Abstract
An example is given where truncation error, caused by finite computer arithmetic, leads to the BFGS variable-metric method becoming stuck, despite the approximated Hessian matrix, the gradient vector, and the search vector satisfying analytical conditions for convergence. A restart criterion to eliminate the problem is suggested.
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References
Broyden, C. G.,The Convergence of a Class of Double-Rank Minimization Algorithms, 2, The New Algorithm, Journal of the Institute for Mathematics and Applications, Vol. 6, pp. 222–231, 1970.
Fletcher, R.,A New Approach to Variable-Metric Algorithms, Computer Journal, Vol. 13, pp. 317–322, 1970.
Goldfarb, D.,A Family of Variable-Metric Algorithms Derived by Variational Means, Mathematics of Computation, Vol. 24, pp. 23–26, 1970.
Shanno, D. F.,Conditioning of Quasi-Newton Methods for Function Minimization, Mathematics of Computation, Vol. 24, pp. 647–656, 1970.
Shanno, D. F., andPhua, K. H.,Inexact Step Lengths and Quasi-Newton Methods, Information Processing 77, Edited by B. Gilchrist, North-Holland, New York, New York, pp. 189–193, 1977.
Powell, M. J. D.,Minimization without Exact Line Searches, Nonlinear Programming, Edited by R. W. Cottle and C. E. Lemke, American Mathematical Society, Providence, Rhode Island, pp. 53–72, 1976.
Powell, M. J. D.,A New Algorithm for Unconstrained Optimization, Nonlinear Programming, Edited by J. B. Rosen, O. L. Mangasarian, and K. Ritter, Academic Press, New York, New York, pp. 31–66, 1970.
Powell, M. J. D.,Nonconvex Minimization Calculations and the Conjugate Gradient Method, University of Cambridge, Cambridge, England, Department of Applied Mathematics and Theoretical Physics, Report No. DAMTP-1983/NA14, 1983.
Polak, E., andRibière, G.,Note sur la Convergence de Methodes de Directions Conjugant, Revue Francaise d'Informatique et Recherche Operationelle, Vol. 16, pp. 35–43, 1969.
Shanno, D. F.,Globally Convergent Conjugate Gradient Algorithms, University of California, Davis, California, Graduate School of Administration, Working Paper No. 84-08, 1984.
Fletcher, R., andReeves, C. M.,Function Minimization by Conjugate Gradients, Computer Journal, Vol. 7, pp. 149–154, 1964.
Beale, E. M. L.,On an Iterative Method of Finding a Local Minimum of a Function of More Than One Variable, Princeton University, Princeton, New Jersey, Statistics Technical Research Group, Technical Report No. 25, 1958.
Shanno, D. F., andPhua, K. H.,Remark on Algorithm 500, Transactions on Mathematical Software, Vol. 6, pp. 618–622, 1980.
Shanno, D. F.,On the Convergence of a New Conjugate Gradient Algorithm, SIAM Journal of Numerical Analysis, Vol. 15, pp. 1247–1257, 1978.
Moré, J. J., Garbow, B. S., andHillstrom, K. E.,Testing Unconstrained Minimization Software, Argonne National Laboratory, Argonne, Illinois, Applied Mathematics Division, Technical Memorandum No. 324, 1978.
Shanno, D. F., andPhua, K. H.,Matrix Conditioning and Nonlinear Optimization, Mathematical Programming, Vol. 14, pp. 149–160, 1978.
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Shanno, D.F. An example of numerical nonconvergence of a variable-metric method. J Optim Theory Appl 46, 87–94 (1985). https://doi.org/10.1007/BF00938762
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DOI: https://doi.org/10.1007/BF00938762