On the Grassmann condition number

TL;DR

This paper introduces new insights into the Grassmann condition number for conic feasibility problems, establishing equivalences with other condition measures and analyzing its relation to the geometry of the problem.

Contribution

It generalizes the Grassmann condition to arbitrary norms, relates it to other condition measures, and connects it to the interior solutions of conic feasibility problems.

Findings

Equivalence between Grassmann distance to ill-posedness and least violated trial solutions.

Relationship established between Grassmann and Renegar's condition measures.

Special case analysis for the one-norm and interior solutions.

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.