Applications of Semi-smooth Newton Methods to Variational Inequalities

  • Kazufumi Ito
  • Karl Kunisch
Part of the International Series of Numerical Mathematics book series (ISNM, volume 155)


This paper discusses semi-smooth Newton methods for solving nonlinear non-smooth equations in Banach spaces. Such investigations are motivated by complementarity problems, variational inequalities and optimal control problems with control or state constraints, for example. The function F(x) for which we desire to find a root is typically Lipschitz continuous but not C 1 regular. The primal-dual active set strategy for the optimization with the inequality constraints is formulated as a semi-smooth Newton method. Sufficient conditions for global convergence assuming diagonal dominance are established. Globalization strategies are also discussed assuming that the merit function |F(x)|2 has appropriate descent directions.


Variational Inequality Optimal Control Problem Global Convergence Descent Direction Merit Function 
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© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Kazufumi Ito
    • 1
  • Karl Kunisch
    • 2
  1. 1.Center for Research in Scientific Computation Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.Institut für Mathematik und wissenschaftliches RechnenUniversität GrazGrazAustria

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