9. Semismooth Newton Methods II: Applications
Society for Industrial and Applied Mathematics eBooks · 2008
In the previous chapter semismooth Newton methods in function spaces were investigated. In was demonstrated that in certain cases the semismooth Newton method is equivalent to the primal-dual active set method. The application to nonlinear complementarity problems was discussed and the necessity of introducing regularization in cases where the Lagrange multiplier associated to the inequality condition has low regularity was demonstrated. In this chapter applications of semismooth Newton methods to nondifferentiable variational problems in function spaces will be treated. They concern image restoration problems regularized by bounded variation functionals in Section 9.1 and frictional contact problems in elasticity in Section 9.2. We shall make use of the Fenchel duality theorem which we recall for further reference; see, e.g., Section 4.3 and [BaPe, EkTe] for details. Let V and Y be Banach spaces with topological duals V* and Y*, respectively. Further, let Λ ∈ ℒ (V, Y) and let be convex, proper, and l.s.c. functionals such that there exists v0 ∈ V with and is continuous at Λv0 Then inf v ∈ V {F (v)+G (Λv) } = sup q ∈ Y* {−F* (−Λ*q) −G* (q) } , 9.0.1 where Λ* ∈ ℒ (Y*, V*) is the adjoint of Λ. See, e.g., Section 4.3, Theorem 4.30 and with The convex conjugates and of and respectively, are defined by F* (v*) = sup v ∈ V { 〈v,v*〉V,V* −F (v) } , and analogously for The conditions imposed on and guarantee that the dual problem, i.e., the problem on the right-hand side of (9.0.1), admits a solution. Furthermore, v¯ ∈ V and q¯ ∈ Y* are solutions to the two optimization problems in (9.0.1) if and only if the extremality conditions −Λ* q ¯ ∈ ∂F ( u ¯ ) , q ¯ ∈ ∂G (Λ u ¯ ) 9.0.2 hold, where denotes the subdifferential of