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Research on Three-Dimensional Extension of Barzilai-Borwein-like Method

Author

Listed:
  • Tianji Wang

    (School of Mathematics, Jilin University, Changchun 130012, China
    These authors contributed equally to this work.)

  • Qingdao Huang

    (School of Mathematics, Jilin University, Changchun 130012, China
    These authors contributed equally to this work.)

Abstract

The Barzilai-Borwein (BB) method usually uses BB stepsize for iteration so as to eliminate the line search step in the steepest descent method. In this paper, we modify the BB stepsize and extend it to solve the optimization problems of three-dimensional quadratic functions. The discussion is divided into two cases. Firstly, we study the case where the coefficient matrix of the quadratic term of quadratic function is a special third-order diagonal matrix and prove that using the new modified stepsize, this case is R -superlinearly convergent. In addition to that, we extend it to n -dimensional case and prove the rate of convergence is R -linear. Secondly, we analyze that the coefficient matrix of the quadratic term of quadratic function is a third-order asymmetric matrix, that is, when the matrix has a double characteristic root and prove the global convergence of this case. The results of numerical experiments show that the modified method is effective for the above two cases.

Suggested Citation

  • Tianji Wang & Qingdao Huang, 2025. "Research on Three-Dimensional Extension of Barzilai-Borwein-like Method," Mathematics, MDPI, vol. 13(2), pages 1-26, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:215-:d:1564132
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    References listed on IDEAS

    as
    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. Yakui Huang & Hongwei Liu, 2016. "Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization," Computational Optimization and Applications, Springer, vol. 65(3), pages 671-698, December.
    3. 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.
    Full references (including those not matched with items on IDEAS)

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