Abstract
This report contains the proofs of four new theorems relating to the behaviour of Broyden's [1] family of variable metric formula for solving unconstrained minimisation problems. In particular, it is shown that if the linear search at each iteration is perfect, then the sequence of points that is generated is independent of the member of the family used at each iteration, provided the matrix remains nonsingular. This result extends Powell's [14] proof of convergence for the original formula on any convex function to all members of the family.
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Dixon, L.C.W. Quasi Newton techniques generate identical points II: The proofs of four new theorems. Mathematical Programming 3, 345–358 (1972). https://doi.org/10.1007/BF01585007
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DOI: https://doi.org/10.1007/BF01585007