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On Optimality of the Parameters of Self-Scaling Memoryless Quasi-Newton Updating Formulae

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  • Saman Babaie-Kafaki

    (Institute for Research in Fundamental Sciences (IPM))

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.

Suggested Citation

  • Saman Babaie-Kafaki, 2015. "On Optimality of the Parameters of Self-Scaling Memoryless Quasi-Newton Updating Formulae," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 91-101, October.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:1:d:10.1007_s10957-015-0724-x
    DOI: 10.1007/s10957-015-0724-x
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    References listed on IDEAS

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    1. J. Z. Zhang & N. Y. Deng & L. H. Chen, 1999. "New Quasi-Newton Equation and Related Methods for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 147-167, July.
    2. Saman Babaie-Kafaki, 2012. "A note on the global convergence theorem of the scaled conjugate gradient algorithms proposed by Andrei," Computational Optimization and Applications, Springer, vol. 52(2), pages 409-414, June.
    3. Shmuel S. Oren, 1974. "Self-Scaling Variable Metric (SSVM) Algorithms," Management Science, INFORMS, vol. 20(5), pages 863-874, January.
    4. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, December.
    5. Saman Babaie-Kafaki, 2012. "A Quadratic Hybridization of Polak–Ribière–Polyak and Fletcher–Reeves Conjugate Gradient Methods," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 916-932, September.
    6. Shmuel S. Oren & David G. Luenberger, 1974. "Self-Scaling Variable Metric (SSVM) Algorithms," Management Science, INFORMS, vol. 20(5), pages 845-862, January.
    7. Andrei, Neculai, 2010. "Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization," European Journal of Operational Research, Elsevier, vol. 204(3), pages 410-420, August.
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    Cited by:

    1. XiaoLiang Dong & Deren Han & Zhifeng Dai & Lixiang Li & Jianguang Zhu, 2018. "An Accelerated Three-Term Conjugate Gradient Method with Sufficient Descent Condition and Conjugacy Condition," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 944-961, December.
    2. Fatemeh Dargahi & Saman Babaie-Kafaki & Zohre Aminifard, 2024. "Eigenvalue Analyses on the Memoryless Davidon–Fletcher–Powell Method Based on a Spectral Secant Equation," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 394-403, January.
    3. Neculai Andrei, 2018. "A Double-Parameter Scaling Broyden–Fletcher–Goldfarb–Shanno Method Based on Minimizing the Measure Function of Byrd and Nocedal for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 191-218, July.

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