Abstract
We consider multi-step quasi-Newton methods for unconstrained optimization. These methods were introduced by Ford and Moghrabi (Appl. Math., vol. 50, pp. 305–323, 1994; Optimization Methods and Software, vol. 2, pp. 357–370, 1993), who showed how interpolating curves could be used to derive a generalization of the Secant Equation (the relation normally employed in the construction of quasi-Newton methods). One of the most successful of these multi-step methods makes use of the current approximation to the Hessian to determine the parameterization of the interpolating curve in the variable-space and, hence, the generalized updating formula. In this paper, we investigate new parameterization techniques to the approximate Hessian, in an attempt to determine a better Hessian approximation at each iteration and, thus, improve the numerical performance of such algorithms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.G. Broyden, “The convergence of a class of double-rank minimization algorithms—Part 2: The new algorithm, ” J. Inst. Math. Applic., vol. 6, pp. 222–231, 1970.
R. Fletcher, “A new approach to variable metric algorithms, ” Comput. J., vol. 13, pp. 317–322, 1970.
J.A. Ford, “Implicit updates in multi-step quasi-Newton methods, ” in press.
J.A. Ford and I.A. Moghrabi, “Alternative parameter choices for multi-step quasi-Newton methods, ” Optimization Methods and Software, vol. 2, pp. 357–370, 1993.
J.A. Ford and I.A. Moghrabi, “Multi-step quasi-Newton methods for optimization, ” J. Comput. Appl. Math., vol. 50, pp. 305–323, 1994.
J.A. Ford and I.A. Moghrabi, “Alternating multi-step quasi-Newton methods for unconstrained optimization, ” J. Comput. Appl. Math., vol. 82, pp. 105–116, 1997.
D. Goldfarb, “A family of variable metric methods derived by variational means, ” Math. Comp., vol. 24, pp. 23–26, 1970.
J.J. More, B.S. Garbow, and K.E. Hillstrom, “Testing unconstrained optimization software, ” ACMTrans. Math. Software, vol. 7, pp. 17–41, 1981.
D.F. Shanno, “Conditioning of quasi-Newton methods for function minimization, ” Math. Comp., vol. 24, pp. 647–656, 1970.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Moghrabi, I. Metric-Based Parameterizations for Multi-Step Unconstrained Optimization. Computational Optimization and Applications 19, 337–345 (2001). https://doi.org/10.1023/A:1011242809975
Issue Date:
DOI: https://doi.org/10.1023/A:1011242809975