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URL:/combinatorics-and-optimization/events/tutte-colloq
 uium-henry-wolkowicz-2
SUMMARY:Tutte colloquium-Henry Wolkowicz
CLASS:PUBLIC
DESCRIPTION:Summary \n\nTITLE:The omega-Condition Number: Applications to P
 reconditioning and\nLow Rank Generalized Jacobian Updating\n\nSPEAKER:\n H
 enry Wolkowicz\n\nAFFILIATION:\n University of À¶Ý®ÊÓƵ\n\nLOCATION:\n MC 
 5501\n\nABSTRACT: Preconditioning is essential in iterative methods for\n
 solving linear systems. It is also the implicit objective in updating\napp
 roximations of Jacobians in optimization methods\, e.g.\,~in\nquasi-Newton
  methods. We study a nonclassic matrix condition number\,\nthe omega-condi
 tion number}\, omega for short. omega is the ratio of:\nthe arithmetic and
  geometric means of the singular values\, rather than\nthe largest and sma
 llest for the classical kappa-condition number. The\nsimple functions in o
 mega allow one to exploit  first order\noptimality conditions. We use thi
 s fact to derive explicit formulae\nfor (i) omega-optimal low rank updatin
 g of generalized Jacobians\narising in the context of nonsmooth Newton met
 hods\; and (ii)\nomega-optimal preconditioners of special structure for  
 iterative\nmethods for linear systems. In the latter context\, we analyze 
 the\nbenefits of omega for (a) improving the clustering of eigenvalues\; (
 b)\nreducing the number of iterations\; and (c) estimating the actual\ncon
 dition of a linear system. Moreover we show strong theoretical\nconnection
 s between the omega-optimal preconditioners and incomplete\nCholesky facto
 rizations\, and highlight the misleading effects arising\nfrom the inverse
  invariance of kappa. Our results confirm the efficacy\nof using the omega
 -condition number compared to the kappa-condition\nnumber.\n\n(Joint work 
 with: Woosuk L. Jung\, David Torregrosa-Belen.)\n\n \n 
DTSTAMP:20250707T175232Z
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