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Beltrán, C.; Dedieu, J.-P.; Malajovich, G.; Shub, M. "Convexity properties of the condition number II" (2012) SIAM Journal on Matrix Analysis and Applications. 33(3):905-939
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Abstract:

In our previous paper [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1491-1506], we studied the condition metric in the space of maximal rank n × m matrices. Here, we show that this condition metric induces a Lipschitz Riemannian structure on that space. After investigating geodesics in such a nonsmooth structure, we show that the inverse of the smallest singular value of a matrix is a log-convex function along geodesics. We also show that a similar result holds for the solution variety of linear systems. Some of our intermediate results such as those on the second covariant derivative or Hessian of a function with symmetries on a manifold, and those on piecewise self-convex functions, are of independent interest. Those results were motivated by our investigations on the complexity of path-following algorithms for solving polynomial systems. © 2012 Society for Industrial and Applied Mathematics.

Registro:

Documento: Artículo
Título:Convexity properties of the condition number II
Autor:Beltrán, C.; Dedieu, J.-P.; Malajovich, G.; Shub, M.
Filiación:Departamento de Matemáticas, Estad. y Comput., Universidad de Cantabria, Santander, Spain
Institut de Mathématiques, Université Paul Sabatier, Toulouse, France
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal Do Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brazil
CONICET, IMAS, Universidad de Buenos Aires, Argentina
CUNY Graduate School, New York, NY, United States
Palabras clave:Condition number; Convexity; Lipschitz Riemannian structure; Self-convexity; Condition numbers; Convexity; Convexity properties; Covariant; Intermediate results; Lipschitz; Log-convex functions; M-matrices; Non-smooth; Path-following algorithm; Piece-wise; Polynomial systems; Riemannian structure; Self-convexity; Singular values; Convex optimization; Functions; Linear systems; Number theory; Matrix algebra
Año:2012
Volumen:33
Número:3
Página de inicio:905
Página de fin:939
DOI: http://dx.doi.org/10.1137/100808885
Título revista:SIAM Journal on Matrix Analysis and Applications
Título revista abreviado:SIAM J. Matrix Anal. Appl.
ISSN:08954798
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08954798_v33_n3_p905_Beltran

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Citas:

---------- APA ----------
Beltrán, C., Dedieu, J.-P., Malajovich, G. & Shub, M. (2012) . Convexity properties of the condition number II. SIAM Journal on Matrix Analysis and Applications, 33(3), 905-939.
http://dx.doi.org/10.1137/100808885
---------- CHICAGO ----------
Beltrán, C., Dedieu, J.-P., Malajovich, G., Shub, M. "Convexity properties of the condition number II" . SIAM Journal on Matrix Analysis and Applications 33, no. 3 (2012) : 905-939.
http://dx.doi.org/10.1137/100808885
---------- MLA ----------
Beltrán, C., Dedieu, J.-P., Malajovich, G., Shub, M. "Convexity properties of the condition number II" . SIAM Journal on Matrix Analysis and Applications, vol. 33, no. 3, 2012, pp. 905-939.
http://dx.doi.org/10.1137/100808885
---------- VANCOUVER ----------
Beltrán, C., Dedieu, J.-P., Malajovich, G., Shub, M. Convexity properties of the condition number II. SIAM J. Matrix Anal. Appl. 2012;33(3):905-939.
http://dx.doi.org/10.1137/100808885