Some properties of the Hessian of the logarithmic barrier function

Research output: Contribution to journalArticle

Abstract

More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly ill-conditioned 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 ill-conditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this ill-conditioning 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 rank-deficiency 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.

Original languageEnglish (US)
Pages (from-to)265-295
Number of pages31
JournalMathematical Programming
Volume67
Issue number1-3
DOIs
StatePublished - Oct 1994

Fingerprint

Logarithmic Barrier Function
Trajectories
Factorization
Linear systems
Subspace
Trajectory
Ill-conditioning
Cholesky factorisation
Hessian matrix
Pivoting
Linear Systems
Entire
Prediction
Approximation

Keywords

  • Barrier Hessian
  • Barrier methods
  • Ill-conditioning
  • Interior methods
  • Rank-revealing Cholesky factorization

ASJC Scopus subject areas

  • Computer Science(all)
  • Computer Graphics and Computer-Aided Design
  • Software
  • Management Science and Operations Research
  • Safety, Risk, Reliability and Quality
  • Mathematics(all)
  • Applied Mathematics

Cite this

Some properties of the Hessian of the logarithmic barrier function. / Wright, Margaret H.

In: Mathematical Programming, Vol. 67, No. 1-3, 10.1994, p. 265-295.

Research output: Contribution to journalArticle

@article{4131beb0b4ba4d83a28cd091c04bec18,
title = "Some properties of the Hessian of the logarithmic barrier function",
abstract = "More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly ill-conditioned 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 ill-conditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this ill-conditioning 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 rank-deficiency 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.",
keywords = "Barrier Hessian, Barrier methods, Ill-conditioning, Interior methods, Rank-revealing Cholesky factorization",
author = "Wright, {Margaret H.}",
year = "1994",
month = "10",
doi = "10.1007/BF01582224",
language = "English (US)",
volume = "67",
pages = "265--295",
journal = "Mathematical Programming",
issn = "0025-5610",
publisher = "Springer-Verlag GmbH and Co. KG",
number = "1-3",

}

TY - JOUR

T1 - Some properties of the Hessian of the logarithmic barrier function

AU - Wright, Margaret H.

PY - 1994/10

Y1 - 1994/10

N2 - More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly ill-conditioned 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 ill-conditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this ill-conditioning 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 rank-deficiency 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.

AB - More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly ill-conditioned 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 ill-conditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this ill-conditioning 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 rank-deficiency 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.

KW - Barrier Hessian

KW - Barrier methods

KW - Ill-conditioning

KW - Interior methods

KW - Rank-revealing Cholesky factorization

UR - http://www.scopus.com/inward/record.url?scp=0001511481&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001511481&partnerID=8YFLogxK

U2 - 10.1007/BF01582224

DO - 10.1007/BF01582224

M3 - Article

AN - SCOPUS:0001511481

VL - 67

SP - 265

EP - 295

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-3

ER -