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Steplength selection in gradient projection methods for box-constrained quadratic programs

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  • Crisci, Serena
  • Ruggiero, Valeria
  • Zanni, Luca

Abstract

The role of the steplength selection strategies in gradient methods has been widely investigated in the last decades. Starting from the work of Barzilai and Borwein (1988), many efficient steplength rules have been designed, that contributed to make the gradient approaches an effective tool for the large-scale optimization problems arising in important real-world applications. Most of these steplength rules have been thought in unconstrained optimization, with the aim of exploiting some second-order information for achieving a fast annihilation of the gradient of the objective function. However, these rules are successfully used also within gradient projection methods for constrained optimization, though, to our knowledge, a detailed analysis of the effects of the constraints on the steplength selections is still not available. In this work we investigate how the presence of the box constraints affects the spectral properties of the Barzilai–Borwein rules in quadratic programming problems. The proposed analysis suggests the introduction of new steplength selection strategies specifically designed for taking account of the active constraints at each iteration. The results of a set of numerical experiments show the effectiveness of the new rules with respect to other state of the art steplength selections and their potential usefulness also in case of box-constrained non-quadratic optimization problems.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:apmaco:v:356:y:2019:i:c:p:312-327
    DOI: 10.1016/j.amc.2019.03.039
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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. 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.
    4. 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.
    5. 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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    Cited by:

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    2. 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).

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