Semi–Smooth Newton Methods for Variational Inequalities of the First Kind
ESAIM: Mathematical Modelling and Numerical Analysis (2010)
- Volume: 37, Issue: 1, page 41-62
- ISSN: 0764-583X
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topIto, Kazufumi, and Kunisch, Karl. "Semi–Smooth Newton Methods for Variational Inequalities of the First Kind." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 41-62. <http://eudml.org/doc/194155>.
@article{Ito2010,
abstract = {
Semi–smooth Newton methods are analyzed for a class of variational
inequalities in infinite dimensions.
It is shown that they are equivalent to certain active set strategies.
Global and local super-linear convergence are
proved. To overcome the phenomenon of finite speed of propagation of
discretized problems a penalty version
is used as the basis for a continuation procedure to speed up convergence.
The choice of the penalty parameter
can be made on the basis of an L∞ estimate
for the penalized solutions.
Unilateral as well as bilateral problems are considered.
},
author = {Ito, Kazufumi, Kunisch, Karl},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Semi-smooth Newton methods; contact problems; variational inequalities; bilateral constraints; superlinear convergence.; semi-smooth Newton methods; superlinear convergence},
language = {eng},
month = {3},
number = {1},
pages = {41-62},
publisher = {EDP Sciences},
title = {Semi–Smooth Newton Methods for Variational Inequalities of the First Kind},
url = {http://eudml.org/doc/194155},
volume = {37},
year = {2010},
}
TY - JOUR
AU - Ito, Kazufumi
AU - Kunisch, Karl
TI - Semi–Smooth Newton Methods for Variational Inequalities of the First Kind
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 1
SP - 41
EP - 62
AB -
Semi–smooth Newton methods are analyzed for a class of variational
inequalities in infinite dimensions.
It is shown that they are equivalent to certain active set strategies.
Global and local super-linear convergence are
proved. To overcome the phenomenon of finite speed of propagation of
discretized problems a penalty version
is used as the basis for a continuation procedure to speed up convergence.
The choice of the penalty parameter
can be made on the basis of an L∞ estimate
for the penalized solutions.
Unilateral as well as bilateral problems are considered.
LA - eng
KW - Semi-smooth Newton methods; contact problems; variational inequalities; bilateral constraints; superlinear convergence.; semi-smooth Newton methods; superlinear convergence
UR - http://eudml.org/doc/194155
ER -
References
top- D.P. Bertsekas, Constrained Optimization and Lagrange Mulitpliers. Academic Press, New York (1982).
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- M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as semi-smooth Newton method. SIAM J. Optim. (to appear).
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- K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces. Nonlinear Anal.41 (2000) 573-589.
- K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim.41 (2000) 343-364.
- D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980).
- D.M. Troianello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987).
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Citations in EuDML Documents
top- Juan Carlos De Los Reyes, Sergio González, Path following methods for steady laminar Bingham flow in cylindrical pipes
- Kazufumi Ito, Karl Kunisch, Semi-smooth Newton methods for the Signorini problem
- Juan Carlos De Los Reyes, Sergio González, Path following methods for steady laminar Bingham flow in cylindrical pipes
- Karl Kunisch, Georg Stadler, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space
- Luise Blank, Martin Butz, Harald Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method
- Karl Kunisch, Georg Stadler, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space
- Luise Blank, Martin Butz, Harald Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method
- Karl Kunisch, Daniel Wachsmuth, Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities
- Karl Kunisch, Daniel Wachsmuth, Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities
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