Applications of Semi-smooth Newton Methods to Variational Inequalities
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 constra
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pplications of Semi-smooth Newton Methods to Variational Inequalities Kazufumi Ito and Karl Kunisch Abstract. 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.
1. Introduction Examples which motivate our study include nonlinear variational inequalities of the form: find x ∈ C such that (f (x), y − x) ≥ 0
for all
y ∈ C,
(1.1)
where C is a closed convex set in a Hilbert space X and f : X → X is C 1 . It can equivalently be written as F (x) = x − P rojC (x − f (x)) = 0,
(1.2)
where P rojC is the projection of X onto C. In particular, let Ω be a bounded domain in RN and if C is a hypercube {x| φ ≤ x ≤ ψ} in X = L2 (Ω), with φ ≤ ψ and the inequalities are defined pointwise, then (1.1) can be expressed as F (x) = µ − max(0, µ + x − ψ) − min(0, µ + x − φ),
µ = −f (x)
(1.3)
The first author is partially supported by the Army Research Office under DAAD19-02-1-039. and the second author is supported in part by the Fonds zur F¨ orderung der wissenschaftlichen Forschung under SFB 03 “Optimierung und Kontrolle”.
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K. Ito and K. Kunisch
where µ ∈ X is the Lagrange multiplier. For example, consider a boundary control problem for the heat equation on a bounded open domain D in R3 ; 1 T α min |y − y¯|2 dx dt + |u|2X (1.4) u∈X 2 0 2 Ω subject to ∂ y = ∆y, y(0, ·) = y0 in D ∂t (1.5) ∂ 3 y + y = u on (0, T ) × ∂D ∂ν with X = L2 ((0, T ) × ∂D), y¯ and y0 ∈ L2 (Ω). Then f : X → X is defined by f (u) = α u + p|∂D where p satisfies the adjoint equation ∂ p + ∆p + y − y¯ = 0, ∂t
p(T, ·) = 0 in D
∂p + 3y 2 p = 0 on ∂D. ∂ν Note that F is a locally Lipschitz continuous functions but is not C 1 , even if f is C 1 . If F is locally Lipschitz continuous on Rm , then according to Rademacher’s theorem, F is differentiable almost everywhere. Let DF denote the set of points at which F is differentiable and let ∂B F (x) be defined by ∂B F (x) = J = lim F (xi ) . (1.6) xi →x, xi ∈DF
We denote by ∂F (x) the generalized derivative in the sense of Clarke, i.e., ∂F (x) = the convex hull of ∂B F (x).
(1.7)
A generalized Newton iteration for solving the nonlinear equation F (x) = 0 is defined by (1.8) xk+1 = xk − Vk−1 F (xk ), where Vk ∈ ∂B F (xk ). In the finite-dimensional case a generalized Jacobian Vk ∈ ∂B F (xk ). Local convergence of {xk } to x∗ , a solution of F (x) = 0, is based on the following concepts; |F (x∗ + h) − F (x∗ ) − V h| = o(|h|),
(1.9)
where V = V (x∗ +
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