Gradient projection methods represent effective tools for solving large-scale constrained optimization problems thanks to their simple implementation and low computational cost per iteration. Despite these good properties, a slow convergence rate can affect gradient projection schemes, especially when high accurate solutions are needed. A strategy to mitigate this drawback consists in properly selecting the values for the steplength along the negative gradient. In this paper, we consider the class of gradient projection methods with line search along the projected arc for box-constrained minimization problems and we analyse different strategies to define the steplength. It is well known in the literature that steplength selection rules able to approximate, at each iteration, the eigenvalues of the inverse of a suitable submatrix of the Hessian of the objective function can improve the performance of gradient projection methods. In this perspective, we propose an automatic hybrid steplength selection technique that employs a proper alternation of standard Barzilai–Borwein rules, when the final active set is not well approximated, and a generalized limited memory strategy based on the Ritz-like values of the Hessian matrix restricted to the inactive constraints, when the final active set is reached. Numerical experiments on quadratic and non-quadratic test problems show the effectiveness of the proposed steplength scheme.

Hybrid limited memory gradient projection methods for box-constrained optimization problems / Crisci, S.; Porta, F.; Ruggiero, V.; Zanni, L.. - In: COMPUTATIONAL OPTIMIZATION AND APPLICATIONS. - ISSN 0926-6003. - (2022), pp. 1-39. [10.1007/s10589-022-00409-4]

Hybrid limited memory gradient projection methods for box-constrained optimization problems

Porta F.;Zanni L.
2022

Abstract

Gradient projection methods represent effective tools for solving large-scale constrained optimization problems thanks to their simple implementation and low computational cost per iteration. Despite these good properties, a slow convergence rate can affect gradient projection schemes, especially when high accurate solutions are needed. A strategy to mitigate this drawback consists in properly selecting the values for the steplength along the negative gradient. In this paper, we consider the class of gradient projection methods with line search along the projected arc for box-constrained minimization problems and we analyse different strategies to define the steplength. It is well known in the literature that steplength selection rules able to approximate, at each iteration, the eigenvalues of the inverse of a suitable submatrix of the Hessian of the objective function can improve the performance of gradient projection methods. In this perspective, we propose an automatic hybrid steplength selection technique that employs a proper alternation of standard Barzilai–Borwein rules, when the final active set is not well approximated, and a generalized limited memory strategy based on the Ritz-like values of the Hessian matrix restricted to the inactive constraints, when the final active set is reached. Numerical experiments on quadratic and non-quadratic test problems show the effectiveness of the proposed steplength scheme.
2022
1
39
Hybrid limited memory gradient projection methods for box-constrained optimization problems / Crisci, S.; Porta, F.; Ruggiero, V.; Zanni, L.. - In: COMPUTATIONAL OPTIMIZATION AND APPLICATIONS. - ISSN 0926-6003. - (2022), pp. 1-39. [10.1007/s10589-022-00409-4]
Crisci, S.; Porta, F.; Ruggiero, V.; Zanni, L.
File in questo prodotto:
File Dimensione Formato  
s10589-022-00409-4.pdf

Open access

Tipologia: Versione pubblicata dall'editore
Dimensione 3.25 MB
Formato Adobe PDF
3.25 MB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

Licenza Creative Commons
I metadati presenti in IRIS UNIMORE sono rilasciati con licenza Creative Commons CC0 1.0 Universal, mentre i file delle pubblicazioni sono rilasciati con licenza Attribuzione 4.0 Internazionale (CC BY 4.0), salvo diversa indicazione.
In caso di violazione di copyright, contattare Supporto Iris

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1287044
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 1
social impact