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On the steplength selection in gradient methods for unconstrained optimization

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  • di Serafino, Daniela
  • Ruggiero, Valeria
  • Toraldo, Gerardo
  • Zanni, Luca

Abstract

The seminal paper by Barzilai and Borwein (1988) has given rise to an extensive investigation, leading to the development of effective gradient methods. Several steplength rules have been first designed for unconstrained quadratic problems and then extended to general nonlinear optimization problems. These rules share the common idea of attempting to capture, in an inexpensive way, some second-order information. However, the convergence theory of the gradient methods using the previous rules does not explain their effectiveness, and a full understanding of their practical behaviour is still missing. In this work we investigate the relationships between the steplengths of a variety of gradient methods and the spectrum of the Hessian of the objective function, providing insight into the computational effectiveness of the methods, for both quadratic and general unconstrained optimization problems. Our study also identifies basic principles for designing effective gradient methods.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:176-195
    DOI: 10.1016/j.amc.2017.07.037
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    References listed on IDEAS

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    1. Birgin, Ernesto G. & Martínez, Jose Mario & Raydan, Marcos, 2014. "Spectral Projected Gradient Methods: Review and Perspectives," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 60(i03).
    2. Y. H. Dai, 2002. "On the Nonmonotone Line Search," Journal of Optimization Theory and Applications, Springer, vol. 112(2), pages 315-330, February.
    3. Hirotugu Akaike, 1959. "On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 11(1), pages 1-16, February.
    4. 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.
    5. Luc Pronzato & Anatoly Zhigljavsky, 2011. "Gradient algorithms for quadratic optimization with fast convergence rates," Computational Optimization and Applications, Springer, vol. 50(3), pages 597-617, December.
    6. 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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    Citations

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    Cited by:

    1. 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.
    2. Yakui Huang & Yu-Hong Dai & Xin-Wei Liu & Hongchao Zhang, 2022. "On the acceleration of the Barzilai–Borwein method," Computational Optimization and Applications, Springer, vol. 81(3), pages 717-740, April.
    3. di Serafino, Daniela & Toraldo, Gerardo & Viola, Marco, 2021. "Using gradient directions to get global convergence of Newton-type methods," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    4. Giulia Ferrandi & Michiel E. Hochstenbach & Nataša Krejić, 2023. "A harmonic framework for stepsize selection in gradient methods," Computational Optimization and Applications, Springer, vol. 85(1), pages 75-106, May.
    5. 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.
    6. Crisci, Serena & Ruggiero, Valeria & Zanni, Luca, 2019. "Steplength selection in gradient projection methods for box-constrained quadratic programs," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 312-327.
    7. Serena Crisci & Federica Porta & Valeria Ruggiero & Luca Zanni, 2023. "Hybrid limited memory gradient projection methods for box-constrained optimization problems," Computational Optimization and Applications, Springer, vol. 84(1), pages 151-189, January.
    8. E. Loli Piccolomini & V. L. Coli & E. Morotti & L. Zanni, 2018. "Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm," Computational Optimization and Applications, Springer, vol. 71(1), pages 171-191, September.
    9. Corsaro, Stefania & De Simone, Valentina & Marino, Zelda, 2021. "Split Bregman iteration for multi-period mean variance portfolio optimization," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    10. Harry Fernando Oviedo Leon, 2019. "A delayed weighted gradient method for strictly convex quadratic minimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 729-746, December.
    11. O. P. Ferreira & M. Lemes & L. F. Prudente, 2022. "On the inexact scaled gradient projection method," Computational Optimization and Applications, Springer, vol. 81(1), pages 91-125, January.
    12. Roberto Andreani & Marcos Raydan, 2021. "Properties of the delayed weighted gradient method," Computational Optimization and Applications, Springer, vol. 78(1), pages 167-180, January.

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