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A coordinate-free condition number for convex programming
Citation key AB-A-Coordinate-Free-Condition-Number-For-Convex-Programming
Author Dennis Amelunxen and Peter Bürgisser
Pages 1029-1041
Year 2012
Journal SIAM Journal on Optimization
Volume 22
Number 3
Abstract We introduce and analyze a natural geometric version of Renegar's condition number $R$ for the homogeneous convex feasibility problem associated with a regular cone $C\subseteq\mathbb R^n$. Let $\mathrm\\\Gr\\\_n,m$ denote the Grassmann manifold of $m$-dimensional linear subspaces of $\mathbb R^n$ and consider the projection distance $d_p(W_1,W_2) := \|\pi_W_1 - \pi_W_2\|$ (spectral norm) between $W_1$ and $W_2$ in $\mathrmGr_n,m$, where $\pi_W_i$ denotes the orthogonal projection onto $W_i$. We call $C_G(W) := \max \ d_p(W,W')^-1 \mid W' \in Sigma_m \$ the Grassmann condition number of $W$ in $\mathrmGr_n,m$, where the set of ill-posed instances $\Sigma_m$ subset $\mathrmGr_n,m$ is defined as the set of linear subspaces touching $C$. We show that if $W = \mathrmim(A^T)$ for a matrix $A$ in $\mathbb R^m× n$, then $C_G(W) łe R(A) łe C_G(W) \kappa(A)$, where $\kappa(A) =\|A\| \|A^\dagger\|$ denotes the matrix condition number. This extends work by Belloni and Freund in Math. Program. 119:95-107 (2009). Furthermore, we show that $C_G(W)$ can as well be characterized in terms of the Riemannian distance metric on $\mathrmGr_n,m$. This differential geometric characterization of $C_G(W)$ is the starting point of the sequel [arXiv:1112.2603] to this paper, where the first probabilistic analysis of Renegar's condition number for an arbitrary regular cone $C$ is achieved.
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