Category 1: Convex and Nonsmooth Optimization Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Category 3: Linear, Cone and Semidefinite Programming Citation: Working Paper. Tepper School of Business. Carnegie Mellon University. Download: [PDF]Entry Submitted: 04/15/2016Entry Accepted: 04/15/2016Entry Last Modified: 04/26/2016Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.