IJPAM: Volume 119, No. 1 (2018)

Title

CURVATURE-BASED QUASI-NEWTON METHODS
FOR OPTIMIZATION

Authors

Issam A.R. Moghrabi
Department of M.I.S.
College of Business Administration
Gulf University for Science and Technology (GUST)
P.O. Box 7207, Hawally 32093, KUWAIT

Abstract

This paper presents a framework model for building minimum curvature Multi-step methods. The Multi-step methods were derived in [6,7] and have consistently outperformed the traditional quasi-Newton methods that satisy the classical linear Secant equation. The methods derived here aim at improving further the Multi-step methods by ensuring that the interpolating curve used in updating the Hessian approximation has minimum a curvature. The model used in the derivation of such methods utilizes a free parameter that is employed as a tuning variable. The idea of minimizing the curvature of the interpolant was introduced in [2]. The encouraging results justify the investigation of these methods further. The algorithms derived are benchmarked against some of the most successful quasi-Newton methods such as the standard BFGS and the methods derived in [2,5,6,7]. The results of the numerical experiments indicate that the improvements obtained are substantially good and that the methods are indeed promising.

History

Received: April 25, 2018
Revised: June 11, 2018
Published: June 13, 2018

AMS Classification, Key Words

AMS Subject Classification: 65K10
Key Words and Phrases: unconstrained optimization, conjugate gradient methods,variable metric methods

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Bibliography

1
I.A.R. Moghrabi and S.Y.Obeid, A new family of multi-step quasi-Newton methods, 6th Int. Coll. on Differential Equations, VSP (1998), 319-326.

2
J.A. Ford and I.A. Moghrabi, Minimum curvature quasi-Newton methods, Computers Math. Applic. 31 (1996),179-186.

3
J.A. Ford and I.A. Moghrabi, Using function-values in multi-step quasi-Newton methods, Journal of Computational and Applied Mathematics 66 (1996), 201-212.

4
J.A. Ford and I.A. Moghrabi, A nonlinear model for function-value multistep methods, Computers & Mathematics with Applications 42 (1996), no. 8-9, 1157-1164.

5
J.A. Ford and I.A. Moghrabi, Further investigation of multi-step quasi-Newton methods, Scientia Iranica 1 (1995), 327-334.

6
J.A. Ford and I.A. Moghrabi, Multi-step quasi-Newton methods for optimization, J. Comput. Appl. Math. 50 (1994), 305-323.

7
J.A. Ford and I.A. Moghrabi, Alternative parameter choices for multi-step quasi-Newton methods, Optim. Meth. Software 2 (1993), 357-370.

8
P. Gill, W. Murray and M. Wright, Numerical Linear Algebra on Optimization, volume I, Addison-Wesley, U.S.A., 1991).

9
R.F. Fletcher, Pratical Methods Of Optimization, John Wiley, Great Britain, 1991.

10
J.A. Ford and A.F. Saadallah, Efficient utilization of function-values in unconstrained minimization, Colloq. Math. Soc. Jan. Bolyai 50 (1986), 539-563.

11
J.E. Dennis and R.B. Schnabel, Minimum change variable metric update formulae, SIAM Review 21 (1979), 443-459.

12
C.G. Broyden, The convergence of a class of double-rank minimization algorithms, J. Inst. Math. Applic. 6 (1970), 76-90 and 222-231.

13
R. Fletcher, A new approach to variable metric algorithms, Comput. J. 13 (1970), 317-322.

14
D. Goldfarb, A family of variable metric methods derived by variational means, Maths. Comp. 24 (1970), 23-26.

15
D.F. Shanno, Conditioning of quasi-Newton methods for function minimization, Maths. Comp. 24 (1970), 647-656 .

16
P. Wolfe, Convergence conditions for ascent methods II: Some corrections, SIAM Rev. 13 (1971), pp. 185-188 .

How to Cite?

DOI: 10.12732/ijpam.v119i1.11 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 119
Issue: 1
Pages: 131 - 143


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