A hybrid scaling parameter for the scaled memoryless BFGS method based on the ℓ_∞ matrix norm

ABSTRACT

An upper bound for condition number of the scaled memoryless Broyden–Fletcher–Goldfarb–Shanno (BFGS) updating formula in the ${\ell }_{\mathrm{\infty }}$ matrix norm is given. Then, in order to increase numerical stability of the related method, the suggested bound is minimized as well and consequently, a class of choices for the scaling parameter is achieved, including the Oren–Spedicato formula as an especial case and guaranteeing the descent property. An approximate value for the most effective member of the suggested class is proposed based on the numerical observations.

KEYWORDS:

• Unconstrained optimization
• scaled memoryless BFGS update
• ${\ell }_{\mathrm{\infty }}$ norm
• condition number
• descent property
• line search

Mathematics Subject Classifications (2010):

• 90C53
• 49M37
• 65K05

Acknowledgements

The author is grateful to Professor Michael Navon for providing the line search code. He also thanks the anonymous referees for their valuable comments and suggestions helped to improve the quality of this work.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by Research Council of Semnan University.