Abstract
Based on eigenvalue analyses, well-structured upper bounds for the condition number of the scaled memoryless quasi-Newton updating formulae Broyden---Fletcher---Goldfarb---Shanno and Davidon---Fletcher---Powell are obtained. Then, it is shown that the scaling parameter proposed by Oren and Spedicato is the unique minimizer of the given upper bound for the condition number of scaled memoryless Broyden---Fletcher---Goldfarb---Shanno update, and the scaling parameter proposed by Oren and Luenberger is the unique minimizer of the given upper bound for the condition number of scaled memoryless Davidon---Fletcher---Powell update. Thus, scaling parameters proposed by Oren et al. may enhance numerical stability of the self-scaling memoryless Broyden---Fletcher---Goldfarb---Shanno and Davidon---Fletcher---Powell methods.
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