<abstract xmlns="http://www.w3.org/1999/xhtml"><p> In this paper, convergence of the sequence generated by the inexact form of the inertial proximal algorithm is studied. This algorithm which is obtained by the discretization of a nonlinear oscillator with damping dynamical system, has been introduced by Alvarez and Attouch (2001) and Jules and Maingé (2002) for the approximation of a zero of a maximal monotone operator. We establish weak and strong convergence results for the inexact inertial proximal algorithm with and without the summability assumption on errors, under different conditions on parameters. Our theorems extend the results on the inertial proximal algorithm established by Alvarez and Attouch (2001) and rules and Maingé (2002) as well as the results on the standard proximal point algorithm established by Brézis and Lions (1978), Lions (1978), Djafari Rouhani and Khatibzadeh (2008) and Khatibzadeh (2012). We also answer questions of Alvarez and Attouch (2001).</p></abstract>

In this paper, convergence of the sequence generated by the inexact form of the inertial proximal algorithm is studied. This algorithm which is obtained by the discretization of a nonlinear oscillator with damping dynamical system, has been introduced by Alvarez and Attouch (2001) and Jules and Maingé (2002) for the approximation of a zero of a maximal monotone operator. We establish weak and strong convergence results for the inexact inertial proximal algorithm with and without the summability assumption on errors, under different conditions on parameters. Our theorems extend the results on the inertial proximal algorithm established by Alvarez and Attouch (2001) and rules and Maingé (2002) as well as the results on the standard proximal point algorithm established by Brézis and Lions (1978), Lions (1978), Djafari Rouhani and Khatibzadeh (2008) and Khatibzadeh (2012). We also answer questions of Alvarez and Attouch (2001).

Inexact Inertial Proximal Algorithm for Maximal Monotone Operators

Department of Mathematics, University of Zanjan, P.O. Box 45195-313, Zanjan,

Zarghan Branch, Islamic Azad University, Zarghan,

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Inexact Inertial Proximal Algorithm for Maximal Monotone Operators"}

In this paper, convergence of the sequence generated by the inexact form of the inertial proximal algorithm is studied. This algorithm which is obtained by the...

Inexact Inertial Proximal Algorithm for Maximal Monotone Operators

Published Online: Apr 22, 2017

Page range: 133 - 146

Received: Jan 01, 2014

Accepted: May 01, 2014

DOI: https://doi.org/10.1515/auom-2015-0031

Keywords
Inertial proximal algorithm, Maximal monotone, Ergodic theorem, Weak convergence, Strong convergence, Subdifferential

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Inexact Inertial Proximal Algorithm for Maximal Monotone Operators

Published Online: Apr 22, 2017

Page range: 133 - 146

Received: Jan 01, 2014

Accepted: May 01, 2014

DOI: https://doi.org/10.1515/auom-2015-0031

KeywordsInertial proximal algorithm, Maximal monotone, Ergodic theorem, Weak convergence, Strong convergence, Subdifferential

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Inertial proximal algorithm, Maximal monotone, Ergodic theorem, Weak convergence, Strong convergence, Subdifferential