New Quasi-Newton Equation And Method Via Higher Order Tensor Models
This thesis introduces a general approach by proposing a new quasi-Newton (QN) equation via fourth order tensor model. To approximate the curvature of the objective function, more available information from the function-values and gradient is employed. The efficiency of the usual QN methods is im...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English English |
| Published: |
2010
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/12435/1/FS_2010_20A.pdf |
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| Summary: | This thesis introduces a general approach by proposing a new quasi-Newton
(QN) equation via fourth order tensor model. To approximate the curvature
of the objective function, more available information from the function-values
and gradient is employed. The efficiency of the usual QN methods is improved
by accelerating the performance of the algorithms without causing more storage
demand.
The presented equation allows the modification of several algorithms involving
QN equations for practical optimization that possess superior convergence prop-
erty. By using a new equation, the BFGS method is modified. This is done
twice by employing two different strategies proposed by Zhang and Xu (2001)
and Wei et al. (2006) to generate positive definite updates. The superiority of
these methods compared to the standard BFGS and the modification proposed
by Wei et al. (2006) is shown. Convergence analysis that gives the local and global convergence property of these methods and numerical results that shows
the advantage of the modified QN methods are presented.
Moreover, a new limited memory QN method to solve large scale unconstrained
optimization is developed based on the modified BFGS updated formula. The
comparison between this new method with that of the method developed by Xiao
et al. (2008) shows better performance in numerical results for the new method.
The global and local convergence properties of the new method on uniformly
convex problems are also analyzed.
The compact limited memory BFGS method is modified to solve the large scale
unconstrained optimization problems. This method is derived from the proposed
new QN update formula. The new method yields a more efficient algorithm
compared to the standard limited memory BFGS with simple bounds (L-BFGS-B) method in the case of solving unconstrained problems. The implementation of
the new proposed method on a set of test problems highlights that the derivation
of this new method is more efficient in performing the standard algorithm. |
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