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On the Grassmann condition number

Javier Pena (jfp***at***andrew.cmu.edu)
Vera Roshchina (vera.roshchina***at***gmail.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: 04/15/2016
Entry Accepted: 04/15/2016
Entry Last Modified: 04/26/2016

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