The aim of this paper is to deepen the convergence analysis of the scaled gradient projection (SGP) method, proposed by Bonettini et al. in a recent paper for constrained smooth optimization. The main feature of SGP is the presence of a variable scaling matrix multiplying the gradient, which may change at each iteration. In the last few years, an extensive numerical experimentation showed that SGP equipped with a suitable choice of the scaling matrix is a very effective tool for solving large scale variational problems arising in image and signal processing. In spite of the very reliable numerical results observed, only a weak convergence theorem is provided establishing that any limit point of the sequence generated by SGP is stationary. Here, under the only assumption that the objective function is convex and that a solution exists, we prove that the sequence generated by SGP converges to a minimum point, if the scaling matrices sequence satisfies a simple and implementable condition. Moreover, assuming that the gradient of the objective function is Lipschitz continuous, we are also able to prove the O(1/k) convergence rate with respect to the objective function values. Finally, we present the results of a numerical experience on some relevant image restoration problems, showing that the proposed scaling matrix selection rule performs well also from the computational point of view.

New convergence results for the scaled gradient projection method / Bonettini, Silvia; Prato, Marco. - In: INVERSE PROBLEMS. - ISSN 0266-5611. - STAMPA. - 31:9(2015), pp. 1-20. [10.1088/0266-5611/31/9/095008]

New convergence results for the scaled gradient projection method

BONETTINI, Silvia;PRATO, Marco
2015

Abstract

The aim of this paper is to deepen the convergence analysis of the scaled gradient projection (SGP) method, proposed by Bonettini et al. in a recent paper for constrained smooth optimization. The main feature of SGP is the presence of a variable scaling matrix multiplying the gradient, which may change at each iteration. In the last few years, an extensive numerical experimentation showed that SGP equipped with a suitable choice of the scaling matrix is a very effective tool for solving large scale variational problems arising in image and signal processing. In spite of the very reliable numerical results observed, only a weak convergence theorem is provided establishing that any limit point of the sequence generated by SGP is stationary. Here, under the only assumption that the objective function is convex and that a solution exists, we prove that the sequence generated by SGP converges to a minimum point, if the scaling matrices sequence satisfies a simple and implementable condition. Moreover, assuming that the gradient of the objective function is Lipschitz continuous, we are also able to prove the O(1/k) convergence rate with respect to the objective function values. Finally, we present the results of a numerical experience on some relevant image restoration problems, showing that the proposed scaling matrix selection rule performs well also from the computational point of view.
31
9
1
20
New convergence results for the scaled gradient projection method / Bonettini, Silvia; Prato, Marco. - In: INVERSE PROBLEMS. - ISSN 0266-5611. - STAMPA. - 31:9(2015), pp. 1-20. [10.1088/0266-5611/31/9/095008]
Bonettini, Silvia; Prato, Marco
File in questo prodotto:
File Dimensione Formato  
PratoNewConverHANDLE 11380-1070364.pdf

accesso aperto

Tipologia: Post-print dell'autore (bozza post referaggio)
Dimensione 410.29 kB
Formato Adobe PDF
410.29 kB Adobe PDF Visualizza/Apri
VOR_New convergence results for the scaled.pdf

non disponibili

Tipologia: Versione dell'editore (versione pubblicata)
Dimensione 1.23 MB
Formato Adobe PDF
1.23 MB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

Caricamento 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: http://hdl.handle.net/11380/1070364
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 37
  • ???jsp.display-item.citation.isi??? 36
social impact