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Article

Keywords:
quasi-Newton methods; unconstrained optimization; conjugate directions; update formulas
Summary:
A survey note whose aim is to establish the heuristics and natural relations in a class of Quasi-Newton methods in optimization problems. It is shown that a particular algorithm of the class is specified by characcterizing some parameters (scalars and matrices) in a general solution of a matrix equation.
References:
[1] C. G. Broyden: Quasi-Newton Methods and heir Application to Function Minimisation. Mathematics of Computation, Vol. 21, pp. 368 - 381, (1967). DOI 10.1090/S0025-5718-1967-0224273-2 | MR 0224273
[2] C. G. Broyden: The Convergence of a Class of Double-Rank Minimisation Algorithms. Journal of the Institute of Mathematics and its Applications, Vol. 6, pp. 79-90, 222-231 (1970). MR 0433870
[3] W. C. Davidon: Variable Metric Method for Minimization. Argonne National Laboratory Rept. ANL-5990 (Rev.) (1959).
[4] W. C. Davidon: Optimally Conditioned Optimization Algorithms without Line Searches. Mathematical Programming, Vol. 9, pp. 1 - 30, (1975). DOI 10.1007/BF01681328 | MR 0383741 | Zbl 0328.90055
[5] J. E. Dennis J. J. Moré: Quasi-Newton Methods, Motivation and Theory. SIAM Review, Vol. 19, pp. 46-89, (1977). DOI 10.1137/1019005 | MR 0445812
[6] R. Fletcher: A New Approach to Variable Metric Algorithms. The Computer Journal, Vol. 13, pp. 317-322, (1970). DOI 10.1093/comjnl/13.3.317
[7] R. Fletcher M. J. D. Powell: A Rapidly Convergent Descent Method for Minimization. The Computer Journal, Vol. 6, pp. 163-168, (1963). DOI 10.1093/comjnl/6.2.163 | MR 0152116
[8] R. Fletcher C. M. Reeves: Function Minimisation by Conjugate Gradients. The Computer Journal, Vol. 7, pp. 149-154, (1964). DOI 10.1093/comjnl/7.2.149 | MR 0187375
[9] D. Goldfarb: A Family of Variable-Metric Methods Derived by Variational Means. Mathematics of Computation, Vol. 24, pp. 23 - 26, (1970). DOI 10.1090/S0025-5718-1970-0258249-6 | MR 0258249 | Zbl 0196.18002
[10] H. Y. Huang: A Unified Approach to Quadratically Convergent Algorithms for Function Minimisation. Journal of Optimization Theory and Applications, Vol. 5, pp. 405 - 423, (1970). DOI 10.1007/BF00927440 | MR 0288939
[11] J. D. Pearson: Variable Metric Methods of Minimisation. The Computer Journal, Vol. 12, pp. 171-179, (1969). DOI 10.1093/comjnl/12.2.171 | MR 0242355 | Zbl 0207.17301
[12] M. J. D. Powell: An Efficient Method of Finding the Minimum of a Function of Several Variables without Calculating Derivatives. The Computer Journal, Vol. 7, pp. 155-162, (1964). DOI 10.1093/comjnl/7.2.155 | MR 0187376
[13] D. F. Shanno: Conditioning of Quasi-Newton Methods for Function Minimization. Mathematics of Computation, Vol. 24, pp. 647-657, (1970). DOI 10.1090/S0025-5718-1970-0274029-X | MR 0274029
[14] H. Yanai: On Conjugate Direction Methods. Seminar Report Vol. 190, Institute for Mathematical Sciences, Kyoto Univ., (1973). (In Japanese)
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