# Semi–smooth Newton methods for variational inequalities of the first kind

- Volume: 37, Issue: 1, page 41-62
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topIto, Kazufumi, and Kunisch, Karl. "Semi–smooth Newton methods for variational inequalities of the first kind." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.1 (2003): 41-62. <http://eudml.org/doc/246035>.

@article{Ito2003,

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 $L^\{\infty \}$ 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 - Modélisation Mathématique et Analyse Numérique},

keywords = {semi-smooth Newton methods; contact problems; variational inequalities; bilateral constraints; superlinear convergence},

language = {eng},

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/246035},

volume = {37},

year = {2003},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2003

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 $L^{\infty }$ 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

UR - http://eudml.org/doc/246035

ER -

## References

top- [1] D.P. Bertsekas, Constrained Optimization and Lagrange Mulitpliers. Academic Press, New York (1982). MR690767
- [2] M. Bergounioux, M. Haddou, M. Hintermüller and K. Kunisch, A comparison of a Moreau–Yosida based active strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11 (2000) 495–521. Zbl1001.49034
- [3] M. Bergounioux, K. Ito and K. Kunisch, Primal–dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37 (1999) 1176–1194. Zbl0937.49017
- [4] Z. Dostal, Box constrained quadratic programming with proportioning and projections. SIAM J. Optim. 7 (1997) 871–887. Zbl0912.65052
- [5] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Verlag, New York (1984). Zbl0536.65054MR737005
- [6] R. Glowinski, J.L. Lions and T. Tremolieres, Analyse Numerique des Inequations Variationnelles. Vol. 1, Dunod, Paris (1976). Zbl0358.65091
- [7] M. Hintermüller, K. Ito and K. Kunisch, The primal–dual active set strategy as semi–smooth Newton method. SIAM J. Optim. (to appear). Zbl1080.90074
- [8] R. Hoppe, Multigrid algorithms for variational inequalities. SIAM J. Numer. Anal. 24 (1987) 1046–1065. Zbl0628.65046
- [9] R. Hoppe and R. Kornhuber, Adaptive multigrid methods for obstacle problems. SIAM J. Numer. Anal. 31 (1994) 301–323. Zbl0806.65064
- [10] K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 573–589. Zbl0971.49014
- [11] K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343–364. Zbl0960.49003
- [12] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). Zbl0457.35001MR567696
- [13] D.M. Troianello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987). Zbl0655.35002
- [14] M. Ulbrich, Semi-smooth Newton methods for operator equations in function space. SIAM J. Optim. (to appear). Zbl1033.49039

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.