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On initial point selection of the steepest descent algorithm for general quadratic functions

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  • Masoud Fatemi

    (K. N. Toosi University of Technology)

Abstract

We prove some new results about the asymptotic behavior of the steepest descent algorithm for general quadratic functions. Some well-known results of this theory are developed and extended to non-convex functions. We propose an efficient strategy for choosing initial points in the algorithm and show that this strategy can dramatically enhance the performance of the method. Furthermore, a modified version of the steepest descent algorithm equipped with a pre-initialization step is introduced. We show that an initial guess near the optimal solution does not necessarily imply fast convergence. We also propose a new approach to investigate the behavior of the method for non-convex quadratic functions. Moreover, some interesting results about the role of initial points in convergence to saddle points are presented. Finally, we investigate the probability of divergence for uniform random initial points.

Suggested Citation

  • Masoud Fatemi, 2022. "On initial point selection of the steepest descent algorithm for general quadratic functions," Computational Optimization and Applications, Springer, vol. 82(2), pages 329-360, June.
    • Handle: RePEc:spr:coopap:v:82:y:2022:i:2:d:10.1007_s10589-022-00372-0
    DOI: 10.1007/s10589-022-00372-0
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    References listed on IDEAS

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    1. Yu-Hong Dai & Yakui Huang & Xin-Wei Liu, 2019. "A family of spectral gradient methods for optimization," Computational Optimization and Applications, Springer, vol. 74(1), pages 43-65, September.
    2. Roberta De Asmundis & Daniela di Serafino & William Hager & Gerardo Toraldo & Hongchao Zhang, 2014. "An efficient gradient method using the Yuan steplength," Computational Optimization and Applications, Springer, vol. 59(3), pages 541-563, December.
    3. Clóvis Gonzaga & Ruana Schneider, 2016. "On the steepest descent algorithm for quadratic functions," Computational Optimization and Applications, Springer, vol. 63(2), pages 523-542, March.
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