# 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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- R. Hoppe and R. Kornhuber, Adaptive multigrid methods for obstacle problems. SIAM J. Numer. Anal.31 (1994) 301-323.
- 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).
- M. Ulbrich, Semi-smooth Newton methods for operator equations in function space. SIAM J. Optim. (to appear).

## 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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