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Analysis of the Barzilai-Borwein Step-Sizes for Problems in Hilbert Spaces

Author

Listed:
  • Behzad Azmi

    (Austrian Academy of Science)

  • Karl Kunisch

    (Austrian Academy of Science
    University of Graz)

Abstract

The Barzilai and Borwein gradient method has received a significant amount of attention in different fields of optimization. This is due to its simplicity, computational cheapness, and efficiency in practice. In this research, based on spectral analysis techniques, root-linear global convergence for the Barzilai and Borwein method is proven for strictly convex quadratic problems posed in infinite-dimensional Hilbert spaces. The applicability of these results is demonstrated for two optimization problems governed by partial differential equations.

Suggested Citation

  • Behzad Azmi & Karl Kunisch, 2020. "Analysis of the Barzilai-Borwein Step-Sizes for Problems in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 819-844, June.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:3:d:10.1007_s10957-020-01677-y
    DOI: 10.1007/s10957-020-01677-y
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    References listed on IDEAS

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    1. Yutao Zheng & Bing Zheng, 2017. "A New Modified Barzilai–Borwein Gradient Method for the Quadratic Minimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 179-186, January.
    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. di Serafino, Daniela & Ruggiero, Valeria & Toraldo, Gerardo & Zanni, Luca, 2018. "On the steplength selection in gradient methods for unconstrained optimization," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 176-195.
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