Symmetric rank-one method and its modifications for unconstrained optimization

The attention of this thesis is on the theoretical and experimental behaviors of some modifications of the symmetric rank-one method, one of the quasi-Newton update for finding the minimum of real valued function f over all vectors x ∈ Rn. Symmetric rank-one update (SR1) is known to have good num...

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Main Author:	Moyi, Aliyu Usman
Format:	Thesis
Language:	English
Published:	2014
Subjects:	Newton-Raphson method
Online Access:	http://psasir.upm.edu.my/id/eprint/70465/1/FS%202014%2044%20IR.pdf
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spelling	my-upm-ir.704652019-10-30T04:17:25Z Symmetric rank-one method and its modifications for unconstrained optimization 2014-06 Moyi, Aliyu Usman The attention of this thesis is on the theoretical and experimental behaviors of some modifications of the symmetric rank-one method, one of the quasi-Newton update for finding the minimum of real valued function f over all vectors x ∈ Rn. Symmetric rank-one update (SR1) is known to have good numerical performance among the quasi-Newton methods for solving unconstrained optimization problems. However, it is well known that the SR1 update may not preserve positive definiteness even when updated from a positive definite approximation and can be undefined with zero denominator. Thus, it is our aim in this thesis to provide effective remedies aimed toward dealing with these well known shortcomings and improve the performance of the update. A new inexact line search strategy in solving unconstrained optimization problems is proposed. This method does not require the evaluation of the objective function. Instead, it forces a reduction in gradient norm on each direction, hence it is suitable for problems when function evaluation is very costly. The convergence properties of this strategy is shown using the Lyapunov function approach. Similarly, we proposed some scaling strategies to overcome the challenges of the SR1 update. Under some mild assumptions, the convergence of these methods is proved. Furthermore, in order to exploit the good properties of the SR1 update in providing quality Hessian approximations, we introduced a three-term conjugate gradient method via the symmetric rank-one update in which a conjugate gradient line search direction is constructed without the computation and storage of matrices and possess the sufficient descent property. Extensive computational experiments performed on standard unconstrained optimization test functions and some real-life optimization problems in order to examine the impact of the proposed methods in comparison with other existing methods has shown significant improvement on the performance of the SR1 method in terms of efficiency and robustness. Newton-Raphson method 2014-06 Thesis http://psasir.upm.edu.my/id/eprint/70465/ http://psasir.upm.edu.my/id/eprint/70465/1/FS%202014%2044%20IR.pdf text en public doctoral Universiti Putra Malaysia Newton-Raphson method
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
topic	Newton-Raphson method
spellingShingle	Newton-Raphson method Moyi, Aliyu Usman Symmetric rank-one method and its modifications for unconstrained optimization
description	The attention of this thesis is on the theoretical and experimental behaviors of some modifications of the symmetric rank-one method, one of the quasi-Newton update for finding the minimum of real valued function f over all vectors x ∈ Rn. Symmetric rank-one update (SR1) is known to have good numerical performance among the quasi-Newton methods for solving unconstrained optimization problems. However, it is well known that the SR1 update may not preserve positive definiteness even when updated from a positive definite approximation and can be undefined with zero denominator. Thus, it is our aim in this thesis to provide effective remedies aimed toward dealing with these well known shortcomings and improve the performance of the update. A new inexact line search strategy in solving unconstrained optimization problems is proposed. This method does not require the evaluation of the objective function. Instead, it forces a reduction in gradient norm on each direction, hence it is suitable for problems when function evaluation is very costly. The convergence properties of this strategy is shown using the Lyapunov function approach. Similarly, we proposed some scaling strategies to overcome the challenges of the SR1 update. Under some mild assumptions, the convergence of these methods is proved. Furthermore, in order to exploit the good properties of the SR1 update in providing quality Hessian approximations, we introduced a three-term conjugate gradient method via the symmetric rank-one update in which a conjugate gradient line search direction is constructed without the computation and storage of matrices and possess the sufficient descent property. Extensive computational experiments performed on standard unconstrained optimization test functions and some real-life optimization problems in order to examine the impact of the proposed methods in comparison with other existing methods has shown significant improvement on the performance of the SR1 method in terms of efficiency and robustness.
format	Thesis
qualification_level	Doctorate
author	Moyi, Aliyu Usman
author_facet	Moyi, Aliyu Usman
author_sort	Moyi, Aliyu Usman
title	Symmetric rank-one method and its modifications for unconstrained optimization
title_short	Symmetric rank-one method and its modifications for unconstrained optimization
title_full	Symmetric rank-one method and its modifications for unconstrained optimization
title_fullStr	Symmetric rank-one method and its modifications for unconstrained optimization
title_full_unstemmed	Symmetric rank-one method and its modifications for unconstrained optimization
title_sort	symmetric rank-one method and its modifications for unconstrained optimization
granting_institution	Universiti Putra Malaysia
publishDate	2014
url	http://psasir.upm.edu.my/id/eprint/70465/1/FS%202014%2044%20IR.pdf
_version_	1747812844251906048

Symmetric rank-one method and its modifications for unconstrained optimization

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