A family of variable-metric methods derived by variational means

Mathematics of Computation - Tập 24 Số 109 - Trang 23-26
Donald Goldfarb

Tóm tắt

A new rank-two variable-metric method is derived using Greenstadt’s variational approach [Math. Comp., this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating matrix. Together with Greenstadt’s method, the new method gives rise to a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. It is equivalent to Broyden’s one-parameter family [Math. Comp., v. 21, 1967, pp. 368–381]. Choices for the inverse of the weighting matrix in the variational approach are given that lead to the derivation of the DFP and rank-one methods directly.

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Tài liệu tham khảo

Broyden, C. G., 1967, Quasi-Newton methods and their application to function minimisation, Math. Comp., 21, 368, 10.2307/2003239

W. C. Davidon, Variable Metric Method for Minimization, A. E. C. Res. and Develop. Report ANL-5990 (Rev. TID-4500, 14th ed.) 1959.

Davidon, William C., 1967, Variance algorithm for minimization, Comput. J., 10, 406, 10.1093/comjnl/10.4.406

Fletcher, R., 1963, A rapidly convergent descent method for minimization, Comput. J., 6, 163, 10.1093/comjnl/6.2.163

Goldfarb, D., 1969, Sufficient conditions for the convergence of a variable metric algorithm, 273

Greenstadt, J., 1970, Variations on variable-metric methods. (With discussion), Math. Comp., 24, 1, 10.2307/2004872

P. Wolfe, Another Variable Metric Method, Working Paper, 1967.

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Mathematics of Computation

Tập 24 Số 109

23-26

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