# An Extension of the Auxiliary Problem Principle to Nonsymmetric Auxiliary Operators

• Volume: 2, page 281-306
• ISSN: 1292-8119

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## Abstract

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To find a zero of a maximal monotone operator, an extension of the Auxiliary Problem Principle to nonsymmetric auxiliary operators is proposed. The main convergence result supposes a relationship between the main operator and the nonsymmetric component of the auxiliary operator. When applied to the particular case of convex-concave functions, this result implies the convergence of the parallel version of the Arrow-Hurwicz algorithm under the assumptions of Lipschitz and partial Dunn properties of the main operator. The latter is systematically enforced by partial regularization. In the strongly monotone case, it is shown that the convergence is linear in the average. Moreover, if the symmetric part of the auxiliary operator is linear, the Lipschitz property of the inverse suffices to ensure a linear convergence rate in the average.

## How to cite

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Renaud, A., and Cohen, G.. "An Extension of the Auxiliary Problem Principle to Nonsymmetric Auxiliary Operators." ESAIM: Control, Optimisation and Calculus of Variations 2 (2010): 281-306. <http://eudml.org/doc/116553>.

@article{Renaud2010,
abstract = { To find a zero of a maximal monotone operator, an extension of the Auxiliary Problem Principle to nonsymmetric auxiliary operators is proposed. The main convergence result supposes a relationship between the main operator and the nonsymmetric component of the auxiliary operator. When applied to the particular case of convex-concave functions, this result implies the convergence of the parallel version of the Arrow-Hurwicz algorithm under the assumptions of Lipschitz and partial Dunn properties of the main operator. The latter is systematically enforced by partial regularization. In the strongly monotone case, it is shown that the convergence is linear in the average. Moreover, if the symmetric part of the auxiliary operator is linear, the Lipschitz property of the inverse suffices to ensure a linear convergence rate in the average. },
author = {Renaud, A., Cohen, G.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Auxiliary Problem Principle / variational inequalities with nonsymmetric operators / convergence of iterative algorithms / partial regularization / rate of convergence.; successive approximations; maximal monotone operator; auxiliary problem principle; nonsymmetric auxiliary operators; convex-concave functions; Arrow-Hurwicz algorithm; partial Dunn properties; partial regularization; Lipschitz property; linear convergence rate in the average},
language = {eng},
month = {3},
pages = {281-306},
publisher = {EDP Sciences},
title = {An Extension of the Auxiliary Problem Principle to Nonsymmetric Auxiliary Operators},
url = {http://eudml.org/doc/116553},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Renaud, A.
AU - Cohen, G.
TI - An Extension of the Auxiliary Problem Principle to Nonsymmetric Auxiliary Operators
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 2
SP - 281
EP - 306
AB - To find a zero of a maximal monotone operator, an extension of the Auxiliary Problem Principle to nonsymmetric auxiliary operators is proposed. The main convergence result supposes a relationship between the main operator and the nonsymmetric component of the auxiliary operator. When applied to the particular case of convex-concave functions, this result implies the convergence of the parallel version of the Arrow-Hurwicz algorithm under the assumptions of Lipschitz and partial Dunn properties of the main operator. The latter is systematically enforced by partial regularization. In the strongly monotone case, it is shown that the convergence is linear in the average. Moreover, if the symmetric part of the auxiliary operator is linear, the Lipschitz property of the inverse suffices to ensure a linear convergence rate in the average.
LA - eng
KW - Auxiliary Problem Principle / variational inequalities with nonsymmetric operators / convergence of iterative algorithms / partial regularization / rate of convergence.; successive approximations; maximal monotone operator; auxiliary problem principle; nonsymmetric auxiliary operators; convex-concave functions; Arrow-Hurwicz algorithm; partial Dunn properties; partial regularization; Lipschitz property; linear convergence rate in the average
UR - http://eudml.org/doc/116553
ER -

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