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A Double-Parameter Scaling Broyden–Fletcher–Goldfarb–Shanno Method Based on Minimizing the Measure Function of Byrd and Nocedal for Unconstrained Optimization

Author

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  • Neculai Andrei

    (Center for Advanced Modeling and Optimization
    Academy of Romanian Scientists)

Abstract

In this paper, the first two terms on the right-hand side of the Broyden–Fletcher–Goldfarb–Shanno update are scaled with a positive parameter, while the third one is also scaled with another positive parameter. These scaling parameters are determined by minimizing the measure function introduced by Byrd and Nocedal (SIAM J Numer Anal 26:727–739, 1989). The obtained algorithm is close to the algorithm based on clustering the eigenvalues of the Broyden–Fletcher–Goldfarb–Shanno approximation of the Hessian and on shifting its large eigenvalues to the left, but it is not superior to it. Under classical assumptions, the convergence is proved by using the trace and the determinant of the iteration matrix. By using a set of 80 unconstrained optimization test problems, it is proved that the algorithm minimizing the measure function of Byrd and Nocedal is more efficient and more robust than some other scaling Broyden–Fletcher–Goldfarb–Shanno algorithms, including the variants of Biggs (J Inst Math Appl 12:337–338, 1973), Yuan (IMA J Numer Anal 11:325–332, 1991), Oren and Luenberger (Manag Sci 20:845–862, 1974) and of Nocedal and Yuan (Math Program 61:19–37, 1993). However, it is less efficient than the algorithms based on clustering the eigenvalues of the iteration matrix and on shifting its large eigenvalues to the left, as shown by Andrei (J Comput Appl Math 332:26–44, 2018, Numer Algorithms 77:413–432, 2018).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:1:d:10.1007_s10957-018-1288-3
    DOI: 10.1007/s10957-018-1288-3
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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.
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    3. 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.
    4. W. Y. Cheng & D. H. Li, 2010. "Spectral Scaling BFGS Method," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 305-319, August.
    5. Neculai Andrei, 2017. "Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology," Springer Optimization and Its Applications, Springer, number 978-3-319-58356-3, June.
    6. Neculai Andrei, 2016. "A New Adaptive Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization," Springer Optimization and Its Applications, in: Boris Goldengorin (ed.), Optimization and Its Applications in Control and Data Sciences, pages 1-16, Springer.
    7. Neculai Andrei, 2017. "Applications of Continuous Nonlinear Optimization," Springer Optimization and Its Applications, in: Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology, chapter 0, pages 47-117, Springer.
    8. M. Al-Baali, 1998. "Numerical Experience with a Class of Self-Scaling Quasi-Newton Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 96(3), pages 533-553, March.
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    Cited by:

    1. S. Cipolla & C. Di Fiore & P. Zellini, 2020. "A variation of Broyden class methods using Householder adaptive transforms," Computational Optimization and Applications, Springer, vol. 77(2), pages 433-463, November.

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