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Semi–Smooth Newton Methods for Variational Inequalitiesof the First Kind

Published online by Cambridge University Press:  15 March 2003

Kazufumi Ito  and
Karl Kunisch
Kazufumi Ito
Affiliation:
Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, USA.
Karl Kunisch
Affiliation:
Institut für Mathematik, Universität Graz, Graz, Austria. [email protected].
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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 areproved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty versionis 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.

Keywords

Semi-smooth Newton methodscontact problemsvariational inequalitiesbilateral constraintssuperlinear convergence.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 1 , January 2003 , pp. 41 - 62
Copyright
© EDP Sciences, SMAI, 2003

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References

D.P. Bertsekas, Constrained Optimization and Lagrange Mulitpliers. Academic Press, New York (1982).
Bergounioux, M., Haddou, M., Hintermüller, M. and Kunisch, K., 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. CrossRef
Bergounioux, M., Ito, K. and Kunisch, K., Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37 (1999) 1176-1194. CrossRef
Dostal, Z., Box constrained quadratic programming with proportioning and projections. SIAM J. Optim. 7 (1997) 871-887. CrossRef
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Verlag, New York (1984).
R. Glowinski, J.L. Lions and T. Tremolieres, Analyse Numerique des Inequations Variationnelles. Vol. 1, Dunod, Paris (1976).
M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as semi-smooth Newton method. SIAM J. Optim. (to appear).
Hoppe, R., Multigrid algorithms for variational inequalities. SIAM J. Numer. Anal. 24 (1987) 1046-1065. CrossRef
Hoppe, R. and Kornhuber, R., Adaptive multigrid methods for obstacle problems. SIAM J. Numer. Anal. 31 (1994) 301-323. CrossRef
Ito, K. and Kunisch, K., Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 573-589. CrossRef
Ito, K. and Kunisch, K., Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. CrossRef
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).