The conjecture is considered that the hybrid GN-BFGS method of Al-Baali and Fletcher [1] is superlinearly convergent. A counterexample is constructed for which the method is only linearly convergent whereas the BFGS method alone would be superlinearly convergent, thus disproving the conjecture. Two new hybrids methods are suggested, and it is shown that these are superlinearly convergent under mild conditions. A wide range of numerical experience is reported and is seen to favour the new methods. Some interesting theoretical properties of a variational Hessian error measure are given.
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Fletcher et al. (1987) studied this question.