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Self-Scaling Variable Metric (SSVM) Algorithms

Author

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  • Shmuel S. Oren

    (Xerox Corporation, Palo Alto, California and Stanford University)

  • David G. Luenberger

    (Stanford University)

Abstract

A new criterion is introduced for comparing the convergence properties of variable metric algorithms, focusing on stepwise descent properties. This criterion is a bound on the rate of decrease in the function value at each iterative step (single-step convergence rate). Using this criterion as a basis for algorithm development leads to the introduction of variable coefficients to rescale the objective function at each iteration, and, correspondingly, to a new class of variable metric algorithms. Effective scaling can be implemented by restricting the parameters in a two-parameter family of variable metric algorithms. Conditions are derived for these parameters that guarantee monotonic improvement in the single-step convergence rate. These conditions are obtained by analyzing the eigenvalue structure of the associated inverse Hessian approximations.

Suggested Citation

  • Shmuel S. Oren & David G. Luenberger, 1974. "Self-Scaling Variable Metric (SSVM) Algorithms," Management Science, INFORMS, vol. 20(5), pages 845-862, January.
    • Handle: RePEc:inm:ormnsc:v:20:y:1974:i:5:p:845-862
    DOI: 10.1287/mnsc.20.5.845
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    Cited by:

    1. Nataj, Sarah & Lui, S.H., 2020. "Superlinear convergence of nonlinear conjugate gradient method and scaled memoryless BFGS method based on assumptions about the initial point," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    2. 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.
    3. C. X. Kou & Y. H. Dai, 2015. "A Modified Self-Scaling Memoryless Broyden–Fletcher–Goldfarb–Shanno Method for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 209-224, April.
    4. Saman Babaie-Kafaki & Reza Ghanbari, 2017. "A class of adaptive Dai–Liao conjugate gradient methods based on the scaled memoryless BFGS update," 4OR, Springer, vol. 15(1), pages 85-92, March.
    5. Fahimeh Biglari & Farideh Mahmoodpur, 2016. "Scaling Damped Limited-Memory Updates for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 177-188, July.
    6. 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.
    7. Martin Buhmann & Dirk Siegel, 2021. "Implementing and modifying Broyden class updates for large scale optimization," Computational Optimization and Applications, Springer, vol. 78(1), pages 181-203, January.
    8. 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.

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