Abstract
Some promising ideas for minimizing a nonlinear function, whose first and second derivatives are given, by a modified Newton method, were introduced by Fiacco and McCormick (Ref. 1). Unfortunately, in developing a method around these ideas, Fiacco and McCormick used a potentially unstable, or even impossible, matrix factorization. Using some recently developed techniques for factorizing an indefinite symmetric matrix, we are able to produce a method which is similar to Fiacco and McCormick's original method, but avoids the difficulties of the original method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons, New York, New York, 1968.
Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Fletcher, R.,An algorithm for Solving Linearly Constrained Optimization Problems, Mathematical Programming, Vol. 2, pp. 133–165, 1972.
Bunch, J. R., andParlett, B. N.,Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations, SIAM Journal on Numerical Analysis, Vol. 8, pp. 639–655, 1971.
Fletcher, R.,Factorizing Symmetric Indefinite Matrices, Linear Algebra and its Applications, Vol. 14, pp. 257–262, 1976.
Goldstein, A. A., andPrice, J. F.,An Effective Algorithm for Minimization, Numerische Mathematik, Vol. 10, pp. 184–189, 1967.
Dixon, L. C. W., andBiggs, M. C.,Meander, a Newton Based Procedure for N-Dimensional Function Minimization, Hatfield Polytechnic, Numerical Optimization Center, Technical Report No. 9, 1970.
Mathews, A., andDavies, D.,A Comparison of Modified Newton Methods for Unconstrained Optimization, Computer Journal, Vol. 14, pp. 293–294, 1971.
Greenstadt, J.,On the Relative Efficiencies of Gradient Methods, Mathematics of Computation, Vol. 21, pp. 360–367, 1967.
Gill, P. E., andMurray, W.,Two Methods for the Solution of Linearly Constrained and Unconstrained Optimization Problems, National Physical Laboratory, Teddington, England, Report No. NAC 25, 1972.
Goldfeld, S. M., Quandt, R. E., andTrotter, H. F.,Maximization by Quadratic Hill-Climbing, Econometrica, Vol. 34, pp. 541–551, 1966.
Hebden, M. D.,An Algorithm for Minimization Using Exact Second Derivatives, Atomic Energy Research Establishment, Harwell, England, Report No. TP 515, 1973.
Author information
Authors and Affiliations
Additional information
Communicated by A. V. Fiacco
Both authors gratefully acknowledge the award of a research fellowship from the British Science Research Council.
Rights and permissions
About this article
Cite this article
Fletcher, R., Freeman, T.L. A modified Newton method for minimization. J Optim Theory Appl 23, 357–372 (1977). https://doi.org/10.1007/BF00933446
Issue Date:
DOI: https://doi.org/10.1007/BF00933446