Convexity properties of the condition number II
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 nonsmoot...
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todo:paper_08954798_v33_n3_p905_Beltran2023-10-03T15:42:22Z Convexity properties of the condition number II Beltrán, C. Dedieu, J.-P. Malajovich, G. Shub, M. 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 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_08954798_v33_n3_p905_Beltran |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
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 |
| spellingShingle |
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 Beltrán, C. Dedieu, J.-P. Malajovich, G. Shub, M. Convexity properties of the condition number II |
| topic_facet |
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 |
| description |
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. |
| format |
JOUR |
| author |
Beltrán, C. Dedieu, J.-P. Malajovich, G. Shub, M. |
| author_facet |
Beltrán, C. Dedieu, J.-P. Malajovich, G. Shub, M. |
| author_sort |
Beltrán, C. |
| title |
Convexity properties of the condition number II |
| title_short |
Convexity properties of the condition number II |
| title_full |
Convexity properties of the condition number II |
| title_fullStr |
Convexity properties of the condition number II |
| title_full_unstemmed |
Convexity properties of the condition number II |
| title_sort |
convexity properties of the condition number ii |
| url |
http://hdl.handle.net/20.500.12110/paper_08954798_v33_n3_p905_Beltran |
| work_keys_str_mv |
AT beltranc convexitypropertiesoftheconditionnumberii AT dedieujp convexitypropertiesoftheconditionnumberii AT malajovichg convexitypropertiesoftheconditionnumberii AT shubm convexitypropertiesoftheconditionnumberii |
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1782025272720097280 |