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URL:https://uwaterloo.ca/combinatorics-and-optimization/events/tutte-colloq
 uium-henry-wolkowicz-2
SUMMARY:Tutte colloquium-Henry Wolkowicz
CLASS:PUBLIC
DESCRIPTION:TITLE:The omega-Condition Number: Applications to Preconditioni
 ng and\nLow Rank Generalized Jacobian Updating\n\nSPEAKER:\n Henry Wolkowi
 cz\n\nAFFILIATION:\n University of Waterloo\n\nLOCATION:\n MC 5501\n\nABST
 RACT: Preconditioning is essential in iterative methods for\nsolving line
 ar systems. It is also the implicit objective in updating\napproximations 
 of Jacobians in optimization methods\, e.g.\,~in\nquasi-Newton methods. We
  study a nonclassic matrix condition number\,\nthe omega-condition number}
 \, omega for short. omega is the ratio of:\nthe arithmetic and geometric m
 eans of the singular values\, rather than\nthe largest and smallest for th
 e classical kappa-condition number. The\nsimple functions in omega allow o
 ne to exploit  first order\noptimality conditions. We use this fact to de
 rive explicit formulae\nfor (i) omega-optimal low rank updating of general
 ized Jacobians\narising in the context of nonsmooth Newton methods\; and (
 ii)\nomega-optimal preconditioners of special structure for  iterative\nm
 ethods for linear systems. In the latter context\, we analyze the\nbenefit
 s of omega for (a) improving the clustering of eigenvalues\; (b)\nreducing
  the number of iterations\; and (c) estimating the actual\ncondition of a 
 linear system. Moreover we show strong theoretical\nconnections between th
 e omega-optimal preconditioners and incomplete\nCholesky factorizations\, 
 and highlight the misleading effects arising\nfrom the inverse invariance 
 of kappa. Our results confirm the efficacy\nof using the omega-condition n
 umber compared to the kappa-condition\nnumber.\n\n(Joint work with: Woosuk
  L. Jung\, David Torregrosa-Belen.)\n\n 
DTSTAMP:20260405T061806Z
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