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Scaling symmetric rank one update for unconstrained optimization

A basic disadvantage to the symmetric rank one (SR1) update is that the SRI update may not preserve positive definiteness when starting with a positive definite approximation. A simple remedy to this problem is to restart the update with the initial approximation mostly the identity matrix whenever...

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Main Authors: Abu Hassan, Malik, Monsi, Mansor, Leong, Wah June
Format: Article
Language:English
Published: Universiti Utara Malaysia 2002
Subjects:
QA Mathematics
Online Access:http://repo.uum.edu.my/761/1/Malik_Abu_Hassan.pdf
http://repo.uum.edu.my/761/
http://ijms.uum.edu.my
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spelling my.uum.repo.7612010-09-30T06:13:46Z http://repo.uum.edu.my/761/ Scaling symmetric rank one update for unconstrained optimization Abu Hassan, Malik Monsi, Mansor Leong, Wah June QA Mathematics A basic disadvantage to the symmetric rank one (SR1) update is that the SRI update may not preserve positive definiteness when starting with a positive definite approximation. A simple remedy to this problem is to restart the update with the initial approximation mostly the identity matrix whenever this difficulty arises. However, numerical experience shows that restart with the identity matrix is not a good choice. Instead of using the identity matrix we used a positive multiple of the identity matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem - maximize the determinant subject to a bound of 1 on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of positive multiple of identity matrix is provided for the SR1 when it is not positive definite. Our experiments indicate that with such simple scale, the effectiveness of the SR1 method is increased dramatically. Universiti Utara Malaysia 2002 Article PeerReviewed application/pdf en http://repo.uum.edu.my/761/1/Malik_Abu_Hassan.pdf Abu Hassan, Malik and Monsi, Mansor and Leong, Wah June (2002) Scaling symmetric rank one update for unconstrained optimization. Analisis, 9 (1&2). pp. 63-76. ISSN 0127-8983 http://ijms.uum.edu.my
institution Universiti Utara Malaysia
building UUM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Utara Malaysia
content_source UUM Institutionali Repository
url_provider http://repo.uum.edu.my/
language English
topic QA Mathematics
spellingShingle QA Mathematics
Abu Hassan, Malik
Monsi, Mansor
Leong, Wah June
Scaling symmetric rank one update for unconstrained optimization
description A basic disadvantage to the symmetric rank one (SR1) update is that the SRI update may not preserve positive definiteness when starting with a positive definite approximation. A simple remedy to this problem is to restart the update with the initial approximation mostly the identity matrix whenever this difficulty arises. However, numerical experience shows that restart with the identity matrix is not a good choice. Instead of using the identity matrix we used a positive multiple of the identity matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem - maximize the determinant subject to a bound of 1 on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of positive multiple of identity matrix is provided for the SR1 when it is not positive definite. Our experiments indicate that with such simple scale, the effectiveness of the SR1 method is increased dramatically.
format Article
author Abu Hassan, Malik
Monsi, Mansor
Leong, Wah June
author_facet Abu Hassan, Malik
Monsi, Mansor
Leong, Wah June
author_sort Abu Hassan, Malik
title Scaling symmetric rank one update for unconstrained optimization
title_short Scaling symmetric rank one update for unconstrained optimization
title_full Scaling symmetric rank one update for unconstrained optimization
title_fullStr Scaling symmetric rank one update for unconstrained optimization
title_full_unstemmed Scaling symmetric rank one update for unconstrained optimization
title_sort scaling symmetric rank one update for unconstrained optimization
publisher Universiti Utara Malaysia
publishDate 2002
url http://repo.uum.edu.my/761/1/Malik_Abu_Hassan.pdf
http://repo.uum.edu.my/761/
http://ijms.uum.edu.my
_version_ 1644277861151932416
score 13.211869

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