Some properties of the Hessian of the logarithmic barrier function
 Margaret H. Wright
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More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly illconditioned at points on the barrier trajectory as the solution is approached. This paper explores some further characteristics of the barrier Hessian. We first show that, except in two special cases, the barrier Hessian is illconditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this illconditioning displays a special structure connected with subspaces defined by the Jacobian of the active constraints. We then indicate how a Cholesky factorization with diagonal pivoting can be used to detect numerical rankdeficiency in the barrier Hessian, and to provide information about the underlying subspaces without making an explicit prediction of the active constraints. Using this subspace information, a close approximation to the Newton direction can be calculated by solving linear systems whose condition reflects that of the original problem.
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 Title
 Some properties of the Hessian of the logarithmic barrier function
 Journal

Mathematical Programming
Volume 67, Issue 13 , pp 265295
 Cover Date
 19941001
 DOI
 10.1007/BF01582224
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Barrier methods
 Interior methods
 Barrier Hessian
 Illconditioning
 Rankrevealing Cholesky factorization
 Industry Sectors
 Authors

 Margaret H. Wright ^{(1)}
 Author Affiliations

 1. AT&T Bell Laboratories, 07974, Murray Hill, New Jersey, USA