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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topRenaud, 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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