Abstract
A Broyden method fails when the free parameter φ takes on a degenerate value; one such value is well known, but this paper shows that many others exist in general. These values have practical significance if the initial Hessian approximation is indefinite. The BFS formula is special in that it avoids these degenerate values. Properties about how the new degenerate values behave and their relationship to the well-known degenerate value are described. A new result is used about a reduced inverse Hessian method, which is equivalent to a Broyden method but is parameter free and provides a simple proof of Dixon's theorem.
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References
Broyden, C. G.,Quasi-Newton Methods and Their Application to Function Minimization, Mathematics of Computation, Vol. 21, pp. 368–381, 1967.
Brodlie, K. W.,Unconstrained Minimization, The State of the Art in Numerical Analysis, Edited by D. A. H. Jacobs, Academic Press, London, England, 1977.
Dixon, L. C. W.,Quasi-Newton Algorithms Generate Identical Points, I, Mathematical Programming, Vol. 2, pp. 383–387, 1972.
Dixon, L. C. W.,Quasi-Newton Techniques Generate Identical Points, II, Mathematical Programming, Vol. 3, pp. 345–358, 1972.
Powell, M. J. D.,Unconstrained Minimization and Extensions for Constraints, Atomic Energy Research Establishment, AERE Report No. TP-495, 1972.
Dixon, L. C. W.,Variable Metric Algorithms, Necessary and Sufficient Conditions for Identical Behavior on Nonquadratic Functions, Journal of Optimization Theory and Applications, Vol. 10, pp. 34–40, 1972.
Fletcher, R.,The Reduced Hessian Method in Variable-Metric Methods, University of Dundee, Biennial Conference on Numerical Analysis, 1977 (unpublished paper).
Sinclair, J. W.,On Quasi-Newton Methods, University of Dundee, Department of Mathematics, PhD Thesis, 1979.
Fletcher, R.,A General Quadratic Programming Algorithm, Journal of the Institute of Mathematics and Its Applications, Vol. 7, pp. 76–91, 1971.
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Communicated by H. Y. Huang
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Fletcher, R., Sinclair, J.W. Degenerate values for broyden methods. J Optim Theory Appl 33, 311–324 (1981). https://doi.org/10.1007/BF00935247
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DOI: https://doi.org/10.1007/BF00935247