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Ehemalige Mitarbeiter

Dr. Dennis Amelunxen

Lupe [1]

Kontakt
City University of Hong Kong
Department of Mathematics
G6613 (Green Zone), 6/F Academic 1
Tat Chee Avenue, Kowloon Tong, Hong Kong
 
Persönliche Homepage
sites.google.com/site/dennisamelunxen/home [2]

Publikationen in der Arbeitsgruppe

A coordinate-free condition number for convex programming
Zitatschlüssel AB-A-Coordinate-Free-Condition-Number-For-Convex-Programming
Autor Dennis Amelunxen and Peter Bürgisser
Seiten 1029-1041
Jahr 2012
Journal SIAM Journal on Optimization
Jahrgang 22
Nummer 3
Zusammenfassung 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 $Csubseteqmathbb R^n$. 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.
Link zur Publikation [3] Link zur Originalpublikation [4] Download Bibtex Eintrag [5]
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