Hostname: page-component-669899f699-ggqkh Total loading time: 0 Render date: 2025-04-26T21:47:14.904Z Has data issue: false hasContentIssue false
  • English
  • Français

A conjugate direction implementation of the BFGS algorithm with automatic scaling

Published online by Cambridge University Press:  17 February 2009

Ian D. Coope
Ian D. Coope
Affiliation:
Department of Mathematics, University of Canterbury, Christchurch, N.Z.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new implementation of the BFGS algorithm for unconstrained optimisation is reported which utilises a conjugate factorisation of the approximating Hessian matrix. The implementation is especially useful when gradient information is estimated by finite difference formulae and it is well suited to machines which are able to exploit parallel processing.

Type
Research Article
Information
The ANZIAM Journal , Volume 31 , Issue 1 , July 1989 , pp. 122 - 134
Copyright
Copyright © Australian Mathematical Society 1989

References

[1] Brodlie, K. W., Gourlay, A. R. and Greenstadt, J., “Rank-one and rank-two corrections to positive definite matrices expressed in product form”, J. Inst. Maths. Applies. 11 (1973) 73–82.CrossRefGoogle Scholar
[2] Broyden, C. G., “The convergence of a class of double rank minimization algorithms. 2. The new algorithm”, J. Inst. Maths. Applies. 6 (1970) 222–231.CrossRefGoogle Scholar
[3] Davidon, W. C., “Optimally conditioned optimization algorithms without line searches”, Math. Prog. 9 (1975) 1–30.Google Scholar
[4] Fletcher, R., “A new approach to variable metric algorithms”, Computer Journal 13 (1970) 317–322.Google Scholar
[5] Gill, P. E., Murray, W., Wright, M. H., Practical Optimization (Academic Press, London, 1982).Google Scholar
[6] Goldfarb, D., “A family of variable metric methods derived by variational means“, Maths of Comp. 24 (1970) 23–26.CrossRefGoogle Scholar
[7] Goldfarb, D. and Idnani, A., “A numerical stable dual method for solving strictly convex quadratic programs”, Math. Prog. 21 (1983) 1–33.CrossRefGoogle Scholar
[8] Han, S-P., “Optimization by updated conjugate subspaces”, in Numerical Analysis (eds. Griffiths, D. F. and Waton, G. A.), (Pitman Research Notes in Mathematics Series 140, Longman Scientific & Technical, Burnt Mill, England, 1986) 82–97.Google Scholar
[9] Osborne, M. R. and Saunders, M. A., “Descent methods for minimization”, in Optimization (eds. Anderssen, R. S., Jennings, L. S., Ryan, D. M.), (University of Queensland Press, St. Lucia, 1972) 221–237.Google Scholar
[10] Powell, M. J. D., “A fast algorithm for nonlinearly constrained optimization calculations”, in Numerical Analysis, Lecture Notes in Mathematics 630 (ed. Watson, G. A.), (Springer-Verlag, 1978) 144–157.Google Scholar
[11] Powell, M. J. D., “Updating conjugate directions by the BFGS formula”, Math. Prog. 38 (1987) 29–46.CrossRefGoogle Scholar
[12] Powell, M. J. D., “Subroutine VA13AD”, Harwell Subroutine Library (1975) U. K. Atomic Energy Research Establishment.Google Scholar
[13] Schendel, U., Introduction to numerical methods for parallel computers (Ellis Horwood, Chichester, 1984).Google Scholar
[14] Shanno, D. F., “Conditioning of quasi-Newton methods for function minimization”, Maths of Comp. 24 (1970) 647–656.CrossRefGoogle Scholar