Globalization of Barzilai and Borwein Method for Unconstrained Optimization

The focus of this thesis is on finding the unconstrained minimizer of a function. Specifically, we will focus on the Barzilai and Borwein (BB) method that is a famous two-point stepsize gradient method. First we briefly give some mathematical background. Then we discuss the (BB) method that is im...

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Main Author:	Farid, Mahboubeh
Format:	Thesis
Language:	English English
Published:	2009
Subjects:	Algorithms Mathematical optimization Convergence
Online Access:	http://psasir.upm.edu.my/id/eprint/10387/1/IPM_2009_10_A.pdf
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id	my-upm-ir.10387
record_format	uketd_dc
spelling	my-upm-ir.103872013-05-27T07:45:52Z Globalization of Barzilai and Borwein Method for Unconstrained Optimization 2009 Farid, Mahboubeh The focus of this thesis is on finding the unconstrained minimizer of a function. Specifically, we will focus on the Barzilai and Borwein (BB) method that is a famous two-point stepsize gradient method. First we briefly give some mathematical background. Then we discuss the (BB) method that is important in the area of optimization. A review of the minimization methods currently available that can be used to solve unconstrained optimization is also given. Due to BB method’s simplicity, low storage and numerical efficiency, the Barzilai and Borwein method has received a good deal of attention in the optimization community but despite all these advances, stepsize of BB method is computed by means of simple approximation of Hessian in the form of scalar multiple of identity and especially the BB method is not monotone, and it is not easy to generalize the method to general nonlinear functions. Due to the presence of these deficiencies, we introduce new gradient-type methods in the frame of BB method including a new gradient method via weak secant equation (quasi-Cauchy relation), improved Hessian approximation and scaling the diagonal updating. The proposed methods are a kind of fixed step gradient method like that of Barzilai and Borwein method. In contrast with the Barzilai and Borwein approach’s in which stepsize is computed by means of simple approximation of the Hessian in the form of scalar multiple of identity, the proposed methods consider approximation of Hessian in diagonal matrix. Incorporate with monotone strategies, the resulting algorithms belong to the class of monotone gradient methods with globally convergence. Numerical results suggest that for non-quadratic minimization problem, the new methods clearly outperform the Barzilai- Borwein method. Finally we comment on some achievement in our researches. Possible extensions are also given to conclude this thesis. Algorithms Mathematical optimization Convergence 2009 Thesis http://psasir.upm.edu.my/id/eprint/10387/ http://psasir.upm.edu.my/id/eprint/10387/1/IPM_2009_10_A.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Algorithms Mathematical optimization Convergence Institute for Mathematical Research English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Algorithms Mathematical optimization Convergence
spellingShingle	Algorithms Mathematical optimization Convergence Farid, Mahboubeh Globalization of Barzilai and Borwein Method for Unconstrained Optimization
description	The focus of this thesis is on finding the unconstrained minimizer of a function. Specifically, we will focus on the Barzilai and Borwein (BB) method that is a famous two-point stepsize gradient method. First we briefly give some mathematical background. Then we discuss the (BB) method that is important in the area of optimization. A review of the minimization methods currently available that can be used to solve unconstrained optimization is also given. Due to BB method’s simplicity, low storage and numerical efficiency, the Barzilai and Borwein method has received a good deal of attention in the optimization community but despite all these advances, stepsize of BB method is computed by means of simple approximation of Hessian in the form of scalar multiple of identity and especially the BB method is not monotone, and it is not easy to generalize the method to general nonlinear functions. Due to the presence of these deficiencies, we introduce new gradient-type methods in the frame of BB method including a new gradient method via weak secant equation (quasi-Cauchy relation), improved Hessian approximation and scaling the diagonal updating. The proposed methods are a kind of fixed step gradient method like that of Barzilai and Borwein method. In contrast with the Barzilai and Borwein approach’s in which stepsize is computed by means of simple approximation of the Hessian in the form of scalar multiple of identity, the proposed methods consider approximation of Hessian in diagonal matrix. Incorporate with monotone strategies, the resulting algorithms belong to the class of monotone gradient methods with globally convergence. Numerical results suggest that for non-quadratic minimization problem, the new methods clearly outperform the Barzilai- Borwein method. Finally we comment on some achievement in our researches. Possible extensions are also given to conclude this thesis.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Farid, Mahboubeh
author_facet	Farid, Mahboubeh
author_sort	Farid, Mahboubeh
title	Globalization of Barzilai and Borwein Method for Unconstrained Optimization
title_short	Globalization of Barzilai and Borwein Method for Unconstrained Optimization
title_full	Globalization of Barzilai and Borwein Method for Unconstrained Optimization
title_fullStr	Globalization of Barzilai and Borwein Method for Unconstrained Optimization
title_full_unstemmed	Globalization of Barzilai and Borwein Method for Unconstrained Optimization
title_sort	globalization of barzilai and borwein method for unconstrained optimization
granting_institution	Universiti Putra Malaysia
granting_department	Institute for Mathematical Research
publishDate	2009
url	http://psasir.upm.edu.my/id/eprint/10387/1/IPM_2009_10_A.pdf
_version_	1747811110843580416

Globalization of Barzilai and Borwein Method for Unconstrained Optimization

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