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

Kazufumi Ito; Karl Kunisch

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

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

Abstract

top
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.

How to cite

top

Ito, 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. [1] D.P. Bertsekas, Constrained Optimization and Lagrange Mulitpliers. Academic Press, New York (1982). MR690767
  2. [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. [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. [4] Z. Dostal, Box constrained quadratic programming with proportioning and projections. SIAM J. Optim. 7 (1997) 871–887. Zbl0912.65052
  5. [5] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Verlag, New York (1984). Zbl0536.65054MR737005
  6. [6] R. Glowinski, J.L. Lions and T. Tremolieres, Analyse Numerique des Inequations Variationnelles. Vol. 1, Dunod, Paris (1976). Zbl0358.65091
  7. [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. [8] R. Hoppe, Multigrid algorithms for variational inequalities. SIAM J. Numer. Anal. 24 (1987) 1046–1065. Zbl0628.65046
  9. [9] R. Hoppe and R. Kornhuber, Adaptive multigrid methods for obstacle problems. SIAM J. Numer. Anal. 31 (1994) 301–323. Zbl0806.65064
  10. [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. [11] K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343–364. Zbl0960.49003
  12. [12] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). Zbl0457.35001MR567696
  13. [13] D.M. Troianello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987). Zbl0655.35002
  14. [14] M. Ulbrich, Semi-smooth Newton methods for operator equations in function space. SIAM J. Optim. (to appear). Zbl1033.49039

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.