In this paper, a generalized variable-metric algorithm is presented. It is shown that this algorithm attains the minimum of a positive-definite, quadratic function in a finite number of steps and that thevariable-metric matrix tends to the inverse of the Hessian matrix. Most known variable-metric algorithms can be derived from this generalized algorithm, and some new algorithms are also obtained.
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Davidon, W. C.,Variable-Metric Method for Minimization, Argonne National Laboratory, Report No. ANL-5990, 1959.
Hestenes, M. R., andStiefel, E.,Methods of Conjugate Gradients for Solving Linear Systems, Journal of Research of the National Bureau of Standards, Vol. 49, No. 6, 1952.
Broyden, C. G.,Quasi-Newton Methods and Their Application to Function Minimization, Mathematics of Computation, Vol. 21, No. 99, 1967.
Fletcher, R., andReeves, C. M.,Function Minimization by Conjugate Gradients, Computer Journal, Vol. 7, No. 2, 1964.
Shah, B. V., Bueher, R. J., andKempthorne, O.,Some Algorithms for Minimizing a Function of Several Variables, SIAM Journal on Applied Mathematics, Vol. 12, No. 1, 1964.
Zoutendijk, G.,Method of Feasible Directions, American Elsevier Publishing Company, New York, 1960.
Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 2, 1963.
Goldfarb, D.,Sufficient Conditions for the Convergence of Variable-Metric Algorithm, Optimization, Edited by R. Fletcher, Academic Press, New York, 1969.
Pearson, J. D.,On Variable-Metric Methods of Minimization, Computer Journal, Vol. 12, No. 2, 1969.
Greenstadt, J.,Variations on the Variable-Metric Methods, Mathematics of Computation, Vol. 24, No. 109, 1970.
Goldfarb, D.,A Family of Variable-Metric Methods Derived by Variational Means, Mathematics of Computation, Vol. 24, No. 109, 1970.
Huang, H. Y.,Unified Approach to Quadratically Convergent Algorithms for Function Minimization, Journal of Optimization Theory and Applications, Vol. 5, No. 6, 1970.
Huang, H. Y., andLevy, A. V.,Numerical Experiments on Quadratically Convergent Algorithms for Function Minimization, Journal of Optimization Theory and Applications, Vol. 6, No. 3, 1970.
Powell, M. J. D.,An Iterative Method for Finding Stationary Values of a Function of Several Variables, Computer Journal, Vol. 5, No. 2, 1962.
Fletcher, R.,A Review of Methods for Unconstrained Optimization, Optimization, Edited by R. Fletcher, Academic Press, New York, 1969.
Penrose, R.,A Generalized Inverse for Matrices, Proceedings of the Cambridge Philosophical Society, Vol. 51, Part 3, 1954.
Murtagh, B. A., andSargent, R. W. H.,A Constrained Minimization Method with Quadratic Convergence, Optimization, Edited by R. Fletcher, Academic Press, New York, 1969.
The author expresses his gratitude to Professor H. Tokumaru for guidance and encouragement.
Communicated by A. Miele
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Adachi, N. On variable-metric algorithms. J Optim Theory Appl 7, 391–410 (1971). https://doi.org/10.1007/BF00931977
- Finite Number
- Generalize Algorithm
- Quadratic Function
- Hessian Matrix