Abstract
Recently. there has been an increasing interest in the self-scaling Broyden family of formulae. These formulae are usually defined by replacing the approximate Hessian matrix B by τ B for some scaling parameter τ. It is clear that if B is replaced by (l/τ)B in a self-scaling formula, then a member of the Broyden family follows. The author will show that tn certain cases this member is not uniquely defined. He will illustrate this point hy addressing new members of the Broyden family and giving a new approach to the self-dual update of Oren and Spedicato. The new members are defined by explicit expression, and satisfy the usual condition which ensures that the current Hessian approximations are maintained positive definite. Because comparison with the BFGS method shows promising numerical results, the corresponding new methods are efficient.
| Original language | English |
|---|---|
| Pages (from-to) | 301-310 |
| Number of pages | 10 |
| Journal | Optimization Methods and Software |
| Volume | 1 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 1992 |
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Keywords
- Broyden's family
- New members of the Broyden family
- Quasi-Newton methods
- Self-scaling formulae
ASJC Scopus subject areas
- Control and Optimization
- Software
- Applied Mathematics
Cite this
Highly efficient Broyden methods of minimization with variable parameter. / Al-Baali, M.
In: Optimization Methods and Software, Vol. 1, No. 4, 1992, p. 301-310.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Highly efficient Broyden methods of minimization with variable parameter
AU - Al-Baali, M.
PY - 1992
Y1 - 1992
N2 - Recently. there has been an increasing interest in the self-scaling Broyden family of formulae. These formulae are usually defined by replacing the approximate Hessian matrix B by τ B for some scaling parameter τ. It is clear that if B is replaced by (l/τ)B in a self-scaling formula, then a member of the Broyden family follows. The author will show that tn certain cases this member is not uniquely defined. He will illustrate this point hy addressing new members of the Broyden family and giving a new approach to the self-dual update of Oren and Spedicato. The new members are defined by explicit expression, and satisfy the usual condition which ensures that the current Hessian approximations are maintained positive definite. Because comparison with the BFGS method shows promising numerical results, the corresponding new methods are efficient.
AB - Recently. there has been an increasing interest in the self-scaling Broyden family of formulae. These formulae are usually defined by replacing the approximate Hessian matrix B by τ B for some scaling parameter τ. It is clear that if B is replaced by (l/τ)B in a self-scaling formula, then a member of the Broyden family follows. The author will show that tn certain cases this member is not uniquely defined. He will illustrate this point hy addressing new members of the Broyden family and giving a new approach to the self-dual update of Oren and Spedicato. The new members are defined by explicit expression, and satisfy the usual condition which ensures that the current Hessian approximations are maintained positive definite. Because comparison with the BFGS method shows promising numerical results, the corresponding new methods are efficient.
KW - Broyden's family
KW - New members of the Broyden family
KW - Quasi-Newton methods
KW - Self-scaling formulae
UR - http://www.scopus.com/inward/record.url?scp=0000930442&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0000930442&partnerID=8YFLogxK
U2 - 10.1080/10556789208805528
DO - 10.1080/10556789208805528
M3 - Article
VL - 1
SP - 301
EP - 310
JO - Optimization Methods and Software
JF - Optimization Methods and Software
SN - 1055-6788
IS - 4
ER -